Ivor Darreg, June 1987
Since Issue No. 9 of the Xenharmonic Bulletin came out some nine years ago, more people all around the country and abroad have joined the Non-Twelve Movement in one way or another.
We are talking about a lateral informal networking of people with some interests in common, on an equal footing: this is not a hierarchical organization with officious Authorities Issuing Orders. Such a vast field of exploration lies so far outside the 12-tone equal temperament that no one lifetime would suffice to develop it.
Until recently, the makers of commercial instruments have not paid attention to our movement, but this attitude has begun to change, and a few electronic instruments or accessories for them with micro-intervallic capability have reached the market.
In addition, the personal computer field has exploded into full fruition during the above-mentioned 9-year span! Many such computers have sound facilities, and merely require proper software to be written to make them capable of xenharmonic composition and performance.
In consequence, some new inquiries and orders for Xenharmonic Bulletins 1 through 10 have come in and all issues continue to be available on order. New issues will come out--at least Nos. 11 & 12 for which ample material is available.
A list of Ivor Darreg's other publications is now available, and since the new studio was started, there are new recordings, so theory is being reduced to practice and xenharmonics is for the PRESENT, not just the future.
Thanks to the cassette, it is now possible for composers to exchange compositions easily, even by mail all across the country! This means that the matters discussed in Xenharmonic Bulletin No. 9, such as the 18-tone scale proposed by Busoni and used by Carrillo and Haba, can now be HEARD instead of just writing ABOUT them in the Bulletins.
Being able to hear and copy the actual sounds of the new scales removes the long-standing roadblock and at lost last permits progress in music itself so we won't have to be slaves of the 19th century.
If anyone asks, Why new scales? the cassettes are the answer and that's that! No need to argue in print anymore.
For the ordinary types of classical and popular music, there has been overcompetition and many people struggling with one another for a place in an overcrowded field. Now, with the new scales open to everyone and the unlimited possibilities of new instruments,there is room for everybody to explore without competing against one another.
It seems everybody has to move these days; nothing is permanent anymore. So some of the addresses in the 1978 Bulletin are changed or now unknown. In this reprint we have inserted some new data that we were able to get.
The good news is that there are now more instruments capable of playing in new scales, so what was written about in Xenharmonic Bulletin #9 has come alive! For instance, quite a number of guitars, both acoustic and electric, were refretted to 31-tone either by Ivor Darreg or by others using his fretting-tables.
Xenharmonic Bulletin No. 10, which came out in December 1981, has some additional data relevant to the systems discussed in Xenharmonic Bulletin No. 9. For instance, there is a Pro-vs.-Con treatment of the 31-tone equal temperament, while the closely similar 1/4-comma meantone temperament is taken up and compared with the other members of of the Meantone Family on pages 8 through 11.
34-and 36-tone are compared on page 23.
1987 happens to be the 80th anniversary of Busoni's Dream of third- and sixth-tones, alluded to in his NEW ESTHETIC OF MUSIC writings of 1907. Haba and Carrillo realized those systems to some extent, as we mentioned in Xenharmonic Bulletin #9. Already we have started to produce some demo recordings comparing Busoni's ideas; and a few recent composers have tried third-tones.
One of the factors holding back new scales, even in 1978, was the impossibility of tuning many of them by ear. Now, there are several well known makes of tuning devices, both light-socket and battery-operated. Some organs and synthesizers have automatic tuning built-in. New computer software can produce more pitches than anyone could ever hear. With the sustained-tone instruments it is also possible to use a direct-readout frequency counter.
For fretted instruments, of course, it is a mere matter of using a fretting-table to place the frets in new positions to produce the new scale.
Then there is the concrete hardware evidence, much more apparent in 1987 than back in 1978, of the pitch-bend levers on most commercial synthesizers. This shows the desire to escape twelve-tone equal tuning, however subconscious that may be. The tremendously important difference is that now the bend level is conspicuously displayed on the synthesizer. Until recently, it was a private matter between violin teacher or the like and the pupil, not to be written down, and the composer was never to write any indication that pitches were to be bent, let alone how much!
After two centuries of the piano, whose pitch you cannot bend, the appearance of a bender on the new keyboards is a very important and quite sudden change in the status quo.
The Xenharmonic Movement is not a new religion or cult! It is the interfacing of art and science to make for progress in music beyond the stasis reached during the 19th Century. The composer needs the status of the painter and the sculptor, no longer being compelled to write every note down and turn it over to performers who have not been trained to perform in any new scales, but rather have been indoctrinated against them.
We need not disturb the music of the past. Let the custodians do their thing while we avail ourselves of the latest recording and tone-making technologies. Their performances should be authentic 19th century while we stay true to the 20th and soon the 21st!
In our previous (July 1976) issue, XENHARMONIC BULLETIN #8, tables and tuning procedures for systems with unit-intervals larger than the quartertone were given--there just wasn't enough space to jam in the equally-important 31-tone system. In case any triakontamonophonists or tricesimoprimalists (that's Greek or Latin academic snobtalk for 31-ians) were wondering, they were not being ignored or neglected; it's just that one has only so much time and energy and also a single issue of this Bulletin should not get too bulky.
Instruments and compositions now exist in the 31-tone system; enough momentum has now been gained to ensure that it is here to stay. This is reinforced by the revival of Meantone Temperament. Both systems favor the major third, being willing to trade off a slight flattening of the fifths to smooth and calm down the major thirds.
This was well exemplified at the concert presented last April in Alhambra (a suburb east of Los Angeles) by the Netherlands organist Anton de Beer and the violinists Bouw Lemkes and Jeanne Vos, who were on tour in the U.S. A special electronic organ called the Archiphone with timbres proper to the Swell manual of a regular organ, had the harmonic constitution necessary to define the intervals of the 31-tone equal temperament--and the continuity with the meantone era of keyboard instruments was demonstrated by using a harpsichord so tuned.
The overall effect was restful, calm, serene: it would not have done for an operatic storm-scene!
Those of you who still have my Xenharmonic Bulletin No. 5 for May, 1975 (it will be reprinted from time to time) may remember the lead article, NEW MOODS.
The 31-tone system commands a different spectrum of emotional expression and and suggestive power than the 12-tone system does. The contrast is great, being mainly that of serenity vs. agitation, calm vs. restlessness.
How so? One reason has to do with 12's sharp major thirds and sixths. In 31 the major third and minor sixth are almost perfect, being out by 0.7 cents--i.e., these 31-tone-system intervals fail to be perfect by the 1714th part of an octave. This discrepancy is inaudible under all but special laboratory conditions. The corresponding discrepancy for the 12-tone intervals of the same names is 20 times as great, and manifests itself as a roughness or harshness--a "reedy growl" as one author called it.
In the 1/4-comma meantone system, these intervals are perfect. The slight 31-tone discrepancy means that there is a circle of 31 major thirds and circle of 31 fifths in the 31-tone system--it is cyclic or closed; whereas in the meantone temperament the circle doesn't quite close and theoretically meantone temperament has an infinite number of tones. In this practical world no-one has all eternity to test that and find out.
For purposes of building and playing guitars or organs or synthesizers or whatever instruments are practicable for meantone or 31-tone, the discrepancy does not matter, because the imperfections of instruments and performance upon them, and the quality of interpretation of compositions will not be different enough to get disturbed about.
A mathematician will care, of course. As a mathematician. I doubt that he could tell the difference, listening to a record or a tape of an actual piece of music.
But the differences between 12-tone and 31-tone performances of the same piece...that is quite a different matter! Many of you have seldom heard a just, pure, beatless major third or even an approach to it. True, the 12-tone system has better fourths and fifths than 31-tone or meantone does; but the price is too high--those fourths and fifths being bought at the cost of degrading all thirds and all sixths, both major and minor.
With the ordinary style of harmony, the result is that 12-tone performances contain a continual unrelieved harshness or grating effect, whenever a major or minor chord is sounded. A single 12-tone major third may not be a crime in itself, but hundreds of them heard during the average concert add up to an overall mood of restlessness and failure to achieve true repose. Because this is subconscious, the average listener doesn't know where there is something wrong.
This wrongness or harshness or restlessness is not an Absolute. There are various ways of alleviating it, practiced by ordinary musicians in the course of performances, and by some instrument-makers also. So when the problem threatens to reach conscious awareness, one of these things is done as an artificial tranquilizer:
1. Heavy vibrato
2. Soft-pedaling (muting)--composer or conductor calling for ppp or pp to subdue the harshness.
3. Auxiliary off-pitch melody-line reinforcements: in the pipe-organ, the voix celeste stop, tuned to beat slightly sharp of the main pitch, and the unda maris stop, tuned to beat slightly flat of the main pitch; in the orchestra many first and second violinists, violists and cellists playing the same part of "warm" it up; in rock groups, trick-boxes which add such pitches or add artificial reverberation.
4. A general dulling of the timbres on 12-tone-tempered instruments: the piano became deficient in high harmonics as it became louder. Many electronic organs have no "edge" or "bite," but instead a muffled blunted tone...although invented quite long ago, the keyboard bowed-string instrument never gained favor because it is intolerable in 12-tone temperament.
5. "Blue" notes and tone-bending.
6. Hurried harried hasty fast speeded-tempi: if you can get the music up to vivace and presto, the short notes won't have time to beat with one another, and animation, aided by the percussion or rhythm section, will distract from what harshness remains. This is why the accordion is always played too fast, and never is used for Bach fugues or lullabies; and why the tremolo of the mandolin and the rapid whizzing runs of the banjo are favored, and the circus orchestrion doesn't ordinarily play largo.
7. The piano has another gimmick: the stretched octave--the tuner does not leave perfect beatless octaves in either the low or high registers of the instrument. Piano bass strings are out of tune with themselves! Result: if ALL intervals INCLUDING THE OCTAVE, are tempered, there are NO just beatless intervals to compare the harsh thirds and sixth on a proper basis, so this somewhat alleviates the effect. (Theoretically, temperaments should leave the octave strictly alone. However, stretched- and squeezed-octave temperaments have been proposed, and would indeed be useful for certain purposes. In later issues we can go into the matter more thoroughly. For the moment, referring specifically to 31-tone equal temperament, the late Prof. Adriaan Fokker had proposed a slightly-shrunken-octave version of 31, and some time ago I recommended that a somewhat stretched-octave version of 31-tone be calculated, viz. the 49th-root-of-3 equal temperament. This was done by Dr. John H. Chalmers, Jr., and recently I have made further calculations to produce tuning tables and comparison-tables for it.
My object in proposing the 49th root of 3--i.e. a tuning-system in which the twelfth or one-octave-plus-a-fifth interval was perfect, and then divided into 49 equal parts, which parts would then be used as if they were thirty-firsts of any octave (i.e., 1/31 of a NORMAL unstretched octave), was that on certain kinds of organs and other possible instruments, a perfect third harmonic would sound clearer and more brilliant. On many organs there is a 2 2/3' stop which has three times the frequency of the nominal pitch. Often this stop is called the Twelfth; sometimes Nazard. The Quintadena stop on some organs sounds both fundamental and twelfth. The 49th root of 3 would result in a slightly-sharp major third and an improvement in the minor third and the fifth and ninth. All these effects would be subtle and perhaps subconcsious.
There is a class of electronic organs, and a smaller class of synthesizers and electronic tuning-aids, on which this kind of temperament based on something else than the perfect octave, would be impossible or extremely difficult. But again, this 49th-root-of-3 scheme would not have very wide use anyway--valuable as a variant for special needs or alteration of the 31-ish mood. It could be approximated on the 31-tone guitar by a very slight moving of the bridge. Therefore we need not bother with a fretting-table for it. Fokker's contracted-octave system could also be so approximated. (Note to mathematicians: yes, I am well aware that an error is involved. But in these cases the error would be inaudible to any Earthbound human being. So let's ignore it, OK?)
Back to the discussion of "artificial tranquilizers for twelve-tone harshness" on the preceding page: a prejudiced or narrow-minded twelvetonist will resort to anything he can dream up to bolster his side of the argument against xenharmonics or non-twelve. By insisting that he piano be the only instrument allowed to be touched during the argument, he can restrict the argument to duller qualities of tone and more importantly to tones which DIE AWAY so that most of the duration is relatively free form the harshness and roughness I have been squawking and grousing about here. The 40-year-long reign of the gear-wheel tone of the Hammond Organ (only recently have its makers abandoned the principle involved) which tried to alleviate the defects of 12-tone equal temperament by means of synthesizing timbres with imitation harmonics of which their gear-wheel mechanism provides only 1 2 3 4 5 6 and 8, without the 7th and with a sharp 5, has had a profound influence on the public ear. If you play only a nylon-string 12-tone guitar, and avoid anything like plectra or fingernails, you can de-fang twelve as much as the piano does, or more.
Ever since the highs-cutting tone-control was invented for radio sets, people have been using it, and even today's hi-fi fans will use this device to take all the sharp edges off whatever comes through their loudspeakers. As a reaction to this practice of several decades, rock people are using fuzzboxes, while classicists have revived the harpsichord with oooodles of high frequency components in its tones.
31-tone and meantone can stand far more penetrating, sharp-edged, strident, acute tones than ca 12-equal. That's why I am spending so much space here on this subject, rather than merely doing a memorial to Christiaan Huygens and an appreciate gesture to Adriaan Fokker for re-discovering the Huygens' important work and doing something about it.
Strident or penetrating tones have a greater percentage of higher harmonics which will define the intervals that do not exist in 12-tone music. Unless these "new" intervals are sounded with sufficient clarity, they will not make the proper impression on new listeners. Their raison d'etre will not be clear. That is, the burden is on us: WHY go beyond 12 tones per octave? Justify the trouble and expense! 31 will cost more than 19 or 22. Is it worth it? Why is it worthy it? You must be ready with adequate answers to such questions. Not tricks of the orator or flippant evasions--good reasons that make sense.
Best is a viva-voce demonstration of the way 31-tone temperament SOUNDS. Of sustained chords in it--the new chords. Of the peculiar MOOD it has, the calm, the restfulness. No amount of verbal description is going to get that over. It must be heard to be believed. 12 notes out of meantone, tuned on a piano, are NOT nearly enough. That presentation can be downright deceitful and misleading. If that is all you have heard of either meantone or 31-tone, you simply don't have all the story yet. It has some value as history, no more.
Frankly, I'm tired of the endless monotonous stupid repetition in book after book of dreadful misinformation about meantone. Are all these people too lazy to look up the better sources? With the revival of the harpsichord and its resumption of full life and growth, and with the reprinting of Alexander J. Ellis' remarks on temperament in the appendix to his translation of the Helmholtz' Sensations of Tone, and with the mention of meantone temperament in various scholarly articles, there is no excuse for the wrong rumors about it to be perpetuated.
The misinformation claims that meantone is UNequal temperament and that it MUST have a WOLF. (No, not Prokofiev's Peter and the Wolf, and not Disc Jockey Wolfman Jack either!) This wolf, so-called because of the howling discord it makes on organs, is a kind of compulsory mistake. If you have a harpsichord, clavichord, piano, or organ with only 12 keys per octave, and try to put it in meantone temperament, you end up with wolves. Let's say you go down to E-flat and up to G-sharp. I.e., the series (it is NOT a circle in meantone!) of Fifths reads Eb Bb F C G D A E B F# C# G#. Then the fifth G# Eb and the fourths Eb G# will be woefully out of tune and distressingly dissonant. The major third B Eb (really a diminished fourth) will be just as bad, if not worse. So unless you have provided yourself with a lot of sheet music that avoids distant keys, you will be more or less compelled to have that instrument put back in 12-tone equal.
That is not really a solution at all. It merely gets rid of the full-grown meantone wolves by giving you a big litter of 12-tone wolf-cubs in exchange. True, the fifths and fourths of 12-tone are nearly perfect, being out by only 1/600 of an octave, but the major thirds are out by 1/87 of an octave, while the minor thirds are out by 1/75 of an octave. Every interval except the octave is out of tune. And the octaves are also out of tune on pianos, except for the middle. This, if you want to know, is what killed off the reed-organ or harmonium. It shows up the defects of 12-tone only too well.
Innumerable compromises between 12-tone and meantone have been invented during the last two centuries or such a matter. This is another matter we must defer to a later issue, since too much digression here would derail your train of thought.
It is mentioned here because it forms an important part of the Opposition's ammunition and will come up whenever somebody advocates meantone temperament or 31-tone equal or certain other non-twelve systems.
Now: meantone temperament is not unequal. It does not favor one key other another key. What happens in the case usually discussed is that one decapitates and curtails it.
12 tones of meantone temperament on a conventional keyboard instrument is not full meantone; it is mutilated meantone, amputated meantone or crippled meantone! The endless infinite LINE, not circle, of meantone fifths does not stop at Eb or Ab, nor at C# or G#.
No matter what the old textbooks on music say, no matter what your music teacher may have taught you, no matter what it says in certain music dictionaries, meantone temperament can be used in all keys. There is no wolf when you have enough of this system, and twelve is not enough.
I deplore the unfair, stacked-cards arguments of the opposition. They take only part of a system and claim it is complete, unequal, and inferior to 12-tone equal.
They go on to make even worse claims: that the case was settled in mid-19th century, that the case is closed forever, and some of them even go on to say that keys (in the sense of tonality vs. atonality, with triads and resolution of dissonances by consonances, and having a tonic or point of repose) are obsolete--that all contemporary composers MUST use twelve-tone serialism. Turn on your radio and listen! Are keys obsolete?
There is no reason why keys and atonality (free atonality and also serialism) can't coexist; indeed they do--and Debussy proved quite a while ago that you can back and forth from tonality to atonality again and again.
Let us return to that infinite line of fifths, from uncountable numbers of flats to inconceivable numbers of sharps. Now obviously an infinite number of tones is not practical, if only because no ear could hear the differences between them.
Suppose we take this line beyond 12, starting from C. When we arrive at A quadruple-sharp, the 31st fifth above C (of course deducting octaves as we go, or alternating fourths and fifths as a tuner does) we find ourselves only 1/200 of an octave short of C, i.e., flat. The 31st flat downward, of course, would be same distance above C (that is, it would sound sharp rather than flat).
So if we added a teeny-weeny interval, the 1714th of an octave mentioned a while ago, an inaudibly small amount, to each major third which is perfect in meantone temperament, 31 of the new major thirds would come out an even number of octaves and land us back where we started, and if we added the far tinier interval of 1/6154 of an octave to each meantone fifth, 31 of them would come out even, closing the circle and getting us back home safe. A-quadruple-sharp would then equal C. Nobody tuning by ear could ever get such accuracy. So let us assume for practical purposes that meantone and 31-tone are bosom twins.
Then there is no reason why we can't consider the 1/4-comma meantone temperament and the 31-tone equal temperament together. The difference is microscopic, and while it is of consequence to a theorist, the practical realization of either on a real musical instrument used for performing worthwhile musical compositions is indistinguishable from the other.
Even if an electronic tuning-device is used to tune the instrument, errors in transferring the tuning to the instrument will be greater than the difference between these two systems. To err is human, and all that.
At the time, 200 years ago or more, when meantone on keyboard instruments was common, there were no frequency-counters or precision tuning-devices, and no acoustical labs with international standards to check with, so what a given tuner did may have been almost anything under the sun. We don't know and can't know, and after 39 years of tuning obstreporous instruments to the much-cruder 12-tone system and checking the tuning of countless pianos and organs and other instruments, if it is that inaccurate today, what must it have been in small towns in the seventeenth or eighteenth centuries?
I realize that many meantone-fans are enthused purely for antiquarian reasons, but they should join forces with the forward-looking 31-ists, since composing in 31 has begun, and enough instruments now exist to show its value, and majority of thirty-one-fans have no objection to playing meantone-period music--indeed many have done so on their own without contacting many meantonists.
The new factors entering the picture are mostly ignored or haven't even reached the antiquarian-meantonists. It's a terrible shame, since their cooperation would lend more clout to the xenharmonic movement.
One such factor is the re-fretting of guitars and other fretted instruments to 31. This brings 31 within the average person's pocketbook and opens up almost the entire guitar literature to 31-tone. WIthout even having to mark the sheet music up! Without having ever again to worry about tuning, since the frets, once calculated, set the temperament PERMANENTLY. And never a wolf! All the so-called "enharmonic pairs," such as C-sharp and D-flat, B-sharp and C, C-double-sharp and D and then E-double-flat, are available right under the hand.
To ask whether a guitar is 31 or meantone is annoying nitpicking. The difference is much smaller than the width of a fret and the unavoidable tolerance is fret-placing and string-tension and bridge-position. The only thing that matters is that 31-tone guitars work and work superbly. It takes a while to learn to play them fast, but this can be simplified by using a 19-tone guitar for the first few months. And you will want one anybody, for its very different mood.
31-tone electronic organs exist. Several keyboard patterns have been invented. The Secor Generalized keyboard uses contrasting colors to identify the keys easily, and duplication of many keys so that no awkward changes of hand-position are required. It is not limited to 31, but capable of pushbutton change among several systems. Ervin WIlson has derived a number of keyboard-patterns that are suitable for the 31-tone system.
A number of 31-tone recordings are available. Last year Motorola Scalatron Inc. released a reel or cassette demonstration of the Generalized Keyboard applied to a special digitally-controlled electronic organ with some synthesizer-type effects, featuring 31, 19, 11, Pythagorean, and harmonic-series tunings, all on the same instrument.
Ervin Wilson has produced a number of 31-tone instruments, with metal bars, metal tubes, bamboo bars, as well as 31-tone guitars, and has compiled scales and harmonic data on the system for a number of years.
The Huygens-Fokker Foundation has a branch in Montebello, California and concentrates on 31-tone music.
The word "meantone" has been used so many times in this article that you may have begun to wonder: just what is a mean tone?
We have to get involved in theory here: you can skip this if it bores you, but a word used so much here has to be explained--the public has a right to know.
For melody without accompaniment, such as a violin alone,the question never arises--at least, it is of small importance. The pythagorean system, in which all intervals other than the octave are arrived at by an infinite line of perfect fifths, is the usual intonation for melody on such instruments, because of its brilliance and sharp leading-tones.
The Harmonic Series--the partials or overtones, are integer multiples of a fundamental frequency. Taking the simplest integers, a scale was constructed for harmonic purposes, called the Just Scale. I have to omit all the ancient history here; we haven't time.
if the perfect major third be used in the diatonic scale to be the best and smoothest harmony, it will turn out to be a member of the harmonic Series but not a member of the Pythagorean Line of Perfect Fifths, even though an excellent imitation can be found by going further in the "wrong" direction. Sorry, no time for that either just now!
That is to say: working with staff-notation or using names or sol-fa syllables of the ordinary kind, it seems as if the C-major scale could be expressed as a line of fifths: C F G D A E B, or the French or Spanish or Italian would say
fa do sol re la mi si
or in staff notation:
Where's the Problem? the conventional musician will ask--he will play it on the nearest piano. So what?
No problem, if all you play is unaccompanied melody. No problem, if you always play scales as fast as you can to get the technical practice on your instrument over with as soon as possible.
But suppose you want an organ to sound its best--no beats, smooth, calm, serene, unruffled. Or a chorus sings slowly and moderately loud, not straining their voices.
The notes of the Pythagorean Line of Perfect Fifths will not do for smooth harmonizing. They will be brilliant but so harsh. In mathematese, the ratios for a Pythagorean triad made up of perfect fifths are: 64:81:96.
The smooth harmonious beatless calm just major triad has ratios: 4:5:6.
Obviously simpler. The difference is in the major thirds, the ratio between the two being 80:81, which is the syntonic or Ptolemy or Didymus comma. Not the same as the Pythagorean comma. The ratio 80:81 works out as about 1/55 of an octave.
Now, with modern inventions like computer music and special electronic organs and other things coming up, it will be possible to use this small interval in justly-intoned (untempered) music in some cases. But for ordinary instruments like the violin and cello it is a dreadful nuisance even though they can do it.
For pianos and harpsichords it is out of the question entirely. For signers it is one cause of "flatting," since it mis-leads them. For pipe-organs it would be incredible expensive. On guitars if the frets for notes a comma apart are provided, they are inconveniently close together. (Approximately one-fifth the distance between adjacent orthodox 12-tone frets.)
That C-major scale up there isn't so simple as you thought! IF you want beatless harmony, you must provide TWO D's a comma apart. If you want to be able to modulate to G major, you must provide an extra A a comma higher than the first A. Otherwise you will get a horrible out-of-0tune fifth betweenteh correct D to be in tune with G, and the correct A to make a perfect major sixth to C. Modulation to G major will require raising the A by a comma so that is is in tune with the first D you provided. And so on infinitely. I will spare you the bad news about minor keys and harmonic sevenths which are flatter than minor sevenths by a brand-new other kind of comma.
I must say this: from C you can go to two different D-minors and at least three different keys of B-flat major.
If you are a violinist, try this:
The B which is right in one dyad is wrong in the other, if your violin is in tune.
Otherwise stated: the interval from C to D is wider than that from D to E or G to A, in an untempered scale. Or, do to re is not the same width as re to mi.
These tones occur in the harmonic series as ## 8, 9, and 10. SO they are as 9/8 and 10/9, which works out to 81/80 which we already met.
What can be done about it? For Pythagorean melodic playing, do nothing--use the wider 9/8 only. For the cheapest and most practical piano-building use twelfths of an octave, a good imitation-Pythagorean.
For those real smooth major thirds, use the Mean Tone--i.e., if the 8:9 and 9:10 intervals are both called whole-tones, take the geometric mean, which is:
Don't panic! It's simply half the width of a perfect major third. Very close to 5/31 of an octave make an interval very close to 4:7. This interval does not occur in the 12-tone scale, so those who grope for it, such as some blues singers, have to "bend" away from 12-tone to get it.
In 31-tone it is provided read-made in any of the 93 keys--31 major, 31 minor, and 31 neutral. 124 keys if you want to count the subminor keys with subminor thirds in them. It will take a long time to exhaust the 31-tone scale. It took nearly three centuries to impoverish and ruin the twelve.
But 31 is not the only new scale. So we can explore other scales for their special moods and personalities to fit our own personalities at this or that time and appeal to this or that group of listeners. 31 is the choice when you want to undo the uptight and relax and need an antidote to the noise-and-bustle overexcitement.
It wouldn't be appropriate for Khacaturian's Sabre Dance or Musorgski's Night on Bald Mountain, or yet the Flight of the Bumblebee. But many of the classical sonatas might gain by being played in 31. The slow movements especially.
What we are driving at is, someone will say--indeed, dive or six people have said--that there is no repertoire for the non-12 systems, or none for 31. What a shameful lack of imagination! Up to the strictest 12-tone tone-row serialism, almost everything written before that point was reached, can be played in 31-tone.
Much of it would be improved; some would lose its verve and punch--that which was meant to be restless and agitated would lose its bite.
No repertoire for 31-tone? Tens of thousands of pieces of existing music, that's all. And new music for it written in the last few years. Even some of the free-atonality pieces could be played in 31-tone to their advantage.
In a previous issue we showed what could be done about the problem of the Leading-Tone--either the seventh degree of the major scale, or the chromatic embellishments and ornamental tones which the violinist sharpens and closes up to their destination-tones, to increase the brilliance.
Some demonstrations of meantone were not readily appreciated because of the flatness of the leading-tone. That's easy--just play the tone 1/31 of an octave higher--never mind what it's called! Never mind either, if this melody-note you are sharpening will therefore clash with a note in the accompaniment which harmonic sense demands you NOT sharpen. It will work anyway! Have a little imagination. Try it; you'll like it.
What's the matter? Are you afraid of Bach's ghost? Or Ebenezer Prout's ghost?
Are you afraid to try something because it is impossible in twelve-tone? Or because it isn't in any book? If I write it out in notes again, as I did int he other issue, you might misunderstand me.
You'll just have to trust me that it is effective in the 31-tone system. To brighten up the "chromatics" and leading tones,you mark up your existing music to remind you to take the next higher thirty-first.
SImilarly, most, but not all of the minor thirds in dominant-sevenths chords on the top of the tetrad--i.e., Bb in the 0-7th chord, should be lowered by a 31st. This gives the 4:5:6:7 chord which is already implicit in the Harmonic Series.
Not blindly doing this to every such chord, but do it unless something in the passage contraindicates it.
Now that you have all those notes, and have escaped the limitations of 12, USE what you have! Otherwise why do it at all? Why tune only 12 notes of meantone and put up with wolves because that's all certain people would do 200 or 300 years ago? This is 1978. We can have ALL of 31, all of 22, all of 41, and freely go from one to another according to the mood. In the next few years, the added cost for non-12 will come down.
So far as guitars are concerned, the cost of non-twelve fretting is trivial. The only obstacle now is psychological. So far as the violin family is concerned, there is NO additional cost at all--it's purely psychological--getting the violinist to hear the non-twelve intervals and play them.
If you wonder how music written for only 12 pitches can be transferred to more (and use more) pitches effectively, I've already told you--alter the leading-tones, alter the ornamental or passing-tones, alter the 7th chords, and there are other possibilities, but those will use up a good many of the 31 pitch-classes you now have. Some modulations in ordinary music go further than they seem to. Sometimes an Ebb is written as D natural and an F double-sharp is written as G natural, to make it easier to read.
This principle applies quite as well to 19-tone. SO that, too, can draw on much of the existing repertoire. 22-tone, also. That is, don't throw away what you already have; don't break completely with your past; explore the kaleidoscopic iridescent variegated effects of taking the same piece through different systems.
I absolutely refuse to honor the argument that we can't move to new tuning-systems because there are no compositions in them, and we can't compose in non-twelve because there are no instruments, and so on around in a vicious circle. That's what they said about color television--they couldn't build broadcasting stations for color because there were no color nets, and they couldn't make programs because there were no color cameras, and they wouldn't build sets in color because there were no programs to watch and no TV stations to broadcast them. But it happened anyway.
I am not writing as an idle dreamer, but on the contrary as one who has composed in the 31-tone system for 15 years and who has instruments in that system and who furthermore exchanges tapes with other composers in that system. I am not writing about the distant future but about what I have already done and tested out and checked against others' opinions and reactions and proved practical.
The other extreme is possible of course: some people who have taken up the 31-tone system refuse to have anyting to do with 19 or 22 or 34. I hope I have shown impartiality in the pages of this Bulletin. We would not be any further ahead if we went to only ONE system other than 12-tone. I insist on the importance of having a range of MOODS, as discussed in Xenharmonic Bulletin No. 5. People have different personalities and some like it quiet and some like it exciting, so will prefer this or that tuning-system.
There are certain combinations of instruments and tuning-systems that are congenial, while others are at corss-purposes. A 31-tone organ is ideal, but a 31-tone piano would be impossible to make, aside from the prohibitive cost. A 31-tone guitar does very well indeed, but a 31-tone banjo would not be worth making, and a 31-tone mandolin would be next-to-impossible to play with the frets so crowded together. The timbre also must be appropriate, as we already mentioned above.
Then there is the tempo of the music--31 suggesting if not demanding a slower range of speeds than 12 or 17 or 19. So all these factors have to be coordinated and considered in each instance. This could be one of the reasons that xenharmonics has been slow in getting off the ground--too much trouble making mechanical or acoustic instruments to designs without precedent; too expensive to go against commercial practice, and unreasonable to demand new techniques of instrument-makers and performers.
With electronics, much of this is alleviated, and no that we are beginning to have automantic tuning, and built-in tuning devices within the instrument, and better keyboard designs and computers to calculate parameters such as frets and string-lengths and design criteria which used to be matters of tedious trial-and-error, many of the formerly-impossible instruments become sway to make and more important, affordable.
With automatic tuning of organs built-in, it is but a step to changeable tuning from system to system, as with the Scalatron. So this is already a reality.
Visually-minded musicians are quick to ask: how do you write it down? There are notations for the thirty-one-tone system. We start with the important and fortunate observation that our ordinary staff-notation, with clefs and sharps and flats, was designed for meantone. It fits meantone temperament hand-in-glove. Just about perfectly. It follows that all the notes of the 31-tone scale can be represented by the conventional notation also. Let us set out the primary names of the 31 tones, as a series of fifths thus:
i.e., E double-sharp would be the same as G double-flat in the 31-tone circle of fifths, and those tones would be 6 cents or 1/200 of an octave apart in the original meantone temperament.
Now it turns out that the serial order, i.e., going ultrachromatically upward degree-by-degree from C, is rather awkward to deal with if expressible only by the names of those notes above--so for this and other reasons, special signs have been invented to offer alternative representations of the tones. I hesitate to call them "enharmonic equivalents," since "enharmonic" in most of these new tuning-systems has other meanings--as for instance the various small intervals that can be used to render the Greek enharmonic tetrachords. Perhaps "synonym equivalents" or simply "synonyms" would do, when two or more names mean the same pitch.
Let us look at Adriaan Fokker's notation, since a number of composers have adopted it, and it has been reproduced in a few music-reference-books.
The double-sharp-sign which will look odd to you beside the usual x, was an older variant; but just because Prof. Fokker preferred it, doesn't mean that we have to adopt it. It simply occurs in some music Mss. and a few fonts of music type, and only occasionally in engraved music.
Since 31sts of an octave, and the diesis shared with meantone, i.e., the interval between GB and Ab in either system, is smaller than a quartertone, it takes FIVE dieses, not four, to span a whole-tone or major second. Hence there is a full diesis between the double-sharp and the adjacent natural, or the double-flat and the adjacent natural.
The new signs aren't new after all; they were invented by Giuseppe Tartini (1692-1770) the celebrated violinist of Devil's Trill fame, and discoverer of difference-tones (beat-tones) which he wrote about in a famous acoustical treatise.
The new signs may be called semiflat, semisharp, sesquiflat, sesquisharp, as with quartertone signs.
George Secor, who now has composed in the 31-tone system and played the Scalatron, had a discussion with M. Joel Mandelbaum, who also composes in the 31-tone system and performs in it; they have decided that the semiflat sign, on the staff in context, is not clear enough: it has been mistaken for a natural in some cases. So it has been suggested that the Couper Quartertone Notation, which is a modification of Ivan Vyshnegradski's, be used also for the 31-tone system. Here are the signs proposed by Mildred Couper back in 1941 and which I have used for quartertones and sometimes for the 22-tone system:
Ivan Vyshnegradski's system differs only in the flats, sesquiflat, and semiflat. In both cases the semi- and sesquisharps are identical with those of the Fokker notation.
This kind of puts me on the spot: if I make the above change for 31, will I get in Dutch with the Netherlands composers who use the Fokker notation? Let's hope not--in any event, I now have pages and pages of Fokker notation that I have neither the energy nor the patience nor the time to re-write, so that much will have to stand as written.
In addition, some composers, such as Prof. Abram M. Plum in Illinois, are using arrows for the semiflats and semisharps, pointing up or down, thus:
This is clear enough and why should this group of composers change? I have been using much smaller arrows for commas when giving examples in just intonation.
Premature attempts to standardize notation or keyboards or nomenclature for 31-tone can only cause more confusion than they should cure. If you want to call G-semiflat, G-minus instead, there is no law against it.
The whole purpose of this article is to promote the sounds of 31-tone,a nd any names or numbers or signs standing for said sounds are purely incidental and only a means to an end. It doesn't matter how they are written or what they are called! All that really matters is that more 31-tone instruments be built, that more computer-music-fans start programming in 31-tone, that the antiquarian meantonists join forces with the 31-tone contemporary composers and performers and instrument-makers, that owners of synthesizers or similar instruments capable of being put into 31-tone try it out, that bowed-instrument players try it out also, that tapes and disks and cassettes in 31-tone be duplicated and circulated so that more people can hear the many new effects.
It is even more important that all the above be done without abandoning 12, 17, 19, 22, 24, 36, 41, 53, just intonation, or inharmonic systems for atonality such as 11, 13,1 4, etc. Diversity and contrast must be kept. Maybe this seems much too complicated for beginners; but it doesn't sound complicated when you listen to xenharmonic compositions or to familiar music played in new scales.
The EXPERIENCE of a new kind of harmony and a new dimension to existing compositions is worth any complication.
A little more about the writing of unit-interval series: if we use only conventional signs such as were shown above to be adequate for denoting the circle of 31 fifths in normal order, then the sequence of thirty-firsts of an octave, or the corresponding and nearly identical notes of meantone, will look like this at the start:
Now, the note-heads will bob up and down, won't they? C, D-double-flat, C-sharp, D-flat, C-double-sharp, D. Many people will prefer the appearance of this instead:
or in sagittarian notation, like this:
so that the steadily-climbing pitches don't appear to be coming back down when they are still going UP.
Likewise for a descending ultrachromatic sequence:
My lack of space here to give several other possible notations does not signify any disapproval of them: if enough readers show enough interest I will return to the subject some time later on. The importance and future of 31-tone are such that more new notations and nomenclature are certain to be proposed and used.
Ervin Wilson has a special nomenclature and uses such names as "G-plus" on occasion; he also has a number of keyboard for 31, some of which are patented, and currently is developing instruments. His work on this and other systems is already influencing others in the field.
Another reason for creating synonyms in notation 31-tone, is that the system gives excellent imitations of 7-based intervals, those whose ratios involve the number 7. For instance,the interval 4:7, which I call the subminor seventh after Alexander J. Ellis and some call the harmonic seventh, is only 1/1200 octave off. The corresponding error for strict meantone is three times as great, 1/400 of an octave--still a barely detectable amount. The errors of 5:7, 6:7, and 7:9 (these intervals occur in the harmonic forms of chords of the seventh and ninth) are also very small.
The errors for these intervals in the 12-tone system are so large it isn't even funny. In classical harmony one pretends that the seventh harmonic does not exist at all, carefully ignoring the greater dissonance of the interval substituted for it, and treating the dominant-seventh chord as a dissonance requiring resolution. I daresay that most professional musician don't even know there are such things as subminor seventh, subminor thirds, supermajor seconds, supermajor thirds, and septimal tritones flanking the augmented fourth and diminished fifth.
By a sort of gentlemen's agreement the fact that a capella choruses often hit on the chord 4:5:6:7 and use it, is never discussed. Sometimes popular musicians will use these septimal intervals "instinctively," but still never writing them down, maybe called them 'blue notes' or 'tone bending.'
We cannot avoid hearing these intervals, because they are present in normal qualities of tone. In clarinets and other reed and brass instruments, they may be quite prominent. Occasionally a train-whistle or an auto-horn or some gearing will emit these intervals. Some piano-makers put the hammer's striking-point at about 1/7 the length to kill the seventh harmonic, but ordinary there is a trace of it in piano tone,and certain strings in the bass register may have it stronger. Some organ-stops have it.
Because it is about 1/6 tone flat of the member of the 12-tone scale nearest it, the fiction has been allowed to grow, that it is out-of-tune, too flat, useless, etc. In the meantone system it is approximated by the augmented sixth, e.g. C to A#, very closely, so on a meantone instrument it will be heard once in a while, willy-nilly.
By using the alternative notations given above, we can call the 4:7 interval a seventh instead of an augmented sixth, which seems to make some people more comfortable. Such is the psychology of prejudice. Non-composers denying composers a useful and fairly consonant interval because it doesn't occur in 12-tone. So:
In 31-tone equal temperament the above pairs mean the same thing--this is also true of meantone temperament if you have carried it out far enough--you don't have to take it to 31 notes, but if you will take it as far as 16 or 19 notes, this approximation to the harmonic or subminor or seventh and its kindred intervals will appear and be usable.
Now we run into more "sacred-cow" tradition and prejudice, alas! The tradition is that the augmented sixth must resolve upward, like so:
which is fine and dandy in 12-tone--indeed, it is part of the 12-tonish bag of tricks and stunts to modulate unexpected to a distant key: you thought you were going to have a dominant seventh to a C-major (or C-minor) chord, but instead you land in F# major, just like that! And what are you going to do about it? --Fooled you that time, eh?
Hence the term "deceptive cadence." If violins or certain other instruments are involved here, they will go Pythagorean--i.e., the augmented sixth will be taken even sharper--in the violin scheme of things, A# is sharper than Bb, and in a case of this kind where dissonance and brilliance are wanted, the difference is often exaggerated. Because it works, that's why.
Now, in 31-tone, as in meantone, the situation is quite the other way around. The above progression, if slavishly read as written, will result in a wishy-washy, tame, much too demure and softened effect, totally lacking in zonk or clout. In these systems, A# is flatter than Bb, and the wide diatonic semitone just is not a leading-tone or scintillating zingy sharp effect. The same would be true in just intonation and in some other temperaments besides 31, notably 19.
The so-called augmented-sixth chord shown above, comes out in 31-tone or meantone as a consonance that does not demand resolution the way the 12-tone or violinists' exaggerated Pythagorean affair does, and it wants to be resolved INTO, not OUT OF. Indeed, with some of the "blues" singing without accompaniment, that really happens.
So, if we want the effect of the above 12-tone example in 31, we must write:
Then we get the effect, despite the allegedly "wrong" notation; and if an old fogy harmony-teacher or conservatory offician objects, tell them to go and [expletive deleted].
If we wish to play existing music in 31, we will have to do some marking-up of music, but it will be confined to special cases like the chords illustrated.
A certain amount of ingenuity will be required to carryout some things which cannot be done in 12-tone and therefore are not even suggested n the originals core. Ornaments, such as trills, turns, mordents, etc., may sound better by microtonal alterations. In the earlier music, much favored by meantone enthusiasts, performers were expected to do their thing--to insert extra ornaments or to vary from the printed score, or to make some modifications. Only in the middle of the 19th century was this creative urge thwarted and the textbooks and music teachers often became nasty martinets, looking for more things to forbid.
Of these fanatical rule-makers, how many became more famous and remembered than the other composers?
Might as well write out a "blues"-like progression, where the subminor seventh or 31-tone/meantone augmented sixth is resolved INTO:
Not sure whether this will help you understand, mislead you, or make you mad at me--but might as well be hanged for a sheep as a lamb.
The systems have toelrable representatives of the eleventh, thirteenth, and some of the higher-up harmonics, as well as the 7th with which we have been dealing. Prof. Fokker went into this matter and produced some observations on the subject.
It would seem logical to take one step at a time--to treat first of the addition of the seventh harmonic and its consequences to the musical palette, then to go on to the next prime number, which is 11, then 13, and so on. WE have devoted so much space to the seventh harmonic because it is the principal element of progress in this direction which has been held up by the extensive use of 12-tone temperament for two centuries.
Now let's backtrack. My explanation of why a certain system is called meantone had to be interrupted for the practical reason that we can't explain four or five things at once, and there are so many loose ends to tie up, and so many misunderstandings to clear up, and so much misinformation to counter, that this article must necessarily be just as ragged as the confused situation surrounding meantone and its twin, 31-tone equal temperament.
What has taken several hundred years to get into a mess cannot be resolved overnight. What has been ignored for over a century cannot be clarified in one issue of journal like ours. All I can do is try to help.
Meantone has been advocated by historians and antiquarians and those whose interests are back in the past, as a revival in the interests of justice to the composers of the time when it was the principal system for keyboard instruments. So the whole truth about meantone, and Christiaan Huygens' admirable discovery of the 31-tone system which makes the meantone idea more useful, and moreover makes it something for the future, and for non-keyboard instruments also, and not just a curiosity for the dead past, has not been coming out.
Not only has the 12-tone gap been so long-lasting, thus putting a wedge between the past of meantone and the future of meantone/31, but the atonalists have tried to make out that any search for more consonant harmony was unnecessary--their method was the ONLY way to progress. Maybe it was the only way to progress beyond the 12-tone tonal key-system dead end; but there are dozens of escapes from that, some of which are tonal and some of which are atonal.
Back on pages 11 and 12, I explained about the minute discrepancy between 1/4-comma meantone and 31-tone equal, so that the two systems sound alike, and can be considered together. Then I showed some of the facts about an interval called the comma, what a mean tone was--the geometric mean of the two whole-tones of just intonation, or, one-half the width of a perfect, smooth, beatless major third. I explained that the scale based on perfect fifths alone,i.e., the usual explanation of the C major scale being made up of the series of fifths F C G D A E B, was not good for harmony, even though it works for unharmonized melody.
But I had to press on to other matters, so now it's time to tie up a few more loose ends. Why one-FOURTH-comma meantone? Are there other meantone systems? Yes, there are, a whole family of them, but no space in this issue to finish up that. Suppose we have a violin and a viola, so that between the two there was a chain of five notes, four perfect fifths:
This is not such a hypothetical situation at that: in a string quartet tuning up, or in an orchestra you will hear the effect of all those fifths, as they get ready to play.
Now, suppose that the low C of the viola and the E-string of the violin were sounded together:
The effect would be harsh and "edgy,' not harmonious at all. The fifth harmonic of the low C is a true two octaves and a major third above it, and it is present in viola tone. The E-string of the violin, correctly tuned by the chain of four fifths, would be a comma sharper, and the two tones would beat at a rate of 8.15 beats per second.
Now, both these E's--the 5th harmonic of tenor C (which can be elicited by itself on a viola by touching that string lightly at ANY fifth of its length--1/5,2/5,3/5,4/5) and the E of the open violin string, obtained by tuning four perfect beatless fifths from that same tenor C, have equal right to be called E. Which one is the REAL E?
Both are! It depends on what context either notes is to be played in. The process could be repeated, getting an infinite number of notes a comma (about 1/55 octave) apart, up and down from this pair of E's, and all of them would have the right to the name! This is the basic dilemma of the un-tempered (natural,pure,just) scale. It has inspired/compelled countless attempts at solution.
People have gone bonkers, bananas, blotto, berserk and galley-west over this problem. It is further complicated by the existence of other kinds of commas, but I will spare you that for the present.
Now that guitars are becoming so very important, if you had a fretless electric guitar and tuned the six strings to perfect beatless fourths and that one perfect beatless pure untempered just major third between the G and B strings, the two E strings would not be exactly two octaves apart.
The first string would be a comma flat respect to the sixth string! If you didn't now that, you might spend days string to iron out the discrepancy. On an electric guitar, that false octave would be pretty hairy. You don't need a fretless guitar to experiment: just use the open strings without using the frets at all.
This forces us to conclude that the guitar, lute, and viol frettings were not designed for a just untempered scale. The attempts to get a 12-tone temperament on fretted instruments thus antedate the present custom of tuning all keyboard instruments to the 12-tone equal temperament.
On guitars, lutes, etc., the 12-tone system was not so harsh as it was on pipe organs, because the fretted instruments had guy strings mostly, there were not uniform, the tone died away quickly, the frets were not accurately s et because they didn't have computers and frequency standards and accurate measuring devices to deal with thousandths of an inch or tenths of a millimeter.
So please don't extrapolate our Age of Ultra-Precision back to the 16th or 17th century, as so many writers of scholarly articles on music theory are wont to do!
The intellectuals of the time might have been able to make calculations, just as they were doing for astronomy, but that does not meant hat the average musician could realize precisely all the results of such calculations on the instruments of the time.
I'm repeating myself, but strictly accurate 12-tone equal temperament on organs or other instruments capable of sustained tones in qualities of tone where the beats and harshness of thirds and sixths were clearly audible, had to wait for mid-19th-century before being really achieved in laboratories or by experts, and then well into the 20th century before the average listener to music could hear enough of such accurate 12-tone tuning (in proportion to the huge amount of inaccurate 12-tone and haphazard practical tunings) to affect him or her emotionally or psychologically.
That is to say, that is why we have endured it so long. We had a considerable variety of accidental deviations from the 12-tone ideal up until the ultraprecision of the present day because mass-producible and built into some of the instruments listeners are exposed to, directly and on records.
So the scholarly debates over Bach and his contemporaries and their instruments are not relevant to 1978 composers trying to break through to restart the musical progress which has been stymied and arrested and thwarted for such an unconscionably long time!
Your problem and mine are note those of Huygens when he discovered 31-tone. Nor those of Bach writing two books of preludes and fugues. Nor is our problem that of the composers seated at their pianos in Europe in 1900. Today we are drowned in an overproduction of mediocre trivial music and innumerable repetitions of the greatness of the past, so can hardly see our way clear.
I am trying to show you how to escape this frustration. Now, that comma over there set thinkers to calculating and figuring and estimating and trying to get rid of it.
Tune the fifths right on a keyboard and the thirds and sixths will be sour so far as harmony is concerned. That's what a comma means in terms of practical consequences.
The statement is made that a string quartet plays in just intonation. That isn't quite so. In the first place, they were taught 12-tone by their teachers and the books, and they were taught to tune in perfect fifths, sometimes called Pythagorean intonation by theorists, but not by any cello or violin teachers I ever met or took lessons from!
They play Mozart or Beethoven or whomever, an occasionally run into a comma discrepancy. Without consciously analyzing it, and certainly without stopping their rehearsal to look up books on acoustics or tables of logarithms, they "intuitively" play around the comma, tempering on the run, so to speak. Do they play in 12-tone equal, then? Nope. It's practical and moreover it's ELASTIC TUNING. So it isn't fair to use string quartets either as propaganda for 12-tone equal or for just intonation.
In this era when we are going to compose on typewriter keyboards for computers to play it, and are going to have all manner of brand-new electronic instruments in our homes and studios to play manually, what string quartets do and what orchestras did, and what piano-tuners do is no longer relevant. So the 17th and 18th and 19th centuries' solutions for the comma problem may be necessary historical background, but must not determine the 1978 and 1985 and 2003 solutions to that problem. If you want to escape to the past, you are free to do so, but that just isn't my bag. I don't even enjoy going back to my childhood in 1925, since I feel much better in the good new days.
If I had been born in 1826, I could have gotten along with plenty of room to compose something different, because there still was unexplored territory of conventional means. Now it all has been conquered and we have to open virgin fields.
Back to those two E's: Alexander J. Ellis in his translation of Helmholtz's work and in his appendix to it, distinguished the two E's by using plain E for the one tuned by fifths and E1 with subscript numeral for the other one a comma lower, which is the fifth harmonic of plain C or a harmonious serene major third to it. Then the C-major scale looks like this:
Before him, other devices such as lines above and below the letters like E, or capital and small letters, were used; or today we might use asterisks or similar marks as E*.
On the staff, if I have to indicate a comma, which is very rarely, I use tiny arrows.
I prefer to record the just-intonation music on tape and evade writing it at all!
I can't afford to hire an orchestra of specially-trained experts to play it and so far, I couldn't afford computer time either. So why write it out?
Just that I have to tell you here how it CAN be written, if necessary.
Commas matter in adagio and largo with brilliant tone-quality and steadily-sustained tones and when you want subtlety. It wouldn't make any difference in a presto agitato or allegro furioso con fuoco.
So in them majority of cases, we want to eliminate the comma somehow. This means sacrificing the thirds for the sake of the fifths, or vice versa.
The atonalist and most serialists say, Scrap the whole affair, and forget classical harmony, and then commas are meaningless. Once in a while, this is the way to go--hundreds of possible atonal systems are usable. Let it be understood that I am not against this.
Back to meantone. Sacrifice the perfects fifths for the sake of the thirds and sixths. Take that 1/55 of an octave discrepancy and cut into four equal pieces. Each will be about 1/220 of an octave, then. Now, since four perfect fifths upward from C (tenor) gave us an E which was about 8.15 herz sharper than the harmonious E1 vibrating exactly 5 times as fast as the Tenor C, let's cut a tiny piece off each fifth so that four of the new fifths give us the smooth harmonious E1 instead of the icky edgy needle-sharp Pythagorean E. We find that although the error of these meantone fifths is about 2.5 times as great as that of the 12-tone fifths, it still doesn't matter in the vast majority of performances of real music. Only when we need brilliance and hard edges and steely glint and bright sparkle, some system with perfect or sharp fifths will be better.
In the new electronic age, we can afford many systems, and do not need to restrict ourselves to just one.
We can even have more than one system in the same piece.
Now: we shrunk all the fifths to get perfect major thirds. No more commas to worry about. Each fifth is 1/4 comma flat, hence this affair is called 1/4-comma meantone. And it so happens that when it is carried out to 31-pitch-classes, the circle almost closes; so 31-tone equal is the same thing for all practical music. We already tempered; why not retemper a teensy-weensy bit? No one will EVER know.
For those of you who love math, you can use your calculators and computers and tables to figure out the other possible meantone temperaments, and compare each with the others. The actually-usable meantone family runs all the way from 1/3 comma (perfect minor thirds and perfect major sixths; major thirds are about as flat as fifths are flat, thus cancelling out the error of the minor third) to 1/6 comma with noticeably-sharp major thirds, and theoretically continues as far as you wish to go, like 1/10-comma with no audible difference from 12-tone-equal and beyond that up to Pythagorean, i.e., zero-comma, and I suppose negative fractions of a comma with sharp fifths to provide a wacky theoretical basis for the 22-tone equal temperament, if your imagination runs that way.
If you like tricky semantics, call 12-tone equal temperament a variety of meantone, since the whole-step is half of a 12-tone imitation major third. Some people would rather play around with weasel-words like that, than listen to music anyway.
As the 1/4-comma meantone approximates 31-tone equal, 1/3-comma meantone approximates 19-tone equal, and 1/5-comma meantone approximates 43-tone equal, for those interested in such things. So it's not necessary to calculate everything when those equal systems are so close as to suffice for any actual trials.
I wouldn't have had to say much about meantone at all, were it not for the woeful misinformation about it in so many books, written by people who usually never heard it, and very rarely if ever have heard an instrument on which meantone was carried out more than 12 notes per octave so as to allow it full function and expressive power with its real, live, useful distinction between D-sharp and E-flat, B-sharp and C, C-double-sharp, D, and E-double-flat, etc.
Non-musicians, non-composers, performers who never play anything other than the 12 ordinary pitch-classes, simply cannot understand why a composer needs non-12 TODAY, regardless of whether anyone needed it 50 or 100 or 150 or 200 years ago--those situations simply are not relevant any more.
Remember: the "wolf" is a compulsory mistake resulting from tuning meantone on an instrument it shouldn't have been tuned on in the first place. If you have a size 10 foot and insist on wearing size 8.5 shoes, you are going to get corns and it's going to hurt. Don't blame the shoe factory. Unlike the piano-factories, they don't insist that you buy something too small.
The real issue here is hardly ever discussed--it involves fitting the ability of the human ear to distinguish differences in pitch to the instruments not capable of fine tuning, instead of the other way around--designing instruments and tunings for them which make EFFICIENT AND FULL USE of the human being's abilities. Why cripple your inborn powers to please manufacturers and non-musician engineers?
A final word about additional resources of the 31-tone system: besides the major and minor triads, 31 has a neutral triad, neither major nor minor, but halfway between, dividing the Fifth in equal halves. This means neutral scales and neutral keys, 31 of each!
There are also subminor triads, with subminor and supermajor thirds, based on the just ratio 6:7:9. And 31 subminor keys and scales.
The supermajor triad is dissonant, so it won't be that important, but if I don't mention it, somebody will squawk.
Since 31 is nearly 30, it has approximately-equal 6-tone and 5-tone scales, which can be used to simulate exotic faraway ethnic music. Since 31 is nearly 32, there is a nearly-equal 8-tone scale. The nearness to 30, 5 x 6, means you can call the unit-intervals "fifth-tones" if you don't want to call them dieses or degrees.
There is no mathematical theory of moods. There is no visual way of showing you the unique flavor of 31-tone harmony. Don't knock it if you haven't heard it. Enough people have heard it to ensure its survival.
The majority of moves are disasters, but this last one, July 1976, just after completing XENHARMONIC BULLETIN NO. 8, was particularly so.
It was a forced move with insufficient notice, because the owners broke their promise to let the editor keep on renting from the new owners. Not certain whether the old address or the present address was burglarized, as it was during the lengthy moving process that two guitars were stolen. Both were electric guitars--one converted to the 19-tone system and the other refretted to the 31-tone system. It would, we hope, be very difficult for the thieves to pawn or sell them, and conversion back to 12-tone would alsob e an onerous process. Photos of both instruments exist.
Phone calls were not referred very long, and the telephone company pursued its relentless policy of reassigning old numbers all too soon--and with the recent deterioration of the postal service, the forwarding of mail ceased after about ten months. This has been a severe loss. Nobody seems to know where Glendale is. For your information: Glendale is a separate of about 140,000 population, just north of Los Angeles a little over 7 miles (12 km) from Los Angeles' business district. It is thus closer than such famous part of Los Angeles as Hollywood or the western part of the Wilshire District.
Unfortunately again, many copies of XENHARMONIC BULLETIN #8 for July 1976 were circulated with the old address and phone number which are absolutely useless for locating us.
Feruccio Busoni's name has come down in music history as a piano virtuoso, as a composer, and also because of editing and transcription. He lived from 1866 to 1924, but what we are concerned with here are some paragraphs in his New Esthetic of Music (German edition, 1907; English edition, 1911) and a brief epilog or commentary published in the German music journal Melos (August 1922), later appearing in a collection of Busoni's writings.
With some sadness, we have to tell you that Busoni had a very interesting and revolutionary concept, but did not carry it to fruition--indeed, the 1922 remarks reflect a timidity dangerously close to retraction on some points. It is almost as though someone had intimidated him, or the Musical Establishment of his time had brought subtle pressure to bear.
Perhaps in retrospect he was afraid of his earlier audacity; more likely, the pressures of routine business, concerts, and travel prevented him from any serious composition in his new system.
His main idea he called the "tripartite tone;" we would now say "third-tones" or the 18-tone system. Of course he was not the only one to investigate this system; Julian Carrillo had a third-tone piano and composed for it, while Alois Haba devoted some space to it in his books, and his 36-tone system contains it.
The point is that Busoni approached the third-tone as a traditional musician of both German and Italian background, steeped in the milieu of his time and enjoying some degree of recognition and respect by the conventional musical community; yet he dared to conceive of a system that went against the mainstream.
Even to imagine third-tones would have been quite an effort then; no wonder he didn't gather a following on this point and didn't have enough encouragement from interested parties. The cost of building an 18- or 36-tone piano would have been formidable then; today it would simply be impossible.
Eventually, Busoni got far enough to conduct a very preliminary experiment: in New York city and aged piano-maker from Italy rebuilt an old harmonium with three manuals (such are very rare) and tuned it to 36 notes per octave, or sixth, tones, giving two rows of third tones separated form one another by ordinary (12-tone) semitones. He tried the 18-tone scale out on an audience who had not seen the instrument, seated in the next room. Their impression was that they had heard an ordinary chromatic scale in semitones.
The 3-x-12 arrangement on the three keyboards was extremely awkward to manage, Busoni said; but these experiments proved that the third-tone interval was large enough to be appreciated as a legitimate melodic interval.
As Carrillo also said, the 18-tone scale has no fourths nor fifths. So Busoni felt obliged to go to 36 tones so that the conventional semitones could be kept. He was contemplating the use of third-tone melody accompanied by more or less conventional semitonal harmony--at least he says this in the 1922 article--this could be in the nature of a retreat from his younger dreams.
Carrillo had a third-tone piano made and presented in concert. I have a tape including a demonstration and actual composition on that piano. The effect is reminiscent of many efforts to carry on where Debussy left off. This because the 6-tone, or whole-tone scale has no fourths or fifths either--that gives a vague sound overall.
George Whitman's Introduction To Microtonal Music, strictly based on Haba, but more narrowminded than he, gives actual exercises for violinists and cellists to practice third-tones.
There was no co-operation among those discovering and using third-tones--no "school" of such composers, no congenial instrument-makers, so it didn't come into general use. However, Busoni's sentences and paragraphs about the tripartite tone have been quoted and even more often cited in footnotes and references and articles on this or that aspect of music. Third-tones were never to be sounds, but it was OK to write about them, commend Busoni for this intelligence, his imaginative foresight,and willingness to indulge in wistful speculation about how nice it would be IF.
Even the occultist/composer Cyril Scott got in the act--the third-tone was supposed to help along the evolution of certain races--not us, of course! To which I must reply, Ug, and double-Ug.
Time here to break in and state that the SIZE of the tempered third-tone, 1/18 octave, is, I agree and many other agree, about right for an expressive melodic interval--cellists and violinists often sharpen leading-tones, so that the distance between F# and G, or B and C, in a melody, is about a third-tone rather than a semitone.
This admitted to be the case, there is a much better solution than 1/3 of 1/6 of an octave, that is, 1/18 of an octave: indeed, there are at least TWO solutions, the 17-tone and 19-tone systems, both of which are incompatible with 6 to 12 tones per octave, indeed 17 and 19 are both prime numbers--BUT they both have fourths and fifths, and therefore it is possible to play familiar tunes in these scales.
it is strange that none of these three composer-theorists happened on either of those two solutions, but then one must remember that conventional musical training tries to keep people from thinking for themselves, and above all forbids them to experiment.
18-tone compositions require some kind of new notation, whereas either 17-tone or 19-tone can be written WITHOUT NEW SIGNS, just the ordinary sharps and flats. So Busoni invented some suggested notations which he reproduced in A New Esthetic of Music, and these will be given in this issue.
Carrillo numbered the 18 tones, 0, through 17 , assimilating this to his general system of numbering the tones of a system and not using note-heads, and one-line instead of a five-lien staff. Haba used signs of his 36-tone notation, in effect agreeing with Busoni on the containment of 18 within 36. Whitman discards Haba's flats, retaining only the sharps, for either 18 or 36, or 24 for that matter. I think that goes against contemporary practice, which seems actually to prefer flats to sharps.
When we go to atonality, or to the use of special instruments which do not have certain partial tones in normal intensity, 18 becomes much more plausible.
For atonality, the absence of fourths and fifths would be an advantage--no longer necessary to avoid implying common chords, the very reason why much "12-tone school" composition sounds so strained and awkward and made up of traditional composers' rejected material! 18 may be regarded as 2 x 9, as well as the more obvious 3 x 6. Indeed, some people have experimented with the 9-tone temperament.
Tuning 18-tone by ear has its difficulties, there being no fourths or fifths, so the ordinary tuner may not want to bother. That could well be why Busoni wanted a 36-tone instrument, and why both Carrillo and Haba went in for 36-tone also. Fretting a guitar to 18 is easy, because one can take a 12-tone guitar and yank out every other fret, making it 6-tone, then interpolate two frets in each gap where one fret had been. So you don't have to be a mathematician, although it helps.
Since 12-tone theory makes much use of the 2-x-6 relation, the "hexachord," extending this to 3-x-6 is hardly any problem. Main objection to 18 is the same as that to 12: it is too symmetrical, and of course 36 is still more symmetrical. Some composers may want such symmetry; I don't like it simply because it conduces to the exhausted trite 12-tone cliches I am so tired of, and which I escape by going in for systems incompatible with 12 or having prime numbers of tones such as 19 or 31. For atonality, primes such as 13 or 23 will break up the 12-tone cliche-patterns, and different amounts of symmetry are available in 14, 15, 16, 20, etc.--i.e., the composer may freely choose more or less symmetry of the scale, instead of being chained to twelve.
36-tone may be used as 3-x-12 or 2-x-18 or 4-x-9 or 6-x-6, and one may "modulate" (we propose the term "transfer") from one to the other.
Haba went so far as to set out a new notation (necessarily complicated) for the twelfth-tone or 72-tone scale. But he did not regard 72 as a means for getting excellent major and minor thirds, harmonic sevenths, and harmonic elevenths, alas and alack!--he merely set it out in skeleton outline as the synthesis of the third-, sixth-, and and quartertone systems. I.e., the lowering of the 12-tone major third by 1/72 of an octave, which makes it restful and harmonious, he would have regarded as a departure or deviation form the sacred-cow 12-tone imitation major third.
In some later issue, we will take up the possibilities of the 72-tone system, but this is not the place for it--it belongs in the domain of computer music and automatically-played instruments where extreme precision can not only be attained, but maintained. Otherwise it would be useless fooling around. Why scare you off? Why leave all of us open to heckling by those who are only too anxious to use reductio ad absurdum and charges of 'quibbling over trifles' and 'impractical demands on listeners' ears' as they keep on doing?
72 tones per octave would surely be microtones, but 18 is not microtonal, if we accept the definition in certain establishment music dictionaries which now define microtones as anything smaller than quartertones. This is my motive for using the term xenharmonic: a term to cover certain systems such as 5 and 7 to the octave with larger intervals than 12 has, but not sounding like 12.
Busoni's harmonium experiment proved that 1/18 octave or a "third-tone" was not perceived as a tiny "bend" or "micro-inflection," so why call 18, or 17, or 19, microtonal? This interval-unit-size is just about right for recruiting neophytes, so third-tones will have their role in the scheme of things.
Names for the 18 pitch-classes? Well, the three exponents of this scale, and undoubtedly some other persons, have made up names: we will quote Busoni's. But I think in a case like this that Julian Carrillo's idea of numbering the tones is better: just call them 0 through 17. Otherwise we imply fifths and fourths that aren't there! The crucial difference between 18 on the one hand, and 17 or 19 on the other! The main reason for using 18 at all is its mood, its ambiguity, its impressionistic vagueness and character.
If 18 be set in a 36-tone framework, of course the tones can be called by their 36-tone names. If that is your pleasure, go ahead. The aspect of suspending some rules of harmony, or warping and twisting them, would be emphasized by assigning numbers, though.
Similar considerations attach to naming the intervals. I suppose 2/3 tone, 7-1/3 tones, etc. would do for some people, or 13o, 8o, etc might not be a bad idea either.
George Secor recently suggested that this degree-sign o be used for the unit-interval of any equal temperament being discussed at the time. If necessary to indicate which temperament's degrees are meant, put that number after the degree-sign as: 10o31. Perhaps we should borrow the mathematicians' style and put the number of tones as a subscript: 7o22.
Numbers of semitones in 12-tone are already in use by the atonal serialists in their theoretical articles, so it shouldn't be too hard to get this practice adopted.
With those temperaments that have representatives of the familiar intervals: thirds (both major and minor), fourths, fifths, and sixths, the usual names can be retained. Such is the case with the systems on either side of 18-tone or third-tone: 17 and 19. No name problem--indeed, no need for special accidental signs, although it is confusing that the 17-tone sharps are higher than their companion flats, while the 19-tone sharps are lower than their adjacent flats. It's an engineer's trade-off--isn't the advantage of retaining familiar notation and being able to read familiar music without alteration while playing it in the new system, worth the confusion?
When we come to 18-tone, it's another matter entirely: we have to have some kind of new nomenclature. There are no fourths or fifths unless you want to call altered, dissonant intervals by familiar names and create an unnecessary confusion.
Sure, there is a brute-force method which will be used in some cases: "map" the 18 tones onto a conventional twelve-tone keyboard and write the name (more than one name is possible, as you know) for that key on the keyboard REGARDLESS Of the actual sound you are going to get from that key when tuned to the 18-tone system. Synthesizer players will do this mapping-onto operation since it means no special, unfamiliar, expensive new keyboard is needed.
Very well; I have no objection to mapping-onto under such circumstances, but I do object to writing it on the staff, because it misleads other reading the examples or articles about 18-tone--even to the point of trying to play it in 12-tone, with dreadful consequences! The band and orchestra custom of "transposing" instruments written in one key and played in another has caused misery and distress for two centuries, and untold mistakes and recopying and rehearsing. This is extending that horrid idea to an even more exasperating extreme. It is asking for trouble and pleading for misunderstanding. The time to do something is NOW: nip this thing in the bud before it becomes a general disaster!
Not too long ago I ran across two new books on synthesizers in the library and here they are, telling you how to retune so that the keyboard of the synthesizer will be transposed up or down like a clarinet in Bb or an alto sax in Eb. Shame on them! A golden opportunity to get rid of this absurd custom when and as new instruments are introduced, and they buck it and spurn it and want to condemn all future musicians to crazy confusion.
This in our progressive USA where we have been leading the world in many ways, and where the Europeans already have had some sense to start writing new scores strictly in C--i.e., parts in the new orchestra scores have no transposed notations. The 20th-century use of atonality has made the idea of a transposing instrument being in the KEY of A or Bb, or F or Eb, really ridiculous, so these human roadblock people are hypocritical, as well as reactionary nuisances.
No, we do not mean that old music has to be rewritten nor that wind players throw out all their old sheet music. These new non-12 tuning systems have not been too practical until electronic and other new kinds of instruments appeared on the scene, so the romantic and classical setups for conventional compositions need not be disturbed. They should remain authentic. Then we would have a useful contrast between them and us. Why provoke the romantics and classicists? It's so much easier to leave them to their own devices and not interfere.
If we were going to have to redesign the symphony orchestra and the piano and pipe-organ and the band instruments and rewrite all the scores and parts and instruction-books, there wouldn't be enough money in the world to pay for it, and the results, even if achieved, would be disappointing; and this drastic, time-consuming re-design would have to be repeated for EACH TUNING-SYSTEM adopted.
No wonder Busoni's ideas didn't get off the ground! The electornic instruments weren't available then. However, be it noted that Busoni was almost unique, in this 1907 publication of his, in recognizing the importance of one of the first electrical (not electronic then, of course) organ, Thaddeus Cahill's Telharmonium, and calling attention to the fact that it was not tied down to the 12-tone equal temperament. He gave this reference: "New Music for an Old World" by Ray Stannard Baker, McClure's Magazine, for July, 1906. Alas! It has taken all of 70 years for the electronic organ to mature and be accepted in its own right, rather than just a cheap shoddy substitute for the pipe-organ.
Just recently, we face another revolution: the electronic organ giving way to the integrated circuit, microprocessor or computer-based instrument, in which the tuning and voicing and other particulars will be automatically programmed in. This is much more important to the progress of music than might seem to the layman.
Now, for Busoni's notation ideas: quoting from p. 32, 1911 English edition of A New Esthetic of Music: "To summarize we may set up either two series of third-tones, with an interval of a semitone between the series; or, the usual semitonic series thrice repeated at the interval of one-third of a tone.
"Merely for the sake of distinction, let us call the first tone C, and the next third-tones C# and Db, and the first semitone (small letters) c, and its following third-tones c# and db; the result is fully explained by the table below:
[In case you wonder about the absence of B-natural, in German, "B" means "B-flat" while B natural is represented by the letter H.]
"A preliminary expedient for notation might be, to draw six lines for the staff, using the spaces for the whole-tones and the lines for the semitones:
"then indicating the third-tones (and sixth-tones) by sharps and flats:
*** "Only a long and careful series of experiments, and a continued training for the ear, can render this unfamiliar material approachable and plastic for the coming generation, and for Art."
(N.B.: There appear to have been several errors in the text of the book at this point: it would not be surprising if they had been errors in translating from German, or some king of non-co-ordination between the type compositor and the music-printer. I have undertaken the risky task of emendation to what I think was meant.)
Before you criticize Busoni's ideas on notation too much, remember that it was all of 71 years ago that it appeared in print, and he probably dreamt them up a while before that. At any rate, this and certain other passages by Busoni have been frequently cited or given as footnote references in many books and articles, but always in a spirit of "it would be nice if we could, but we must lay all such notions aside." The above was only a preliminary expedient after all.
The notation proposals just cited were of course based upon the whole-tone scale already contained in the 12-tone temperament by taking every other tone. Here is a passage from Busoni's 1922 fueilliton:
"The whole tone scale with Debussy, and with Liszt before him, is like an anticipation of the whole-tone interval being filled in with the not-yet-existent tripartite-tone intervals. In this expectation,t he semitone is passed over. But only in the melodic parts; the accompanying harmony remains the one that has long been established. Therefore one does not destroy, one builds up! Time automatically rubs off what is wrong and unnecessary and automatically takes up what it good and beautiful in order to keep it. And the great and beautiful thrive."
There have been a number of 12-tone notation-reform schemes based on the 6- or whole-tone scale--there is also a movement for a 12-tone keyboard on a 6-&-6 principle instead of the familiar 7-&-5. Once an electronic organ with such a keyboard was exhibited at a meeting. None of us could play a familiar melody or scale on it!
The third-tone system retains the whole-tone scale's mood. In that sense it is the extrapolation or filling-in of the whole-tone scale. But if it were to be used only with semitonal harmony, it would not be free to develop very far--it would remain something like today's "tone-bending" that keeps being pulled back to the 12.
Today we have many more resources, even if the ordinary musical worlds has kept on spurning them. In the case of a scale like 18-tone, with unfamiliar intervals, some of which lack the definition or clarity of our fourths and fifths, the answer may lie in "tailoring" special qualities of tone to fit. This requires a close cooperation between scientists, engineers, musical-instrument makers, musicians, and composers that so far has not been implemented. It involves the partial or complete removal of certain components of a musical tone that interfere with the free functioning of a system such as 18-tone, and the possible insertion of new components which may be inharmonic partials so far as ordinary practice is concerned. It is a much more difficult problem than the synthesis of musical tones for "normal" systems having familiar intervals or new intervals that happen to be members of the Harmonic Series.
Naturally, at the time Busoni dreamt of third-tones, the Piano was Emperor--its tones were Law. Atonality had not yet taken over the place in serious music that it occupies today, so the resistance to such an affair as a fifthless scale was very strong. Nothing like the 12-tone circle of keys for tonal music exists in 18. The 36 keys of the sixth-tone scale are in 3 circles of fifths which never intersect.
Not quite as severe a situation as the two non-intersecting circles of the quarter-tone scale, however: the harmonic seventh of ratio 4:7 is closely approximated in the 36-tone equal temperament and conceivably it could be employed to bridge the chasms between the three non-meeting circles. It could today; but in Busoni's time that would have been unimaginable.
The haphazard motley assortment of instruments used in the traditional concert-hall were too strong a roadblock to the drastic revolution that the third-tone system would entail with most of them.
It had to wait for new instruments, and now we have some. Busoni called attention to the fact that violins could play third-tones--indeed they often do, by sharpening leading-tones in a melody. But you dare not ask a cellist or violinist to tune his instrument to the non-fifths of the 18-tone system! It's going to be hard enough to expect any 18-tone guitar players to tune their instruments to non-fourths.
Personally, I don't see as bright a future for the 18- and 36-tone systems as I see for 17, 19, 22, 31, 34, and 41, or for just intonation in various forms. It's not different enough from 12 and it still contains the now-pretty-much-exhausted 6-tone or whole-tone scale, with its mood, and it is just as symmetrical as 12, being 3x3x2 instead of 3x2x2. What such symmetry means, is that a series of major thirds does not pass through ALL the tones of the system, a series of whole-tones or ninths-of-an-octave does not either. But in 19-tone, any interval less than an octave MUST traverse all 19 tones before it returns to starting-point. This means more variety.
18 and 36 will have their place, now that automatically-tuned and rapidly-switching-from-one-system-to-the-other instruments exist, but it will be less important in the Scheme of Things.
Haba wrote some sixth-tone compositions, so that is a good reason for discussing sixth-tones. One of Carillo's pianos was in sixth-tones. A guitar for the 36-tone system is difficult to negotiate. It should have quite a long neck, i.e., long effective string-length, in order to get the frets far enough apart. The practical obstacle to this, of course is that the wound strings for guitars do not come in lengths much over one meter (39 3/8") so would have to be made to order at fantastic expense. As the length in use increases, the tuning must be lower in proportion. With special tunings and restricted bass compass, plain strings can be used, permitting unlimited length. Fret-tables for 18- and 36-tone will be given in this issue.
Three pianos tuned as sixth-tone apart have been tried. The only direct information I have was from Mildred Couper of Santa Barbara, California in 1942. She said that she tried this out, but it was very unsatisfactory in all the attempts, so she went back to two pianos tuned a quartertone apart and produced a number of compositions in the 24-tone system.
Modern computer and electronic-organ and synthesizer the techniques make any talk of three pianos for 36-tone anticlimactic and completely obsolete. Three players would be required, and any third-tone themes such as Busoni was hoping for would be awkwardly divided up among the three players in almost ridiculous fashion. The presence of sympathetic resonances in each of the three pianos so far as its own 12-tone set is concerned, coupled with the lack of any sympathetic response from one piano to the other two, prevents integration of all 36 tones, defeating the purpose and destroying all those expectations.
The natural human desire to conserve is not always an unmixed virtue: in this case, composers have been hamstrung and hogtied by instrument-manufacturers, commercial interests, and performers' inertia, besides the very real technical problems any performer faces in reprogramming him or herself. Insistence on keeping the 12-tone system system on the new instruments blocks progress, because we have then 12 familiar sounds and X number of utter strangers, and the contrast between members and non-members obstinately remains. Since millions of 12-tone tempered instruments will continue to exist, there is no need of making the non-12 or xenharmonic instruments conform to them or conserve the 12-tone tradition; on the contrary, a contrast is desirable.
Many complaints have been uttered in the last 50 years or so about "quartertones tacked on," "super-twelve," "half-inches instead of whole inches," "non-structural deviations and/or mere embellishments," "florid microtonal inflections of the melodic line," and so on.
It boils down to xenophobia, or fear of the unknown. Performers and instrument-makers cannot understand why a composer might want out of the 12-tone squirrelcage. I say 'squirrelcage' because of certain 12-tone-scale patterns, such as the augmented-fifth chord, the diminished-seventh chord, the whole-tone scale which is also a chord-of-seconds, the 1/2-octave or augmented-fourth/diminished-fifth, which symmetrical series go round and round and round and have been just about exhausted by several generations of composers. They can't IMAGINE that which they never allow themselves to HEAR. So this produces the logical-enough desire to retain the original twelve tones as a security-blanket to hold onto while timidly poking any arm or leg into the unknown non-twelvular territory.
Having spent some years in non-twelve composition, instrument-building, design, and performance, I can assure you that you won't get sick or catch cold or get wet or suffer any damage to your 12-tone tuning or performing skills. On the contrary, they will improve. There is nothing to dread or be anxious about or be afraid of. Xenharmonics won't bite. It will expand and liberate your imagination and reduce boredom and monotony.
Time now to reproduce Alois Haba's microtonal signs. They are in use, either as shown or modified, by quote a number of composers:
If you want to invent your own brand of accidentals, you have lots of company. Far too many systems exist for all microtonal accidentals to be standardized, and besides, there are ideas such as Augusto Novaro's and Julian Carrillo's to eliminate accidentals altogether.
What I have quoted in this article must not be taken as condemnation or approval on my part; I merely report.
Ventures beyond the confines of the twelve-tone scale necessarily involve new instruments, as well as the modification and redesign of certain existing instruments. Unless more people become involved in this endeavor, there will not be enough progress in the field. It is not sufficient to write and talk ABOUT non-twelve--the world has had far too much of that, to the point of boredom and retrogression. There would be no point whatsoever in starting and continuing the XENHARMONIC BULLETIN, if no sounds of the new systems were ever heard.
In case the Bulletin's policy is not clear to you, the emphasis is on those tunings which do not SOUND like 12-tone equal temperament, rather than on slight variants or subtle differences which require CONCENTRATED ATTENTION to notice at all.
Within this bulletin and certain other publications previous and forthcoming, there is a STIPULATION: the term microtones means the intervals less than quartertones--less than 50 cents or 1/24 octave. From 1/22 octave through 1/24--from 55 cents or hundredths of a twelve-tone semitone to 50 cents, will be considered a "Gray Area" to which the term microtones applies in an attenuated sort of way.
Some people insist that microtones are any interval less than a twelfth of an octave, 100 cents, the usual semitone. Let them do so for their own writings. But let them so stipulate! And preferably say WHY they mean it that way.
The other day I had to consult the standard standby Grove's Dictionary of Music & Musicians. (5th edition in this case.) The brief article on MICROTONES clearly and unambiguously says "smaller than quartertones;" so does a similar article in a recent music dictionary. Scholes' Oxford Companion to Music says the contrary, "small than the semitone."
The point here is that XENHARMONICS is not a mere synonym for MICROTONES. IF that were so, it would be another irritating superfluous word. The equal temperaments with larger intervals than the semitone--5,7,8,9,10 and 11 to the octave--are xenharmonic because they do not sound like 12-tone and have new intervals in them; likewise the Pelog and Slendro, which have large intervals as units.
The scale of the Highland Bagpipe is xenharmonic because of the unusual intervals some of its 9 notes make with the drones.
On the other hand it is possible to have microtones WITHOUT xenharmonics--certain expressive deviations by violinists, vocalists, etc. from a 12-tone accompaniment which are too small to disrupt the 12-tone patterns, noticeably or even deliberately by small intervals--the voix celeste, vox angelica, unda maris, etc. of the organ are tuned a tiny bit sharp or flat, but used only for 12-tone melody or harmony.
The deviations are REALLY micro-- 1/200 octave and like that: vibrato will not be treated here because it needs a very extensive analysis.
We suggest "minitones" for the less-than-semitone and down to quartertone. Current computerese jargon and even general slang have made the maxi-, mini-, macro-, micro- notions fairly popular and understood.
So much for that. Now to instruments: Graham Johnson of Hollywood, finding out about the stolen guitars, provided a bright-red electric guitar, which has been converted to 31-tone.
A suitable replacement for the 19-tone stolen guitar has not been found, at the time of writing this.
In May 1977, Susan Rawcliffe, who has been designing and making wind instruments, brought an ocarina which can, among other things, play the Japanese scale with the characteristic minor second.
In November 1977, John Reuschel, builder of harpsichords, brought a 5-stringed instrument of the monochord/sonometer order, which he calls the Tune-A-Fish, since it is in the shape of a stylized fish such as used for planking or as wall-decoration. Monochords have long been used for transferring tuning from one instrument to another, as well as teaching frequency-ratios and string-lengths.
In December 1977, Ervin Wilson and Glen Prior brought a two-octave set of metal tubes in the 17-tone system. They had been building such instruments int he 31-tone system for some time, and the successful results with this latest effort have inspired them to make another 17-tone instrument for themselves.
In October 1977, Glen Prior presented the largest Megalyra in the course of a recital at Los Angeles City College on the border of Hollywood. The audience received it with great enthusiasm.
Updating on the Megalyra: there now are 5 of these instruments, and more will undoubtedly be built before long. Some description of the idea has been given before: basically, it is an extrapolationf o the steel-guitar principle--a board with steel strings and magnetic pickup (and/or contact microphone), the usual guitar frets being replaced by painted, inlaid, or stencilled lines,and the various pitches obtained by using a "steel" which is a metal bar or tube serving as slidable movable bridge cutting off this or that vibrating length.
Now: the tradition (if I may use that word for something about 50 years old) Hawaiian or steel guitar has been restricted to such tawdry trite boring trivial music most of the time that serious musicians and composers became justifiably disgusted. Indeed, during the 1940's or such a matter and the early 50's it suffered eclipse and was almost a dead duck to most people. What could one expect from an affair with six strings tuned to an A-major chord and often played by incompetents who slid around aimlessly? This misuse gave the instrument a very bad name, and hardly anyone analyzed the situation or had any imagination.
Above all, no one seemed to recognize its close similarity with that highly respectable early keyboard instrument, the CLAVICHORD.
Both instruments have vibrato as an essential power of expression, and both depend on the intimate contact of a movable piece of metal with metal strings.
In 1940, I built an amplifying clavichord 8'4" long (2.54 m) and still have tape-recordings of it, and the resemblance to the steel guitar is indisputable. Unfortunately and alas! The clavichord is now rusting away in storage because there is no place to keep it here or elsewhere, and furthermore it was severely damaged by numerous moves and unavoidable weathering and deterioration. [ The amplifying clavichord was completely destroyed by 1985. Open-reel tape recordings were transferred to DAT as of 1994. -- mclaren ]
So now is about the right time and place to suggest that someone or a group sponsor the making of amplifying clavichords, and of large psalteries on its pattern, with xenharmonic tunings such as 19- or 22-tone, for which its timbre is excellent.
I am not keen on 12-tone clavichords because after 39 years of tuning 12-tone I am thoroughly tired of it, but those who insist on twelve can engage me as consultant and supervisor of the construction. The traditional clavichord is so extremely soft-toned that its gentle voice is lost in this noisy environment, whereas amplifying clavichords have not only plenty of volume, but extensive variety of timbre.
With many new keyboard designs, there is no excuse at all for anyone wasting any more time on 12-tone of meantone or any of the numerous compromises between meantone and 12-tone equal temperament, nor any of the desperate so-called "well-tempered" or unequal and incomplete schemes, as already stated in the article on 31-tone.
Such tunings and pseudoproblem involving them, and that dreary tiresome question of what did Bach really use? are NOT the concern of xenharmonics. In xenharmonics we are concerned with what composers can do TODAY, right NOW, to sound different from what has gone before--to use a new musical vocabulary.
The first steel guitars were in the familiar hourglass silhouette of the classical guitar--it took some time, even after solid-body instruments solely for amplification were built, to get away from this now useless design, and to start increasing the number of strings.
One bypath already taken by now is the so-called "pedal steel," with from 10 to 20 strings or more,and built as a less-portable instrument on a stand, with pedals and knee-levers to alter the tension of four or more of the strings to change the available chords. We have to mention this (mainly "country & western") instrument to show two features about it that differ from our design principles: the alterations in tuning are by 12-tone semitones; the 12-tone fret-line pattern not only is stencilled or silk-screened onto the non-fingerboard underneath the strings; it is used as an essential component of the special tablature for pedal-steels--the number of the fret-line -- so many semitones above the note to which the open string in question happens to be tuned at the moment--takes the place of lines and spaces (degrees) on our conventional five-line staff. That is to say: they pretend that those lines are real frets like a Spanish guitar preventing the instrument from sounding non-12-tone pitches! This is a subconscious psychological roadblock but very real nevertheless.
So I built my first Megalyra without any fret-lines, but people soon demanded them. I was counting on my own cello experience, rightly or wrongly. Likewise with the two Kosmolyra instruments, which have a variety of chords on their four sides, all of which are strung, and the Hobnailed Newel Post, likewise a 4-sided chordal instrument.
What was the best course for the Megalyra and to retrofit onto the other instruments? Seeing as how I was involved n none-12. And already having designed the Megalyra for certain purposes: 1) to be capable of playing existing music both solos and the bass part in orchestras; 2) to fill a gap in the scheme of things--the "electric bass" does not go down to Contra C, 33 Hz, but only to E near 42 Hz; 3) to play unisons and octaves as the pedals of an organ do, even twelfths & double octaves; 4) to have chorus-effect; --i.e., by playing in unisons and octaves to simulate the whole group of cellos and contrabasses in orchestras, or the corresponding wind instruments in a band; 5) to make certain features of the organ, guitar, clavichord, piano, and harpsichord available in flexible, non-keyboard form.
This is the age of COLOR. Drab black and gray and dark browns are somewhat passe, unless accented with bits of color. So, put color to work as the painting consultants do in our now environment. Not mere decoration but functional.
Red sharps and blue flats. Black or white naturals, depending on the ground-color of that particular instrument. (There are yellow, blue, and green Megalyrai so far--maybe red, orange and purple before we get through.) Yellow harmonic nodal points--where to elicit the harmonics 5th to 16th. Orange on the yellow instrument. Green 12-tone short fret-lines, unless the instrument is green, in which case, black.
The main fret-lines extending most of the way across the board, are a selection of 19 to 25 notes of just intonation per octave. Any more would defeat their own purpose.
The 12-tone lines are there for two purposes: so the conventionally-trained musician can continue playing with 12-tone instruments and won't get lost; and as a very inexpensive means of converting the instrument into an education comparison-chart showing clearly how much each just interval differs form 12-tone. The 12-tone lines are like dashes or hyphens on the near-the-players side of the board.
Both sides of the Megalyra are strung, so the basic design is repeated on both sides, but farther apart on the side having longer strings, of course.
Non-12 temperaments can be estimated fairly well from the two sets of lines; if not, removable plastic charts can be attached for those who need them.
UNEXPECTED BONUS: These instruments were equipped with square endpieces to permit their being laid down in any of 4 position, so that the strings will always be clear of the table, floor, or rug. Immediately it was found they would stand erect, saving storage space--and then, lo and behold, they looked like Mondrian Totem-Poles in cages. With overtones of Kandinsky and Miro. I had unique abstract sculptures on my hands, whether the instruments worked or not.
Amplification is via magnetic pickups--each set of strings has pickups in different positions for bright and subdued timbre. Contact microphones can also be used.
Since these instruments descend to the lowest C on the piano keyboard and could be taken an octave still lower--they range in length from 2 m (6 ft. 7 inch.) to 2.49 m (8 ft. 2 in.) and longer ones are possible, the steel-guitar principle at last has dignity, solidity, and depth at full volume or down to a whisper.
Also, going beyond original dreams & expectations, the strings may be plucked, struck with padded or hard mallets, or struck with the steel itself, duplicating the clavichord principle exactly, and a more piano-like tone is obtained with a "wooden steel" i.e., striking the strings with wooden bars and holding them down firmly.
Further elaborations of the idea will come by actual use of these instruments by real people in various situations. It's too early to freeze the design, although an optimum median size and basic tunings now exist as guidelines.
Those who want to learn the Megalyra will start with existing cello, double-bass, tuba, trombone instruction-books, according to the teacher's judgment, and probably supplemented with the special pedalboard study books for organ students. So at the beginning it won't be necessary to prepare special method-books until it is time for going into xenharmonics. That is, in the case of the Megalyra, the problem of repertoire is already solved. Both conservative and popular orchestrations already have suitable parts, and besides that there are solo works quite suitable for the instrument--it wasn't designed in an ivor tower for Utopians, but with full awareness of the current situation from which it has to start.
Not only that, but it had to be practical to make these instruments without elaborate and expensive special equipment.
In later issues the chordal instruments will be discussed--at present we are conducting a sort of survey to find out what designs are best. Experiments so far prove that the steel-guitar idea can be taken very far indeed from the primitive stage it had remained in for so long, and also: it intersects with some other instrument-ideas such as psaltery, dulcimer, clavichord, etc., so there will be enough work to keep many inventors busy for a long time to come.
The important point is that these instruments are here right now and they work: they are not vague future plans but solid hardware reality, so one of the obstacles to just intonation has disappeared without making it necessary to have them incompatible with existing music or not playable with other ordinary instruments.
So the time has come to make deals and form business associations of whatever kind on however small a scale at first, and to farm out whatever ideas I have no time or energy to complete. As a composer primarily, I have to put the taping and copying recordings of compositions first, so other things have to be delegated in whatever way proves practicable.
Some other kinds of instruments have to be "dedicated"--e.g. a Spanish guitar is fretted in one system,usually an equal temperament, so cannot be used in another. Even there, it is possible to make do: a 24-tone (quartertone) guitar can be used as 22-tone by moving the bridge, creating a theoretical error, which only bothers int he higher register. An 18-tone guitar (refer back to the Busoni article) can be used as 17 or 19 by moving the bridge, again with theoretical errors which only become serious beyond the octave from the open string. I have such instruments and use them.
Jonathan Glasier of San Diego (southwest corner of California) came up to this area and demonstrated a special psaltery capable of just and other tunings. He plans more instruments soon, and has had a cassette copied.
As a result of a letter published in the July-August issue of Synapse, a magazine dealing with electronic music and synthesizers, I have gained a number of correspondents and one visitor so far, Jasun Martz of Tujunga.
The letter dealt with applying new tuning-systems to synthesizers. Everett Hafner of EMSA in Northampton, Massachusetts, said he had been modifying synthesizers for just that purpose. In this case, UNdoing a built-in restriction to 12-tone in that kind of synthesizer.
I call that "Exorcising the Piano's Ghost."
[ The following addresses are no longer valid. All these people have moved, and Ivor has no current addresses for them. They are included for historical accuracy only. -- mclaren]
I heard from David Vosh in Maryland who has a group and has a synthesizer capable of non-12, but who didn't know anyone else was actually working with non-12 seriously until he saw my letter. He and his group thus felt extremely isolated.
His friend, Prentice Whiddon, wrote me about his experiences: he heard notes in his mind which were not on the piano or 12-tone instruments, and the music teachers etc. denied there were such, so he had to go in for musique concrete; and says he didn't really hear those odd notes that ran through his head until he listened to the tape that I sent to David Vosh.
Prentice Whiddon is a laboratory glassblower and so is exploring the possibilities of making all manner of new instruments out of glass in one form or another. Both these people wondered how many non-twelvists there were: so here is your chance to tell them.
In November 1977 I was taken to hear Tom Stone's demonstration of a special guitar modification--the instrument he used was a nylon-string "classical" equipped with an aluminum slide-channel of his design. Special interchangeable fingerboards which mate with this device by sliding over it and locking-in, permit the one guitar to be used for a variety of tunings. Those demonstrated included an Arabic scale,a just scale for one key and its nearest relations, 12 tones of meantone, 12-tone equal, and fretless. Other will be available later on. The Sharmachord was also demonstrated-this name is derived from a Sanskrit word--using a segment of the Harmonic Series (16 to 32) and having a special tuning starting from contra A. The effect is very subdued and tranquil. A variant of the 12-tone, using ratios based on 2:3 and 16:19, was also demonstrated.
[Alas, Tom Stone stopped manufacturing interchangeable fretboards around 1993. -- mclaren]
In 1941, Ivor Darreg, residing at the time in Santa Monica, was referred by the composer John Cage to Mrs. Mildred Couper then of Santa Barbara CA, as being involved with the quartertone system. Two of her pieces for 2 pianos tunes 1/4 tone apart were published by the then firm of New Music Editions--I have not seen them for about 30 years, but presume they are on a good many library shelves about the country still. Quartertonists should try to hunt out and revive these pioneering compositions.
Mildred Couper loaned me the booklet in French by Ivan Vyshnegradsky (this name has many transliterated spellings, some of which begin with W or Wi-; and the original Russian spelling uses letters not available in English) for some six months, during which time I translated it into English for my own reference and as propaganda for musical progress. During the war and having to move several times, it was of course impossible to get it published, and for some years it was lost (the English translation, I mean) and it was found and lost again several times between 1944 and 1974. When I did have it on hand, nasty critics scolded me for "being ahead of my time;" what they said about Vyshegradsky, Carrillo, Haba and other quartertonists is not printable.
Manuel d'Harmonie a quarte de ton, or Quartertone Harmony Manual as we might call it in English, was published in Paris in 1933, and actually is rather conservative in tone. But we have to start SOMEWHERE, and for certain types of people this is the best way. In 1974 I resolved to publish my translation after revision and writing a brief preface and commentary.
After getting a few xerox copies of it off to my friends, Motorola Scalatron Inc. in Illinois offered to print up some copies, which they did; and sent me a few. (See next item.) In 1978, it is now 45 long agonizing tedious years since Vyshnegradsky wrote the booklet--only just that he get some posthumous recognition.
This XENHARMONIC BULLETIN has contained articles on the quartertone system in a number of previous issues, and will continue to do so in future numbers. Surely after all these years have passed, it should now be safe to promote quartertones. One favorable factor has been the revival of Ives' quartertone pieces, long neglected.
The greatest stumbling-block has been the unavailable of suitable instruments at affordable prices and with the proper tone qualities to define the new intervals. As I have told you hundreds of times already, piano-tone-quality is not suitable because it doesn't contain enough 7th and 11th harmonics. Harpsichord tone does. The tones of some electronic organs do. Electric guitars do, IF and ONLY if they have pickups located about 1/10 to 1/15 the active string-length. That is seldom the case. The violin family does. On synthesizers, one has to adjust for high upper-harmonic content.
If sufficient interest develops, plans are to produce articles continuing from the point where Vyshnegradski left off, and stating the quartertone possibilities in the contemporary environment with all the facilities now available. Meanwhile, a suite, FIVE SUBMINOR SKETCHES by Ivor Darreg is available as scored by violin and using Couper notation. [See the scores section of this website - mclaren]
Melody is more important than harmony in the 24-tone system, for reasons given in other issues of this Bulletin. Those seriously interested in harmonic progress should at least consider the 22-tone temperament. Back in 1941 when I was translating, and also composing in quartertones, and planning that my amplifying clavichord would be equipped with 6 octaves of quartertones, I didn't even know there WAS a twenty-two-tone system. Few people do now!
THE SOUNDS OF NON-TWELVE
At last report, a demonstration tape or cassette (as you prefer) from Motorola Scalatron Inc. [ This tape is no longer available: the Scalatron vanished when integrated circuits proliferated in electronic musical intruments circa 1978 -- mclaren ] Played by George Secor of Chicago on the generalized-keyboard Scalatron, this tape contains examples of Pythagorean, 19, 31, just intonation with harmonic series, and atonal 11. They also had informative literature.
Shortly before moving to Glendale, I was asked to write up some attachments to an electronic device called a Lim Tuner. It contained a binary switch-controlled countdown. While I had it a couple of days, I discovered I could play its non-keyboard.
The slide-switches were never designed for any such thing; the idea was to set up, in binary one's and zero's, the divisor of a large number (or radio-frequency, if you wish to call it so), then the device would output an audio-frequency square wave (clarinet-like tone with rough edges) a definite submultiple of the original, and thus these settings would have definite ratios among themselves.
See what I mean by imagination and attitude? Often the inventor and the builders of a device never think of many things their device can do well, because of their mental blocks or narrow-rut habit-patterns, not the physical properties of the materials in the machine. Sometimes, invention consists in seeing an already-existing thing from a different viewpoint!
I bring this up because the teaching of music has fossilized and gotten rigid over a long period of time, such that creativity is presented as obsolete and the masterworks of the past are considered not only final, but complete--a revealed dogmatic 'religion,' rather than a living, growing field in which the present and future generations continue to work and produce new items and attitudes and modes of action.
So I got out the tape-recorders and took down the sounds produced, and then superposed them in two parts, once some acceptable themes had been played.
The basic ordering of the pitches is in a subharmonic series:
1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
and the result was suspiciously like the key of F minor in ordinary music. The arpeggio or harmonic implied framework is what some people now call the sub-seventh, D semi-sharp F Ab C or a minor chord with a subminor third added below it. The sub-eleventh, or F-sharp-plus, puts in a an occasional appearance. This kind of thing works passably well in the 31-tone scale, which is one reason for including it in this issue.
This composition is in just intonation, because ratios among most of the tones are exact by countdown.
Unlike the violinist or singer, the performer on an instrument of this sort is never anxious about being on pitch. So an entirely different attitude toward the just scale can be taken; therein lies the novelty of this piece. Neither the seeking of the right pitch, nor the constraints operating when a keyboard is used in a just tuning.
Since this issue has discussed the 31-tone system so extensively, and the 36-tone system to some extent, it would look like neglect or prejudice if some space were not devoted to another usable system which lies between them.
Sort of a stepchild or more accurately, a poor little orphan! Indeed, I know of no compositions in 34-tone except a couple of preliminary attempts of my own. I know of no fair treatment of the system except the passing mention by M. Joel Mandelbaum in his Doctoral Thesis on Multiple Divisions of the Octave and the Nineteen-Tone Temperament (Indiana University 1961) still not yet published in book form.
I might not even have learned about 34-tone, but for that Thesis. The reasons for this neglect are several: 31-tone has such great capabilities and potentialities that it overshadows any other system near it. Once instruments have been dedicated to 31-tone, the economic facts of life deter dedicating another expensive instrument to something "nearby" which from mathematical theory doesn't seem worth added expenditure.
Now that we have computers, tables, midget calculators in every home, and the necessary data to re-fret guitars--and synthesizers and the exciting future of computer music--and built-in automatic retuning of electronic organs--the financial restraints no longer kill off our dreams.
From mathematical calculations, the fact that 31- and 34-tone have different MOODS in actual music composed or played in them, cannot be predicted. Calculations show that 34-tone does not have the excellent approximation to the 7th harmonic that 31 has, so again certain people dismiss it unheard. Fair enough, if you were still forced to have just one sole non-twelve system; but that is no longer the case. Sometimes you need subminor sevenths and subminor thirds; sometimes you don't, or a less-accurate imitation will work.
This arithmetical fact has much bearing, in both ways, on he 34-tone tuning. 34 is to 17 as 24 is to 12. Otherwise stated, 34-tone is a Second-Order System. Two circles of 17 fifths each, which have no tone in common. That's like 24.
The major thirds of 34 are excellent, while the major thirds of 17 are inharmonious even though melodically brilliant. Since both the major thirds and the fifths of 34 are slightly sharp, the error of the minor third partially cancels out--with the result that the errors of major and minor thirds and sixths are practically the same.
In one respect the two non-intersecting circles of fifths in 34 do not matter as much as those of 24 (quartertone) because while there is no circle of major thirds going through all 23 tones of 12 nor all 24 tones of 24, there is such a circle of major thirds in 34 which MUST go through ALL 34 tones before returning to the starting-point, which means that many of the conventional means of modulation will reach all 34 keys, both major and minor. TO some extent this mitigates the theoretical disadvantages of having two, rather than one, circle of fifths.
34 also has the neutral thirds and other intervals and chords enjoyed by 24 and 31. Probably the brilliant mood of 17 will show through in most 34-tone music, and a compromise affair in which 17-tone melodies are accompanied with 34-tone harmonies will get considerable use. Cf. the Busoni notion about 18-tone being accompanied by the 12-tone harmony inherent in 36-tone.
Since any compositions previously done in 34-tone apparently never were published or even written about, I felt free to invent a special notation for extending the obvious notation for 17 to include 34.
Unless someone can come up with something provably superior, and worthwhile 34-tone compositions in it, I think the following will stand:
17-tone is notated with NO extra signs, but the circle of 17 fifths, since they are 4 cents sharp, results in strange equivalences: E#=Gb, and in the sharps being sharper than their adjacent flats, in contradiction to 19 and 31. A few pitches:
Other aspects of using 17-tone without going into 34 at all will be discussed in future issues of this Bulletin.
Since 17 is contained in 34, it would be irritating and disruptive to upset the above scheme when using 34. So: two new signs, up-dagger and down-dagger, simply the familiar printer's obelisk or dagger, erect and inverted, pointing up and pointing down, are used with an without the # and b we already use with the special 17-tone meanings given above--these meanings are nothing more than slight exaggerations of the Pythagorean meanings of sharp and flat signs already familiar and even preferred by violinists.
Just like the arrows already used by many composers by pitch-bending, if the point of the dagger points down, the pitch is to be lowered 1/34 octave, and if the point of the (inverted) dagger points up, the pitch of the tone is to be raised 1/34 octave. This will create equivalents or synonyms, but the relation of the good major thirds of 34 to the fifths will result in the lowering of sharps and raising of flats, much more than the other way around. Very much like the comma in just intonation, or the single degree in systems like 41 or 53.
Hence a few pitches:
If nobody but me ever uses this scheme, I won't cry; I won't even mope or sulk. You can always just number the tones 0 through 33 as Carrillo does for 30 and 36 and thus evade the confusion among systems in respect to the meanings of 'sharp and flat.' Or you can resort to graph-paper. Or forget about writing down and use a tape-recorder. All I care about is that some composers try 34-tone. It deserves a better fate than it has received all these centuries.
34 lies on the upper limit of guitar-playing difficulty, of course. Therefore a longer neck and active string-length than standard is recommended. In most cases people would want to approach 34 by way of 17. Players of bowed instruments should be able to follow along in 34 without too much trouble.
I don't want to repeat the sad mistake which has retarded the development of 19, 24, 31, and some other systems by setting up rules of harmony and melody-writing in 17 or 34, so won't make rules to cramp your style. I hope nobody else writes regulations in advance either. Principles of composition and adaptation into 17 and 34 must be arrived at through actual experiment on real instruments with people listening and feeding back their reactions.
The second issue of a new magazine called INTERVAL has just appeared. Edited by Jonathan Glasier, it bears the subtitle "A Microtonal Newsletter" and features letters, articles, illustrations, and reviews of concerts and books.
Subscription price has been set at $12 per year, and communications may be addressed to INTERVAL [ Address obsolete - INTERVAL magazine and INTERVAL foundation are now defunct -- mclaren]
The stress is on practice rather than theory, and there will be How-To articles as well as illustrated descriptions of new instruments and techniques of performance.
The terms "interval" and "microtonal" are meant to imply, much as this Bulletin has been saying for some time, that the moment in history has finally arrived to escape the wornout 12-tone equal temperament -- and this means new intervals, and many such intervals happen to be micro;
The 8 preceding issues of the Xenharmonic Bulletin have been working off a backlog of accumulated in£ormation which necessarily has involved getting technical at times, and very often necessitated going into careful details. INTERVAL, on the other hand, is a brand-new magazine focussing on current un-twelvular happenings and the people who have now begun to interact with one another to extend the general tonal vocabulary.
Thus the term "newsletter" has some relevance, although this is a professionally printed magazine, not a fly-by-night mimeo sheet or hastily-thrown-together flyer:
Because the San Diego area where this new publication started was the abode of the late composer Harry Partch, and the Foundation in his memory and some instruments and memorabilia are there, the first issues of INTERVAL necessarily have a Partchian aspect. Partch's system represented such a departure from "tradition" and the stuffy conventions of concert-halls, that his followers have had to exert extra effort to ensure the survival of his work and they have developed considerable ingenuity in building new instruments to carry on his ideas from where he left off.
Two new instruments by Cris Forster of San Francisco are typical of this action, and are described in the first two issues. Ample constructional and tuning details are given in the respective issues.
Jonathan Glasier is active in improvisational and performance groups and is producing copy-cassettes of some of these.
He observes that, without actual performance and sufficient HEARING of the new scales, chords, and intervals involved in the non-twelve Systems, xenharmonics would never get off the ground. One way to get something started without further delay is to improvise. Far from a mere ego-trip, this includes group improvisation where people prevent each other from stagnating at awkward dead spots. When improvisation is conducted in new tuning-systems, many of the stock cliches become impossible.
I have pointed out frequently in this Bulletin and elsewhere that music is sounds, rather than silent ink-mark notation. It is dangerous to write too much at first in non-12 tuning systems, without hearing all the new interrelations between the tones. The twelve-tone rules and frozen practices must not be allowed to kill spontaneity.
A new set of auditory memory-images and attitudes must be installed in listeners as well as performers. The quickest way to do this is to improvise in non-twelve. Since the new instuments are being developed along with new scales for them, group improvisation becomes a means of discovering how to fit new and old instruments together, and what techniques on the new instruments are going to work.
Interval will also carry announcements of new activities, a small amount of advertising, and an extremely important feature called the Tape Exchange -- until now, contemporary composers have been virtually unable to hear one another's works. Without the sounds of the new intervals, all writing and printing would be worthless.
* * *
NEW GUITAR PROGRESS
The February, 1978 issue of GUITAR PLAYER magazine, on pp. 24 sqq., contained an article by Ivor Darreg on Non-Twelve-Tone Guitars, which unleashed a flood of correspondence from distant places--and is still doing so, the latest missive being from Australia.
This correspondence has resulted in playing this typewriter far more than any of my instruments the last several months. Definitely it can now be said that enough persons want untwelve tunings that we really have a movement going.
The guitar is suited to systems of moderate numbers of tones, and re-fretting is much less expensive than building special keyboard instruments, so the average guitarist can realize the new sounds before curiosity and interest have any chance to fade.
Fretting tables for 19-, 22-, and 31-tone guitars of standard size with 650 mm active string-length ("scale length" as it is called in the trade) are included in the article. The Editors of Guitar Player added considerable introductory material and included the Bibliography.
A surprising number of readers save back numbers, and evidently get around to answering, even 7 months later! So we are publishing a few names and addresses in this Xenharmonic Bulletin 9 so that you can help establish a Xenharmonic Guitar Network. The ease of refretting, the portability of guitars, and the happy fact that 95% of standard guitar literature is performable in the 19-tone and 31-tone systems without any marking-up of the sheet-music, make the guitar an ideal entry into non-twelve. There must be about 45-50% of the existing literature playable int he 22-tone system--as the only obstacle is the visual notation problem. A different fretting is nowhere near the obstacle that a new keyboard, or the mapping of a xenharmonic system onto a 12-tone standard keyboard, presents to the newcomer.
The 24-tone or quartertone guitar is also easy to acquire, since all the 12-tone frets remain in place; but it requires composing and improvising new music for it. The 17-tone guitar is probably the easiest refretting to get accustomed to, but 17 favors melody so much over harmony that again, new compositions and adaptations from one-note-at-a-time instruments such as violin solos will be required. A considerable amount of existing music could be played on a 34 tone guitar, but would need some arranging.
Choice of systems depends mostly on the mood of the music to be played, and on one's own personality--some people prefer 22 and some people prefer 19; and no doubt psychologists will put this in their Personality Tests someday. However, a decade of interviewing and opinion-gathering has shown that one can recommend a definite order of proceeding through the systems practical on the guitar: first 19, then 22, then 31, then 17, then 24, then any others whether harmonic or inharmonic, up to the practical technical limit of close-togetherness of frets.
Note carefully: if you are not into guitars, but want to start with keyboard instruments, it is practical to start right in with the 31 tone system, if that is your cup of tea. It's just that on the guitar, 31 is definitely far more difficult than 19, and you will be held back by purely technical problems unless you practice in 19-tone first. I found this out the HARD way. This is not propaganda against 31 or 41 or 53. In later issues of this Bulletin, the many-toned systems will get their due. There are promising futures for 41, 43, 46, 50, 53, 65, 72, 77, 87, and some other systems of many tones per octave--mostly by way of computers, automatic players, special electronic organs, and the like.
While it is possible to fret guitars to UNequal temperaments and to the just-intonation system carried out to say 20 or 30 tones, the standard tuning entails splitting frets and this increases the expense and difficulty. So that phase of non-12 was not stressed in the article. Instead, a 19-tone Prelude was published in toto, and some examples of harmony and notation for 19, 22, 24, and 31 were included.
Also, while there are some fretLESS guitars and basses, this was necessarily passed over quickly, as for the average student and newcomer to xenharmonics, just tunings are more advanced technically and require greater care on the part of the performer.
So far as I am concerned personally, I am using the Hawaiian or Steel guitar principle of a sliding bar which is a movable bridge and can be also a hammer on the clavichord principle, to make just intonation practical and inexpensive and accessible to the average interested person. This is being done via like Megalyra, Kosmolyra, Hobnailed Newel Post, and some new Drone Instruments. You will be kept up-to-date in future Xenharmonic Bulletins.
Guitars would be suitable for Busoni's and Carrillo's 18-tone or third-tone system, but more difficult for the 36-tone or sixth-tone system-this latter would require extra-long strings.
But the most important thing abut the February 1978 GUITAR PLAYER article is that is what CAUSED quite a number of people to re-fret their guitars and start composing and performing in non-twelve. The lid is off Pandora's Box and the 12-tone Squirrelcage lies in ruins and it is much too late for anyone to stop us.
Because of the guitar vogue, non-twelve is open to everyone, not just esoteric specialists or persevering scholars.
The new frets SYSTEMATIZE and legitimize the so-called "bent" tones, which now can be logically integrated into coherent melodic and harmonic schemes.
* * *
One of the respondents to the February 1978 Guitar Player article was Gary Morrison.
He had fretted a guitar and a bass to the 10-tone system, and then produced flutes having the proper finger-holes for the ten-tone equal temperament. He won prizes at the Houston Science Fair and later at the Science Far in Anaheim CA, for which the first award had made him eligible, so, through emergency transportation arrangements provided by Glen Prior, he and his physics teacher were able to visit both Ervin WIlson and Ivor Darreg and see their instruments. (For those living in the crowded Eastern U.S., distances among suburbs and cities on the West Coast are extreme--in this case over 30 miles.)
The ten-tone scale is implied by the close approximations to equal pentatonic or five-tone equal temperament found in various cultures (such as the African Amadinda marimba) and also by the use of decimal subdivisions in scientific work. It has a very different mood from 12-tone. Gary Morrison has produced some cassettes demonstrating it.
When they were trying out decimal time and ten-day weeks during the French Revolution, ten-tone tuning was attempted, but information on this is extremely difficult to come by.
Morrison is now using a 19-tone guitar and some other temperaments. Recently he had a friend of his who works with computers do a performance of Ivor Darreg's second guitar prelude in 19-tone.
Dennis Genovese has developed microtonal capability for synthesizers. This is an important point, because, in the last few years the manufacturers and designers of commercial synthesizers, as well as the various experts who have been writing articles and books about synthesizers and publishing magazines and how-to data for those who roll their own, have been for the most part retrogressing to a piano-idolizing stance, even locking the 12-tone system or a very close approach to it, permanently into their synthesizer circuitry.
So it is now time for those in the xenharmonic movement to show interest in any persons or firms who are experimenting, developing, or working with non-twelve synthesizers and/or electronic organs. It is up to us to make our desires known in this regard, even if we can't acquire such an instrument for some time to come. That is, our silence about the unhappy situation of so many synthesizers being permanently restricted to 12 tones per octave will be taken by many manufacturers, engineers, textbook writers, article writers, and professional societies as confirming that no demand whatsoever for other than 12 tones per octave exists!
Recently, Dennis Genovese organized a Hawaii Electronic Music Group, and more information abut his will be included in the next Xenharmonic Bulletin. Dey Martin, one of those connected with this group, visited Ivor Darreg recently, and several other composers and musicians on the West Coast, before returning to Hawaii.
Xenharmonic LESS-Than-Twelve
George Secor recently produced some experimental temperaments and more about this will be published later as his research progresses.
On a tape which I had a chance to hear recently, he improvised in 9-tone equal temperament, producing some strange effects. Now nobody can call that "microtones" without a severe semantic wrench. This, of course, is relevant to our initial article in this issue about Busoni's third-tone and sixth-tone experiments.
In the course of his Motorola Scalatron demonstration tape some time ago, George Secor has a fairly long piece in 11-tone equal temperament, which displays its suitability for atonality, and because eleven is prime, the overly-symmetrical patterning of atonal 12-tone compositions does not occur.
Currently he is exploring certain unequal temperaments such as an UNequal 17, now realizable on new electronic instruments although tuning them by ear on conventional instruments would be impossible or nearly so.
Cris Forster has been involved with the piano for some time, and not too long ago because interested in the instruments and compositions of the late Harry Partch.
His practical experience, both with performing the piano literature and with rebuilding pianos, provided an excellent standpoint from which to carry on certain implications of Partch's system and instruments.
In Mid-July, at the studio of Ervin Wilson in Los Angeles, Cris Forster presented a recital including readings from Walt Whitman, accompanying himself on instruments that he had built.
Most unusual of these is the Chrysalis, which has two disk-shaped sounding boards, built as a wheel, and this wheel can be revolved on its axle to where the desired notes are, and it will stay in any position, as it is carefully balanced. On each wooden disk are stretched 82 strings, tuned by guitar-type tuning-gears to just ratios in an extended version of Partch's system, going to ratios involving the number 13.
The wheel is mounted by its axle on a special stand so that the strings are at a proper height for performance.
Forster's version of the Harmonic Canon is much sturdier and heavier than Partch's various forms of this instrument--this invention of Partch's has 48 strings movable bridges for accurate measuring-off of fractional string-lengths. Thus is the strings are tuned to unison for their full lengths, the movable bridges can be set at a measured fraction of that length to get other pitches bearing a definite ratio to the full length. Partch's Harmonic Canons and similar instruments used thin strings, such as guitar B-strings; Forster's uses heavier strings looking more like those on the piano. The frame is necessarily heavier accordingly. The effect of this new Harmonic Canon was at times like the una corda pedal on a grand piano--but in this case no longer subject to 12-tone temperament.
The Forster version of the Diamond Marimba used ratios involving 13, again going beyond Partch's 11 limit. This is a peculiar arrangement of tones, making easy certain patterns impossible on ordinary instruments, while some familiar patterns become difficult or impossible on this.
Cris Forster presented a special composition, ASCENT OF THE PHOENIX, taking advantage of the potentialities of this tone-arrangement.
The large living-room was exactly the right size for the volume of these acoustic instruments and for the audience.
Those of you who have, or who at least have read, the English translation of Helmholtz's SENSATIONS OF TONE will have run across both the comments of Helmholtz himself and of the translator, Alexander J. Ellis, on the 53-tone equal temperament and its extreme closeness to just intonation. The fifths are out by a fifteenth of a cent, which is such a tiny deviation that one could not count the beats by ear. THe major thirds are out by only a small amount. 1.4 cents or 1/852 of an octave, far too small to bother about in musical performances. Intervals determined by higher harmonics do have greater errors.
The Appendix on Theory of Tuning which Ellis added to the Helmholtz work, contains impassioned pleases for adopting the 53 system, but this came at a point in history coinciding with the zenith of the piano's development and the fuzzy mushy lush style of the Romantic period, and worse yet, the most agonizing point of the Divorce between Art and Science.
So, quite literally, no-one was listening. In the century since, unconscionable slanderous invective has been poured out on Ellis. As atonality and serialism were born and flourished, the disdain of 53 and just intonation grew. Thus nothing was ever done about trying out 53-tone for any worthwhile musical purposes. It stayed silent on library shelves.
Until just recently! Larry Hanson became interested in this system, unearthed some forgotten information abut it, and had Ervin Wilson and Glen Prior build three sets of aluminum instruments, each with two octaves of the 53-tone system.
It now is possible to experiment. Other instruments will follow later on. When they do, of course this will be discussed in future Bulletins.
Instead of getting rid of the comma as many other temperaments do (refer back to pages 12 & 18, this issue, where the comma is explained), the 53 system flaunts it. SUbtle effects in melodies are possible, and certain familiar intervals come in different varieties, a comma apart. The 53-tone comma is a compromise between the Didymus and Pythagorean commas.
The XENHARMONIC BULLETIN has not yet dealt with many-toned temperaments, and it will take one or two more issues before the proper time to discuss 53-tone in depth is reached. But there has been no practical playable realization of the 53-tone system for a century, and no one has revived the experimental instruments of that time, so this story can't wait.
When we get to the territory beyond 31, automatic performances and the computer become increasingly important, and will ensure the future of such systems.
As detailed in Xenharmonic Bulletin No. 8 when certain routines for tuning by ear were set out, it became impractical to push one's accuracy beyond a certain point. Novice piano-tuners are discouraged and often give up entirely, when they fail to close the circle of 12 fifths. The usual 12-tone piano-tuning routines do not in fact produce an equal temperament, but a tempered temperament i.e., a second-order compromise.
This will be true for other than 12-tone temperaments, and by the time we reach the 53-tone system, the frequency stablility of musical instruments available at the time Ellis and Helmholtz wrote their arguments on behalf of 53-tone was too poor to hold a fine tunings, whoever well the job might have been done.
Today, those obstacles no longer exist: we have crystal control and other frequency-maintaining means which can serve as clocks, so great is their accuracy. Electronic tuning devices abound and their price is coming down. Sustained tones, such as those of organs and synthesizers, can further be checked with frequency counters having direct digital readout. Thus the stability of an electronic tuning device AND of the instruments it tunes can be frequently monitored by an independent standard.
Counting circuits developed for computers have been adapted to providing a set of stable tempered pitches for electronic organs and the new digital synthesizers.
Further developments are on the drawing-boards and in the laboratories. 53-tone thus becomes as easy as any other usable tuning-system. The problems which remain pertain to musical notation, performance techniques customary today, and adaptation of existing printed music to 53tone.
It is too early for me to make any statement on the mood of this system, as I have had only limited experience hearing and performing in it. Since the small intervals are extremely subtle, it is safe to assert that compositions specifically for 53 will feature slow tempi and sustained tones.
Conventional 12-tone-system harmony is inadequate to deal with commas, so a great amount of experimentation will be necessary. This, or course, is understandable to theorists and acousticians, but very difficult to bring home to musicians and composers, who have so long been told not to think of experimenting.
The Pythagorean tuning, which uses exact perfect fifths in an infinite chain in both directions, is extremely close to the 53-tone equal temperament. Thus violinists and other string players would not be estranged by the excellent imitation of Pythagorean intonation which the 53 system provides.
In 1876, R.H.M. Bosanquet published a treatise on musical intervals and temperament in which his keyboard layout for 53 was depicted. This design has been the basis for a number of special keyboards in recent years, some of which are generalized: they can be used for many tuning-systems merely by reassigning the digitals. So there is no need to worry about keyboard problems.
With modern technology and the availability of great precision, it is no longer necessary to concentrate on just one system for music--we can have all the systems we can use. Now that instruments exist, the road is clear.
After building several of the Kosmolyra and Megalyra instruments, a new possibility appeared out of the blue. Glen Prior plays the Highland Bagpipe, which has a 9-tone scale on its melody-reed or chanter, and three drones tuned to B-flat second line bass staff and the second and third drones to B-flat top space bass staff. This instrument has stubbornly resisted conforming to the 12-tone temperament--it has mainly just intervals and some slight variants all its own.
Music for the bagpipe is written all in naturals, and the pitch which seems to be B-flat based on A=435 is called "A."
What if a steel-guitar-type instrument were made, which was about the same length as the megalyrai, and it had unison and octave strings on one side of the board with fret-lines for the bagpipe scale, and a set of drone strings on the other side? It could be amplified with a big amplifier, and could play all the regular bagpipe tunes. So I made such an instruemnt. It is called Drone I for the time being. Played softly, it sounds with a sitar-buzz just like something fresh off the boat from India, no Scotland. So an extra set of drone strings tuned to E-flat was installed. It can play Indian tunes which often are used with a soft drone.
The drone principle is helpful when someone wishes to learn new intervals--if one component of the interval is held constant, it serves as a point of reference--it usually is the tonic, but may be some other scale-degrees.
The vielle or hurdy-gurdy, and some country dulcimers and bowed instruments, have drone strings, so the principle is shared by many continents and cultures.
A second drone instrument was then built, called simply Drone II for the time being. This has long, very thin strings on the front side, 8 tuned in unison to Tenor C, and 4 somewhat the strings tuned an octave lower. A long steel can span all 12 strings, or a smaller number of them if so desired. This gives a very rich chorus effect. The magnetic pickups are located so as to accentuate the high harmonics, for a "thin" reedy tone which helps define xenharmonic intervals.
The fret-line-pattern is the same as that on the Megalyra, two octaves each of 12-tone and just intonation carried out to at least 19 pitches, and harmonic nodes up to the 16th.
On the reverse side are 10 strings, in 5 pairs, tuned in octaves, C D F G A (or this could be altered to several other pentatonic patterns) and these five drones will take care of a very wide range of folks and exotic tunes. It is not necessary use these drones at all--they are optional--but having them available can be very helpful on many occasions.
Just recently a duplicate of the above, Drone III, was completed.
These instruments will be available for use by interested parties from time to time and as many variations on this idea are possible, considerable room for experimentation exists. Using the drone is like picking up your magnifying glass--the differences between close-together pitches are "magnified" and show up clearer.
COMPOSITIONS AND INSTRUMENTS!
Brian Hartzler composed a 19-tone guitar piece after refretting a guitar to that system. Soon he will be going on to further systems. Martin Sweidel, of Cincinnati, Ohio, composed a 19-tone guitar piece some time ago and authorized me to take a few copies of it. He has also worked int he 31-tone system.
Lists of Ivor Darreg's compositions will be updated and kept available. They include a number of systems.
Someone should father up the forgotten non-12-tone compositions and make a bibliography. Few were published in any quantity.
For information on 31-tone compositions and the progress of the 31-tone movement in the Netherlands, the U.S. address is:
C. Albanese, Representative, Huygens-Fokker Foundation, 816 Carmelita Avenue, Montebello CA 90640. [This address is obsolete - mclaren]
Or you may write directly abroad:
Huygens-Fokker Foundation, Weesperzijde 23, 1091 EC AMSTERDAM, the Netherlands. [I've changed this to the current address so this address is valid - mclaren]
For information on his several instruments built in the 31-tone system, Ervin M. Wilson, 844 North Avenue 65, Los Angeles CA 90042. [This address is still valid - mclaren]
Glen Prior is composing for 31-tone instruments--one piece is based on Wilson's "hexany" arrangement of the 31 tones, which creates an unusual melodic and harmonic patterning. That is, while, as stated in the 31-tone article in this issue, 31-tone embraces the regular key-system and indeed expands it, it also permits new orderings of tone-patterns inconceivable in the 12 system.
Dr. Abram M. Plum, School of Music, Illinois Wesleyan University, Bloomington Illinois, 61701, has composed in the 31-tone system and recently produced a new score. [Dr. Plum is reportedly deceased - mclaren]
Webster College, 470 East Lockwood, Saint Louis MO 63119, under the direction of its President, Leigh Gerdine, has been promoting the 31-tone system for some time. [Leigh Gerdine has now retired - mclaren]
We can't cram everybody in xenharmonics into this issue, but we tried. Next Bulletin out soon enough, and more there.
In April of this year, Ivor Darreg was taken down to San Diego with 11 of his instruments and introduced around by Jonathan Glasier, who arranged a lecture/recital at Calliope's Coffeehouse before a capacity audience. This was also a reunion, in that Mr. and Mr. John Metzger attended the concert--Ivor Darreg lived in San Diego in 1937-38, 40 years ago, and played cello and electronic keyboard oboe in John Metzger's orchestra.
The recital began with Darreg's earlier piano compositions, some of which were composed back then, and proceeded to the Megalyra and Kosmolyra in just intonation, and guitars in 19, 22, and 31. A 5-string amplifying cello was demonstrated, and the "zero" system, Futurist intoned noise, was taken care of by the electric keyboard drum (capable of polyrhythm).
John Glasier Sr. joined Jonathan Glasier in 17-tone and Pythagorean improvisations to exemplify that aspect of xenharmonics.
The lecture portions were interspersed, and featured live comparisons.
In XENHARMONIC BULLETIN No. 7, six fretting tables were given, for the 17-, 19-, 22-, 24-, 31-, and 34-tone systems, all for the standard 650 mm (25 5/8") active string length (scale-length is a common alternative term) which appears to have been decided on by some guitar-makers in Spain in the last century, and has been approximated in other countries to a considerable extent. These tables continue to be available.
It should be obvious enough that, the more frets per octave, the closer together they have to be placed. This means that the frets cannot be carried up as high as they should be, if the number of tones per octave is increased beyond a certain point and one still insists on ordinary scale-lengths. It also means that some other fretted instruments, such as the mandolin, will not be able to have frets beyond 19 or 22 tones per octave, unless one is willing to leave much of the fingerboard unfretted. (This was in fact the case with many of the old viol family, which did not carry frets as far as one octave, so it remains a realistic possibility.)
Electric basses, the rare mando-cello, and some other fretted instruments have longer strings than guitars; also, most readers of this Xenharmonic Bulletin will have, or plan on having, a monochord or other experimental string instrument with measuring-rulers or scales on it; also, these fretting-tables can be used for fret-lines or charts of the steel-guitar type. Furthermore, in planning string-length layouts for psalteries, harpsichords, harps, hammer-dulcimers, etc., these tables give proper lengths for a considerable portion of the compass, and can easily be transplanted.
Tables for constructing charts of laying out fret-lines for instruments of the Kosmolyra and Megalyra types, as long as 2 meters or more, are available on special order.
There are so many active string-lengths in use that it is impractical to publish all tables for all lengths. With an electronic calculator it is easy enough to take the table for 1 meter and multiply it by the actual wanted scale-length. To get the distance from the nut instead of that from the bridge, each entry must be subtracted from the total length. On many instruments, it is not practicable to assume a fixed bridge-position, or it is not practicable to insert frets or lay them out while the bridge is in its final position on the instrment.
IMPORTANT: on many guitars, it is necessary to correct the nut-position by moving it toward the bridge 1 or 1.5 mm. This depends on the individual instrument, and ordinarily is not necessary if the guitar has a special zeroth fret. It is more important on a 31-tone guitar than on a 17, for instance. These tables DO NOT include any such correction, since there are a number of fretted instruments not requiring a correction. Whether to deduct the distance when laying-out frets, or actually to move the nut by taking that much of the top end of the fingerboard, is up to the craftsman. Mainly: don't write angry letters about the tables being wrong!
The 34-tone table can also be used for 17-tone by taking only alternate frets. Likewise the 36-tone table can also be used for 18-tone or (ugh!) 12. A 36-tone guitars could be used as 34 by moving the bridge, or vice versa--the error is small in actual performance, since other errors are also involved, of similar magnitude.
After consulting with interested parties, we propose a scheme of scale-lengths as follows: 650 mm, 720, 810, 915, and 1000. This gives approximately 9:8 or whole-tone steps per increment, and can be extended downward as well as upward, to reduce the too great number of fretted-instrument active string-lengths now in use. Reasons for those numbers: 650 mm is already fairly standard internationally as one of the lengths. 720 and 810 stand in 9:8 ratio. 915 is very close to the yard (exactly 914.4) so there may be existing instruments with that length, and there may be monochords etc. which were designed to be used with a 36" yardstick. 1 meter is a standard for monochords, sonometers, etc., and was the basis for John H. Chalmers Jr.'s computer printouts, from which these tables are derived.
There is no longer any point giving tables in inches, since the metric conversion will slowly but inevitably take over, and millimeter rulers and tapes are now easily obtainable. These tables are for 1980 and 1990 people rather than 1900 or 1920.
For the past century and more, an amazing amount of misinformation and incomplete information usually misleadingly presented, has circulated in music textbooks and articles, on the subject of meantone temperament. It just has to be one of the greatest scandals in music history. Meantone temperament, in a very incomplete form, only 12 notes of it to fit the standard keyboard layout, was used on keyboard instruments for a considerable period of time before the 19th century. With the recent revival of the harpsichord and the "Baroque" design of pipe organs, and the pre-Bach keyboard literature to go with them, there also has been a revival of meantone temperament.
Unfortunately, the antiquarians and traditionalists have carefully ignored Christiaan Huygens' writings on the 31-tone equal temperament, which for all fractical purposes of musical performance, is not audibly different from the 1/4-comma meantone temperament. Therefore it becomes necessary to show the difference between the two systems, and how very small it is. Hence the frequency-tables for both systems in this issue, and the fretting-tables for 31-tone, so that guitars, monochords, viols, and other fretted instruments may be put in 31-tone to perform the earlier music as well as contemporary compositions, and to make demonstrations, comparisons, and calculations possible.
31-tone-equal is cyclic--there is a circle of 31 fifths, 31 major thirds, and 31 subminor sevenths. 1/4-comma meantone is linear--an infinite series of fifths extends forever in both directions from the starting note, ordinarily considered as C. On a comparison chart such as Ivor Darreg's wall-chart No. 3, certain pitches of 31 equal are represented by close-together (about 1/200 octave apart) pairs of meantone pitches. This 6-cent failure of the meantone to close a circle of fifths, happens to be similar to the 5.4-cent flatness of the fifths in meantone, so that by carrying 1/4-comma meantone far enough, nearly perfect fifths could be obtained; but of course this is out of the question on practical musical instruments, although it might be an interesting computermusic stunt.
Major thirds are perfect, beatless, exact 4:5 ratio in meantone, while the major third in 31-tone is 0.7 cent sharp, causing slow beats. For example, the major third on Middle a would beat 0.91 of a time per second -- say 10 times in 11 seconds. Since very few chords are sustained as long as a second in ordinary musical performances, this discrepancy is negligible in actual practice. Two musicians trying to play the same pitch, or a just major third, on anything short of a computer, are hardly ever going to hit it off that well.
The flatness of the 31-tone fifth is 0.1958 of a cent (a 6129th of an octave) less than that of the --1/4 comma-meantone fifth. As if anyone cares. The meantone fifth on Middle C would beat 2.447 times a second, while the corresponding 31-tone fifth would beat "only" 2.1654 times a second, giving a difference of 0.2816. As a tuner of some 39 years' experience I can assure you that such minutiae have no practical importance.
Admittedly, a greater difference exists between the meantone and 31-tone versions of the subminor or harmonic seventh (just ratio, 4:7), which often is notated as an augmented sixth. Basing this interval on C, beats for meantone, 3.231 per second; for 31-tone, 0.7119 per second; dif£erence, ~2.519. Even so, this is a tempest in a teapot. However, if I didn't reveal it, hecklers might make a big fuss about the matter.
Let's look at it this way: We now live in an electronic and computer age of fantastic precision. Just intonation is now practical and attainable as never before. This means that hairsplitting quibbling over a tiny difference between two temperaments is now pointless. That is to say, objecting to the simulation of 1/4-comma meantone by 31-tone equal is tantamount to demanding an accurate amount of error. What an absurdity! If, for any reason, such errors are not tolerable, why, you can have just intervals at no extra cost.