The place of QUARTERTONES in Today's Xenharmonics
(1958?) [Note from Monzo: Clearly this was written in the 1980s]
[Typographical errors and omissions corrected 2020.0731 by Joseph Monzo.]
For a century and a half, the 12-tone equal temperament has been locked in place by keyboard instruments, such as piano and organ. During that time, the previous use of meantone temperament and other tuning-systems has been virtually forgotten, even though actual performance is often in other systems than 12-equal.
During that same period, the tone-quality of the piano evolved away from a thing and delicate timbre reminiscent of the harpsichord to a heavier and duller quality--this was all subconscious, of course, but it achieved the objective of making the tempered intervals of 12-tone more acceptable: but this means that the piano timbre became less and less suitable for quartertone intervals.
Nobody noticed this, for nobody during the 19th century had much incentive to explore new tuning systems on pianos--other instruments were available, especially from exotic and distant lands. It was not until the 20th century that several people, such as Vyshnegradsky and Haba and Carrillo, began wondering what would happen if the conventional semitone were split in half. Not being scientists or mathematicians primarily, but composers brought up on the masterworks of the Romantic Period, they tried the simplest and most obvious method of increasing tonal resources without altering the way keyboard instruments were supposed to be tuned: just double the number of tones per octave, and since pianos were ubiquitous, why not tune two pianos a quartertone apart? In turn this led to two obvious further developments: Haba, Barth, and others had two grand pianos built stacked on top of one another, which must have been dreadfully hard on piano movers! And Julian Carrillo had a special piano made (one of a long series of fractional-tone pianos) where a standard keyboard has quartertones mapped onto it, this of course, cutting the pitch-range in half but making all notes equally convenient to play. Much easier on the movers.
Around the 1920's a number of news items about quartertone pianos and compositions for quartertone piano surfaced. Sporadically a few more such appeared in magazines and publications later. But it never set the world on fire. Inventors patented a few quartertone instruments, lost in the dusty archives of Patent Offices. Now and then someone published a new notation system for quartertones--there must be several hundred by now, and no hope of any agreement. Why this stagnation, then? Are things any different in the Eighties?
...Well, yes. Since the Great Depression of the 1930's, there has been a very very gradual decline of the Piano, but that is now accelerating. Hitherto it has proceeded so gradually that hardly anyone noticed it. All pianos were deteriorating at about the same rate, and those in some high places like conservatories and concert halls were well cared-for, so the Image did not deteriorate, just the real instruments in most homes and other places. The price of new pianos started to climb, at first slowly, now to fantastic incredible levels way beyond the means of the average musician or student. Thus the "crossover," or point where the price of new pianos exceeded that of electronic keyboards, has already occurred and the disparity continues to widen as we progress through the 1980s. In addition, the guitar in various forms has contended for No. 1 Musical Instrument and is heard everywhere and everywhen. The electronic organ hasn't done too badly either; and now the Synthesizer and various applications of (small and therefore affordable) computers to musical purposes have greatly advanced in just the last couple of years.
With the advanace of the guitar to more prominence in the musical world, the revival of the harpsichord has also escalated, and to my mind this is no mere coincidence. It may mark a public change in preferred timbres. Many of the electronic keyboards offered as substitutes for the piano (their portability is irresistable!) also provide harpsichord imitations, some of which are excellent.
Although no orthodox music textbook ever mentions it, at least none seen to date, this shift from dull to brilliant tone-quality, involving an increasing proportion of higher harmonics, is crucial to the Quartertone Question. Even the ascendency of the guitar alone, without help from the harpsichord and its imitations, would be quite sufficient to tip the balance and remove the chief roadblock to quartertones.
Let's get acoustically and electronically technical for a while and inquire into the scientific reasons. The earliest pianos had thin strings and rather hard hammers. Thhis caused a greater percentage of higher-order harmonics (the string more easily dividing into small fractions for the subsidiary or overtone vibrations). The early pianos, however, had a thin weak tone, and couldn't compete with the huge symphony orchestras and large auditoriums and halls asked for by such composers as Wagner and Richard Strauss and the late 19th century Russians and the growth in size of orchestras and halls and audiences during the Twentieth Century. To get a louder piano tone, it was necessary to use thicker strings. To control the new dynamic range and to please the aficionados of the romantic style, the hitherto hard narrow hammers were widened and thickly padded with more layers of felt. The striking-point of these hammers was said to have moved over to one-seventh the vibrating length of the strings to eliminate the 7th harmonic from the tone. Actually, this is only a half-truth; in the low bass it would be impractical anyhow to move it in that far from the end. The actual place might be one-eighth or one-ninth in many cases, the fact does remain that the 7th harmonic is reduced from what it is in harpsichords or was in early pianos, and the 11th harmonic which helps define some quartertone-system intervals is reduced to insignificance. This is more than enough to explain the fizzling and lack of enthusiasm for the quartertone-piano experiments and for the many compositions written for two pianos a quartertone apart.
Indeed, it would have had more wet-blanket influence by 1930 than it would have had in 1860 or possibly even 1890, as piano tone kept on getting duller and mellower through the decades--we can presume this process eventually slowed down and in recent years there have been some pianos with a brilliant tone--perhaps this will become a trend as the piano tries to hold on to its 19th-century glory. Twentieth-century piano composition has called for hard percussive effects rather than mellow "singing" or "impressionistic" dreaminess.
The fact remains that the average piano today favors the ordinary musical intervals as against the new intervals approximated by certain of those in the quartertone system; therefore it subconsciously influences the public and the professionals alike against the idea of trying non-12 harmony.
So if you are new to this idea, you might consider harpsichords or electronic imitation-harpsichords rather than getting two orthodox pianos for experimenting with quartertones. That blind alley as we have indicated above, has already been explored and found wanting, so heed our advice.
Happily, there's much more. Let us talk about synthesizers and electronic organs, and the possibilities of computers and other electronic marvels. (At today's impossible prices, only a powerful multinational corporation could afford a pipe-organ for quartertones, so that sort of kicks the whole idea out of court.)
Synthesizers have introduced many musicians to the concept of dealing with waveforms directly. Instead of having to take what some long-dead piano manufacturer forced you to have or go without a piano entirely, you now have a choice; you have control. The synthesizers is use till recently employ subtractive synthesis, which means providing certain waveforms and then altering them by removal of components. New digital synthesizers may also employ additive synthesis and other methods as well. The waveforms called square and sawtooth from the way they look on an electronic technician's oscilloscope contain harmonics desireable for a quartertone instrument. The square wave contains odd harmonics only, with gradually decreasing intensity--1 3 5 7 9 11 13... and if the corners are rounded off, the tone gets smoother and more like a clarinet. The sawtooth wave contains all harmonics even and odd, again in gradually decreasing intensity, and when enough sharp edges are rounded-off, approaches the steady parts of cello and violin tones.
The subminor seventh, ratio 4:7, the subminor third, ratio 6:7, the supermajor second, ratio 7:8, are roughly imitated by quartertone intervals, whereas their 12-tone or semitone imitations are too far off to be taken seriously. The semiaugmented fourth, ratio 8:11, is almost perfectly imitated by quartertones, and the neutral sixth, ratio 8:13, is at least passable. This suggests using timbres which possess enough of the harmonics cited in their ratio-numbers to define those intervals and make them more acceptable. Dull qualities of tone with few harmonics will blur everything and reduce the dissonance-vs.-consonance contrast and thus the impact of any harmonic schemes in the music. If all you want is laid-back, soothing background stuff, go ahead; but you can get that from 'Muzak' or supermarket speaker systems.
Advocates of xenharmonic tunings other than quartertones will probably object here that the quartertone system does not afford good approximations to the true (just or pure) untempered major and minor thirds and sixths, ratios 4:5, 5:6, 3:5, 5:8. Quartertones do not improve these intervals at all over what the semitone system or 12-tone equal temperament does. Accordingly, or rather discordantly, the advocates of just intonation and of such temperaments as 19, 22, 31, 41 and 53 tones per octave will refuse to have anything to do with the quartertone (24-tone) system and attack it with all the fervor of the Inquisition of the witch-hunt! (Much of Yasser's Theory of Evolving Tonality is devoted to attacking quartertones and quartertonists; many of the followers of Harry Partch consider quartertones some kind of heresy or unmixed evil; and so on and on.)
They miss a number of important points: it might have been true 100 or 50 years ago that if you spent your money on a special harmonium or organ that could play just intonation or some tuning almost as good as just, that you could not afford a quartertone instrument, and would not have the space to put it in even if you could afford to have it made. So you had an agonizing choice--only one new system, and which one was it to be? And whatever you chose, you were stuck with it for life, so had to spend the rest of your life defending it. And attacking those who weren't so smart as to make exactly the same choice as yours.
Hardly anyone realized that different tuning systems have MOODS, each one having a unique mood that the others do not have, so even the inharmonic scales and those with large errors in the most used intervals are still valuable for their unique emotional coloring and the CONTRAST with every other mood. How could they know? No one could afford instruments in many systems.
Now all is quite different: we have electronic tuning devices that permit accurate tuning of instruments by even those with tin ears. We have the possibility of building in the tuning-means within the instrument itself, or with computers, of putting it in a PROM of other storage-device so that one does not have to tune the instrument; or the tuning can be a program like any other computer program--a software item that can be changed at will. WIth the fretted instruments one can use computer-compiled fretting-tables and locate the frets so that he tuning-system is permanently fixed.
There is a further special case with quartertones: certain portable keyboard instruments are now on the market at a reasonable price--now they may be locked into 12-tone at the factory by a built-in "chip" or some packaged circuit permanently mounted, so that it is difficult to modify them. Yet nearly all of those instruments are equipped with means for raising and lowering the pitch overall--i.e., it will still remain in 12-tone equal temperament but the pitch of say A can be higher or lower than 440 Hz--a quartertone higher or lower or in some cases a minor third higher or lower and anything in between. Then one can get two such instruments and put one a quartertone higher than the other and stack them together and--instant quartertones!
Several makes and models of such highly portable keyboards now being available, that makes writing this article worthwhile at this time. Quartertones are a way for certain types of people to enter xenharmonics, the unexplored tonal area beyond ordinary twelve. Having tasted this new experience, they well may progress further, to such systems as 17, 19, 22, 31 and other numbers of notes per octave.
This is the point at which to bring up the subject of tuning. For about a century, there has been a standard set of tuning-routines for twelve-tone equal temperament. While on the surface it looks like something dreamt up by physicists, acousticians, and/or mathematical wizards, it is really a rote-procedure, taught and learnt as such, with little or no explanation to the budding tuner of WHY one has to memorize this procedure by drudgery and conscientious repetition.
The electronic instruments already mentioned above will make the ordinary piano- and organ-tuning routines virtually obsolete in the long run. According to Alexander J. Ellis in his appendix to Helmholtz's Sensation of Tone, various versions of meantone temperaments were tuned on harpsichords, pianos, and organs through the end of the eighteenth century, with closer and closer approximations to equal temperament as the 19th century ran its course. England was something of a holdout, as compared with the Continent. So the subconscious ideal of equal spacing of 12 tones on keyboard instruments was not fully in place till the 19th century was well advanced. However, fretted instruments such as guitar and viols and lute had much closer approaches to equal temperament long before the keyboards went equal. This has some bearing on the idea of splitting the equal semitone into equal halves.
Briefly, the tuning-routines in use today are based on the circle of fifths in 12-tone equal temperament, by a slight distortion of each fifth or fourth (using them alternately so as to stay within one octave) which is a flattening of the fifth or sharpening of the fourth by 1/600 octave so that the twelfth such tuning will close the circle perfectly. This is done by memorizing the sound of the beats of the distorted intervals. The major and minor thirds and sixths beat much faster than the fifths and fourths, and are used for checking. There are routines based on the beats of thirds and sixths where the fifths and fourths are used for checking. What usually happens is that the temperament is not strictly equal--a certain amount of error is acceptable. So for the frets on standard guitars and banjos--they were not always in the theoretical positions and the old methods for placing them would give somewhat unequal results, but nobody would ordinarily be offended thereby. Some of the new electronic methods for getting 12-tone-equal have small errors--they will be based on countdown division of a very high inaudible frequency such as 2 MHz. Their deviations are usually less than those of traditional tuners, and may even be inaudibly small.
The problem with two pianos a quartertone apart will be that the piano-tuner does not have and wouldn't learn if he had access to it, a method tuning quartertone intervals. He could use a second tuning-fork a quartertone higher or lower than standard pitch and go through his routine a second time, quite as if the first instrument tuned did not exist anymore. Some tuners would attempt to guess a quartertone difference by ear. I might as well restate my hint: use the ubiquitous alternative-current power-line hum, heard on radios, amplifiers, TV sets, etc. as the quartertone tuning standard, since it is almost exactly a quartertone between B-flat and B-natural when A is 440 Hz. Generally this hum is twice the power frequency, so is 120 Hz. But most new electronic tuning-devices can be adjusted a quartertone above and a quartertone below their center pitches for each of the 12 semitones, so that the tuner of pianos and organs and harpsichords no longer needs the temperament-setting routine if he has one of these new gizmos.
The only interval excellently approximated by the introduction of quartertones is the 11th harmonic and its reduction by removing octaves: 1:11, 2:11, 4:11, 8:11. The eleven-quartertone interval is 550 cents (5.5 semitones) while the 8:11 or semiaugmented fourth is 551.3 cents, and this tiny discrepancy of 1.3 cents is too small to hear, although under favorable conditions with sustained strident electronic or reed-organ tones, the beats of this tiny error could be heard and counted. When we get down to thousandths of an octave, that's hair-splitting, man! You could learn this if taught in person to listen for the beats under those favorable circumstances, but it would be next to impossible to use this method for tuning two pianos a quartertone apart. The piano has come to have a quality which blurs the definition or defocusses the quartertone intervals.
With electronic instruments and refretting of guitars, we just don't have to worry about that anymore. Borrow a tuning-device and set the electronic instrument, or take a regular orthodox guitar and insert frets between those already there, slightly sharpwards of the half-way point between two existing frets. No problem.
Because the added new notes are not related by any familiar intervals, there is no quartertone circle of 24 fifths, but two circles of 12 fifths each which never meet. This means novel methods of modulating to the new keys have to be invented. Ivan Vyshnegradsky in his 1933 Manual of Quartertone Harmony (original in French) showed some of these. Other composers have experimented, such as Charles Ives.
It follows logically enough from the above that mistuning of the quartertones is not going to have very serious consequences. Don't worry about it! Enjoy yourself. The tolerance for mistuning 12 is very very wide. Equally wide is the tolerance for 24.
Since the additional intervals furnished by inserting the quartertones are less consonant than those already available with 12-tone, this does not enlarge the harmonic resources to double what 12 gives. If harmony were the only consideration, then the contentions of those using such temperaments as 19, 22 and 31 tones per octave would stand: they have more harmonic resources than quartertones (24) and the idea of quartertones would be played down and considered more historical.
However, the melodic resources of quartertones are very important and there is a tie-in with Ancient Greek because of what was called the Enharmonic Genus. There is, in particular, a tetrachord figure running downwards as the Greeks reckoned, major third or two whole-steps, quarter-step, say A F F semi-flat E or C A semi-flat A 3/4-flat G.
This was a very powerful mood-effect and it is a shame it has remained unheard and disused for some two millenia. This tetrachord pattern would be enough in and of itself to make quartertones worth the trouble. Many variations are possible in it. While we are about it, it works in 22-tone also, furnishing a possible pivot-point or means of transferring from either system to the other.
12-tone is symmetrical, so 24-tone is more symmetrical. 24 is divisible by 2, 3, 4, 6, 8 and 12. In addition to the whole-tone scale of 6 tones per octave we get the three-quarter-tone scale of 8 tones per octave.
Some composers and theorists have even taken the usual system of key-signatures and frightened newcomers away by writing out the signatures for the new quartertone keys bristling with unfamiliar accidentals and implying a double of the amount of practice time one would have to spend at whatever instrument. This hasn't done the Quartertone Cause any good. Maybe--blessing in disguise--the half-century since the last flurry of quartertone experimentation has weaned most advanced composers away from key-signatures and gotten them into the habit of putting accidentals in when necessary, even if the music is "tonal"--sometimes this is done just to make it look fashionable just like the serialists and the Avant-Gardists. It saves on naturals and has become acceptable. It also has greatly reduced the need for use of double-sharps and double-flats when using the 12-tone equal temperament, and therefore with the quartertone system this would also be the case. (In some other new tunings, such as the 19 and 31 systems, double-sharps and double-flats will be quite necessary.)
If one writes for two pianos a quartertone apart, this is going to mean employing two pianists in almost all cases, but that carries a disagreeable consequence: neither performer has all 24 tones of the system at his or her fingertips--and it seems "unfair" or "not quite proper" to to keep one pianist of the duo busier than the other; in turn this leads to dense texturing--too many notes in chords or in passagework, since the composer worries about one pianists having to do all the work in a certain measure while the other one is resting or has only one or two notes to play! This fact alone may have accounted for much of [the] failure of quartertones to set the world on fire or to catch on. Too many notes at once blur the tonality feeling if the music is tonal; they make atonal music hard to follow; they suggest a rather contrived expedient called "bi-atonality" which is using the two sets of 12 tones each in the ordinary atonal fashion and having a duel rather than a cop-operative enterprise. It cramps the composer's style because such a figure as the Enharmonic Tetrachord pattern mentioned previously becomes an awkward passage of the melody between the two pianists requiring very careful timing to get legato and phrasing. If, as is usually the case, the two pianos are different makes or models, and one is in better condition than the other, the continual automatic changes of timbre become very distracting.
With new electronic instruments, there is no need for such troubles. Either a single keyboard can have quartertones "mapped onto" it, or two keyboards can be placed very close together--because of the recent growth of synthesizers and other electronic keyboards, new racks for holding several keyboards, many quartertone keyboard designs have been invented and in these days of electronic computer consoles and all kinds of new office equipment, it becomes quite practical to realize such designs to suit one's hand and preferred technique habits.
There are other possible keyboard configurations that could be tried: one is 8 + 8 + 8, three rows of 8 keys per octave, breaking up the diatonic-scale white-key pattern and the pentatonic-scale black-key pattern. It should also be mentioned that there is a whole movement called Six-Six which has been promoting whole-tone-scale 6 + 6 patterns for keyboards for some years now, issuing a magazine and also promoting special notations for 12-tone music based on a 6 + 6 rather than a 7-white + 5-black principle--obviously it is no big deal to expand these schemes to 24 tones/octave. However, there is the 8 + 8 + 8 idea also, which might be desireable for certain composers. We might digress here for a moment and bring up the remark by the late Harry Partch in one of his lectures (where he was roundly denouncing Serialism and the concentration on atonal use of 12 pitches per octave) "Who wants a 12-tone row?"--well, he does have a point since few listeners could keep track of a series of 24-tone--but, for serialists, how about an 8-tone row? Why wouldn't that work if they want atonality or serialism? And there is little trouble laying out such a keyboard. The two pianists would have troubles, but we simply don't need pianos anymore for composing in quartertones.
Whereas the whole-tone scale has only two transpositions in 12, the 8-tone scale has three transpositions in 24, which should give more variety to quartertone compositions on a 3 x 8 = 24 basis. (The obvious 4 x 6 = 24 would suffer from the problem that the 2 x 6 = 12 has been exhausted and done to death ad nauseam and the doubling from 12 to 24 would not help enough in many cases.)
The conventional major-and-minor chord system is a mirror-image affair, with the fifth being unequally divided. Adding quartertones, we get the half-of-a-Fifth or Neutral third, producing a symmetrical chord which is neither major nor minor, and with that a whole array of new keys and scales which we can also call "neutral" although they may sound rather belligerent at times. This is a considerable color gain. Besides neutral thirds we get neutral seconds, neutral sevenths, neutral sixths, and the ambiguous interval of one-eighth octave, three quartertones.
But please, let's not spoil it by instisting on writing out all the neutral scales and schemes and making people use key-signatures and practice all those new scales and whatever new exercises the sadistic teachers can dream up!
Since much of the real gain from adding quartertones will be in melodic resources, we should mention trills and turns and mordents and other ornaments--more kinds of ornamentation and the possibility of widening or narrowing a trill while it is going on. Timid composers and those having misgivings about getting into quartertones might very well initiate themselves through such ornamentation. Surely you can see by taking a little thought here, that a quartertone or three-quartertone trill would be very difficult to bring off by two pianists at the two pianos, or one pianist at two grand pianos stack one on top of the other! Yet on a synthesizer with quartertones mapped onto the keyboard, it becomes very simple and moreover very effective.
Quartertones on the cello are quite practical, even though one might have to invent some new fingering-patterns--a relatively simple matter, however much it may jar the prejudices of orthodox teachers. Are you man or mouse? Do you worry that much about What Other People Might Think? On the violin, the spacing becomes closer, but that does not make it impossible. On the guitar, one simply intercalates new frets between the existing ones and such a guitar is useful in getting violinists, cellists, and wind-instrument players used to finding the quartertone intervals.
Some books on woodwinds have actually given quartertone fingerings. The late trumpet player, Don Ellis, added an extra valve to the trumpet and wrote a book of quartertone practice materials. The French Horn, dealing as it does with higher members of the Harmonic Series, does not need extra valves to negotiate quartertones.
What do we call the new intervals? Some people will ask.
Unfortunately, disagreement over nomenclature has created endless quarrelsome squabbles and greatly retarded the development of quartertone music, and this has even spilled over into controversy about non-12 tunings other than quartertones. Such publications as Howard Risatti's New Music Catalog and Karkoschka's Das Schriftbild der neuen Musik and a few others each list a number of quartertone notations, while keeping discreetly silent on what to call the intervals and what to name the pitches however notated. No matter what I say here about it, I am certain to get nasty snide remarks and hot polemics.
Now really, it's a tempest in a teapot. The new sounds are all that matter, whatever they are called, and standardization is impossible because too many composer and instrument-makers and theorists and performers--all of these presumably of equal authority and standing--have not been able to agree for more than a century. Don't waste your time and energy trying to standardize when you need it to compose!
We might start with the practice of Julian Carrillo of Mexico, who advocated the numbering of the tones in any equal-tempered system whatever, the starting note, usually taken as C, being zero. Then the intervals' names are merely the number of quartertones they contain. Several people have used this system. Whether you care to or not, might depend on your predeliction for or against atonality or serialism. It is well-known that many articles on serialism use numbering, and it is usually 0 through 11 rather than 1 through 12. So one may use through 23 for quartertones and call the intervals "7 quartertones" or "21 units" or just "17" and appear fashionably in keeping with the computer age.
The disadvantage of Carrillo's idea is of course that if you are going from semitones to quartertones, and even more if you are also interested in other divisions of the octave such as 17 19 22 31, the same kind or class of interval is going to keep changing names as you move to another system and back to 12-tone again. The interval called a perfect fifth, for example, is 7 in 12-tone, 14 in 24-tone, 11 in 19-tone, 18 in 31-tone, and in such systems as 18 tone (Busoni's Carrillo's, and Haba's third-tones) the fifth simply does not exist! If we opt for just intonation, then we have no number unless we fly to ratios such as 2:3 or 3:2 or use cents (hundredths of an ordinary semitone) and call it 702.
So there is some argument for retaining the old names such as Fifth, perhaps capitalizing it if you wish to keep it clearly apart from the number 5 and from the fraction 1/5. This, in turn, depends on whether you want to be an exclusive quartertonist or to be eclectic and have general names for interval-classes across the possible tuning-systems. These numbers and old names are very confusing in different languages, and there is a sharp division between the Germanic countries using letters A B C D E F G (H) to name notes while Latin countries use do re mi fa sol la si for fixed pitches in opposition to the English or German "movable doh." (They use numerals 1 through 8--confusion worse confounded!)
Ivan Vyshnegradsky (also spelt Wisschnegradsky and other ways) had a set of names for quartertone intervals in French, not quite agreeing with other quartertonists' names. We need not go into detail here.
I will make a set of recommendations for interval names based on the customary 12-tone nomenclature of English-speakers. In each case, prefaced by the number of quartertones in the interval: 0, unison or prime; 1, quartertone or quarter-step or semiaugmented prime; 2, semitone or half-step or augmented prime or minor second; 3, sesquiaugmented prime or neutral second; 4, whole-step or major second; 5, semiaugmented second or subminor third; 6, minor third (sometimes augmented second); 7, neutral third; 8, major third; 9, supermajor third or semidiminished fourth; diminished fifth; 13, sesquiaugmented fourth or semidiminished fifth; 14, fifth; 15, semiaugmented fifth; 16, minor sixth; 17, neutral sixth; 18, major sixth; 19, semiaugmented sixth or subminor seventh; 20, minor seventh; 21, neutral seventh; 22, major seventh; 23, supermajor seventh or semiaugmented seventh or semidiminished octave; 24, octave.
[Note from Monzo: There is clearly a section missing from this webpage between 9 and 13 degrees; if anyone has a printed copy of Darreg's paper please contact me so that it may be corrected. Here I present Darreg's nomenclature in tabular format, with my surmised interpolation for the missing part, and three other additions for completeness.]
deg. interval name other names often used 0 prime unison 1 semiaugmented prime quartertone quarter-step 2 augmented prime / minor second semitone half-step 3 sesquiaugmented prime / neutral second 4 major second (whole-tone) whole-step 5 semiaugmented second / subminor third 6 augmented second / minor third 7 neutral third 8 major third 9 supermajor third / semidiminished fourth (10 fourth) (11 semiaugmented fourth) (12) diminished fifth 13 sesquiaugmented fourth / semidiminished fifth 14 fifth 15 semiaugmented fifth 16 (augmented fifth /) minor sixth 17 neutral sixth 18 major sixth 19 semiaugmented sixth / subminor seventh 20 (augmented sixth /) minor seventh 21 neutral seventh 22 major seventh 23 supermajor seventh / semiaugmented seventh / semidiminished octave 24 octave
But remember there is and can be no real standard set of interval names at the present time, and that makes the numbering [system] very attractive. Another factor here is that with our present strong trend toward computer music proper, as well as the extensive use of computer-controlled synthesizers, the necessary codes for programming the machines and for discussing the realization of the quartertone system on them must involve numbers, so might as well take the leap now. Dyed-in-the-wool 12-tonists may wish to use things like "5 1/2" instead of "11 quartertones" but you will excuse me if I shrink from this particular controversy.
Now we get to the thorny topic of names for the pitches. Unlike the 17, 19, 31-tone systems, it is absolutely necessary to invent additional accidentals for the quartertone system, and inventions abound. So we will have room here for a very few of the symbols actually used by various composers. My own recommendation is for Mildred Couper's modification of Ivan Vyshegradski's system, as follows: natural, semiflat, flat, sesquiflat, double-flat; natural, semisharp, sharp, sesquisharp, double-sharp. Thus the tone between A and B-flat may be called either A-semisharp or B-sesquiflat (sesqui- being the Latin prefix for 1.5) and written out: A or B. The pitch between B and C may be called either B-semisharp or C-semiflat and written either B or C . This will antagonize the users of arrows and admirers of Haba and Carrillo and Vyshnegradski and Barth and Penderecki, but I have to use something, and nobody has any more authority to object than anybody else and we just don't have the time or energy to fuss over names.
There are literally hundreds of new musical notations for 12-tone and for other systems, a very few special notations for quartertones, and various kinds of accidentals used for taking a 12-based system such as the Keyboard Staff and making a quartertone notation out of it. Guitar chord-diagrams are easily doubled in size to accommodate quartertone frets and the semisharp and semiflat symbols can be used to name the chords, as well as the abbreviation Neu for neutral chords. Guitar tablatures number the frets or fret-lines on steel guitars, so there is no problem renumbering and putting in the intercalary frets or fret-lines. Nor is graph notation any special problem.
Personally I should like to keep the arrow notation often used for quartertones, for other purposes, mainly the indication of a mere bend, i.e., an arbitrary or indefinite or discretionary raising or lowering of the pitch. Just because you go in for quartertones and nominally or usually employ 24 equally-spaced pitches per octave, that does not stop you from being free to deviate from these quartertone levels. You may bend all you want to, and so why not keep arrows for that purpose? There is no law against it. We have to remind the theorists and the would-be standardizers and the self-appointed Authorities and the exclusively-keyboard-players that there is a big world out there of violins and singers and trombones and 'humoring the tone' and glides and slides and Hawaiian guitars and what all. Even the hardest-boiled of Serialists have to leave a tiny loophole when they [use] "pitch-classes" rather like the linguistic specialists who deal in phonemes and allophones.
Might as well discuss this phonemes-vs.-allophones thing as this linguistic analogy applies to music. Another name for the dichotomy is Etic-vs.-Emic. In linguistics, a phoneme is a category of sounds used in a given language--a class of sounds such as vowels or consonants that is lumped together by speakers of that language, such that any shade of a certain kind of sound is recognized as that phoneme--minute variations are ignored, so that it takes a major difference to make a speech-sound a non-member of the given phoneme. But different languages and different dialects of the same language, such as Western American speech and New England speech and the Deep South and the London speech in Great Britain, have different opinions of what fine shadings of sound (called allophones) are, or are not, members of a given phoneme. There is even a zero (0) when some speakers do not pronounce a certain sound while other speakers of the same language do: Western American here vs. Southern British heah. A phonetician will pay close attention to such matters and to very fine shades of sound which the average listener (consciously!) ignores. The phonemic specialist will ignore all these shades and give them just one covering letter or symbol. "They do not serve to distinguish one word from another" will be his argument. While the average listener conciously is unaware of such fine shades, he or she will be subconsciously aware and this will influence the listener's attitude toward the speaker--it could mean money or a job in many cases!
When we double the number of pitch-classes by going from 12 per octave to 24, from semitones to quartertones, it is something like creating another language or dialect of music. So the listeners may or may not be willing to heed the new distinctions. The average person's hearing in the middle of the audible spectrum is capable of distinguishing extremely small intervals, under the proper conditions. Speech uses such small intervals usually in a gliding fashion. Going to quartertones, we cut the area of each "toneme" or "pitch-class" in half, making greater demands on the listeners. They have been "programmed" or "conditioned" to ignore less than a semitone difference and so may have to be "deprogrammed" to undo this damage. Some authors of theory and harmony books have made a big to-do over this, condemning quartertones literally without any hearing.
Actually, it's not any big deal. A little common sense on the part of composers and performers will make things much easier. Slow down. The ability to discriminate small intervals depends on the tempo. Use timbres rich in harmonics. That helps define or brighten-up the new intervals. Tend toward leanness rather than lushness. Smaller ensembles, fewer notes per page (i.e., not so many notes per chord), wider spacing of chords, not so much doubling of parts, go easy on the effect-boxes. Vibrato is all right if and only if it is narrower. No need to stop vibrato, just be more discreet with it. Pitch-bending and deviations from strict 24-tone-equal tuning are all right, so long as one is aware of the halving of the areas. Tighter toleratnces in the engineering sense--this does not mean no tolerances.
The quartertone is still not really a micro-tone--it is still a usable melodic interval, even if some people will class it with the really micro-intervals like 1/31 octave or 1/41 or 1/53. The clarity of a quartertone pattern does depend on the instrument used. Extremely loud levels like the rock groups or extremely soft playing will make it more difficult to hear the quartertones, of course. When we come to polyphony, more than one tone sounding at a time, the new quartertone intervals contrast with their neighbors. For many people quartertones are the boundary-line where there still is a difference in character of the adjacent intervals such as a normal Fifth and a semiaugmented Fifth.
Quartertone frets are still easy enough to negotiate by most guitarists, and so melodic these and harmonic considerations make the system a logical boundary between the wider intervals such as those of the 17, 18, 19, and 22 systems, and the really micro intervals of the 31, 34, 36 and just-intonation systems. Whether to call quartertones microtones is up to you--no use arguing the point. The various music dictionaries and reference works do not agree about it. This suggests devising some other terms to cover the "grey areas" and get around the disputed points. One such is xenharmonics, meaning that music which does not sound like the 12-tone equal temperament to the average musically-trained listener. A scale such as 5 or 8 or 13 notes per octave sounds quite strange, but is hardly microtonal. In this case quartertones, since they contain and use the 12 pitches of 12-tone equal temperament are themselves a borderline case, and the way in which they are used will determine whether a given quartertone performance is more or less xenharmonic. Another proposed term is novasonic, involving new sounds. Most quartertone performances would qualify for that!
Without intending to cast aspersions on the many quartertone notations that have been used to date, I shall give here a table of the quartertone intervals in staff-notation with the Couper quartertone signs, and below that the number of quartertones in the interval, counting inclusively from C, taken as zero, to the upper note. I omit the names of the intervals since my proposals have already been given and there is too much disagreement about that. With the rapid trend toward use of synthesizers and computers and the obvious opportunity for expanded use of atonality in the quartertone system, coupled with the new freedom to go from one tuning-system to another at the touch of a button or the typing-in of a program instruction, it should be obvious that many people who once try quartertones will also get interested in other tuning-systems such as the harmonic systems 19, 22, and 31, or the non-harmonic systems such as 13 or 23. There, they will have to use numbers since names are less useful in non-harmonic scales. So, going from quartertone staff--notation to the numbers is logical enough to be worth setting out here. (It is true that some sets of numbers for quartertones will begin at 1 instead of 0, such as numbering the notes inside an electronic organ or the keys on a quartertone keyboard. Also, because of the long-term use of 12-tone a certain number of quartertone keyboards etc. might carry designations like 3.5 or 7.5. [Note from Monzo: Darreg's notation example is not available.]
Since quartertones contain the ordinary 12 or semitone system, the new intervals all have synonyms, at least two possible ways of writing each, as a semisharp and sesquiflat, semisharp and semiflat in the case of the notes between E & F and B & C; as a sesquisharp and semiflat in other cases, as shown above. This ties in with the equivalence of such notes as G-sharp and A-flat in 12-tone equal temperament. This will also be the case in 36-tone or the sixth-tone system. Then remember that in the 17, 19, 22 and 31-tone systems, among others, such pairs of names or pairs of sharps and flats will be different pitches, not synonyms for the same pitch. We disapprove of using the 12-tone term "enharmonic" for synonyms of that kind, since we need 'enharmonic' in the Ancient Greek sense of a system of tones involving small intervals on the order of a quartertone. It's all right to use "enharmonic" with the meaning of equivalent pitch but different name, when one is confining oneself to the ordinary semitone or 12-tone equal temperament, but when the quartertone system offers such excellent imitations of the ancient Greek enharmonic genus, it would be too confusing to retain the 12-tone-equal-temperament sense of the term. The reason why quartertones, but not 17, 19, 22, or 31 have these synonyms, is that there is no 24-tone circle of fifths, but two circles of fifths a quartertone apart which never intersect. In the other systems mentioned, there is only one circle of fifths using up all its tones.
Getting back to the key-signature system discussed earlier on page 6 & 7, when writing pieces of any complexity in quartertones, there are only 7 natural pitches, denoted without accidentals or the ordinary accidentals. Thus even if you invent a key signature for one of the new keys like E-semisharp-neutral, you wouldn't remain strictly within 7 pitches of that key as indicated by its signature, and all the 7 or 9 kinds of quartertone accidentals would be frequently cancelling this signature and making it very hard to remember. So the wisest course is to abolish all signatures and never have to remember them. It might even be wise to adopt the convention of some modern composers that every note must be read natural unless preceded by a semisharp, sharp, sesquisharp, semiflat, flat, or sesquiflat. That gets rid of nearly all naturals. You probably are too timid to emulate Carrillo and abolish the 5-line staff and note-heads and number all the tones? Dream about that anyway. All his numbers are in the table above.
If you can get to hear the quartertone intervals, you will hear that each has its personality or individuality and that it doesn't sound merely like a variety of its neighbors. Most of the new intervals are dissonant, but some of them such as the subminor third of 5 quartertones and the subminor seventh of 19 quartertones are fairly consonant. The semiaugmented fourth of 11 quartertones is another almost-consonance in the treble register, or when spaced out by adding one or more octaves. (That reinforces the the suggestion above about wider spacing between voices.)
In conclusion, we have published more quartertone information in Xenharmonic Bulletins 7 & 8, and elsewhere.
These fretting tables, frequency tables, and beat-tables continue to be available separately.
The time has now come to pose questions: what does one get in return for doubling the number of tones in the octave, as composed with the standard 12-tone equal temperament? How do quartertones compare with other possible equal divisions of the octave? Does the 24-tone temperament approximate any of the intervals in just intonation? How well, in comparison with other temperaments?
The gain in using quartertones is chiefly melodic. The powerful emotional effect of the Ancient Greek Enharmonic Tetrachord and its variations, of the Ancient Greek Enharmonic Genus and the obvious developments one can derive from it; something which we have been starved for during 2000 years or so. Some exotic cultures have continued to use intervals of the this sort--they are found in the Near East, for instance.
Your imagination is surely equal to the task of constructing a wealth of quartertone scales, and trying them on for size, and selecting a few for actual composition and performance and improvisations. Not much need here to spell it all out, since that risks cramping your style and implying that something not set forth in notation is a no-no. During the past two centuries, too many authors of books on music have assumed too much authority and laid down too many rules and constraints. This has almost stopped any progress in music for quite a while.
The tape recorder will be a valuable tool in such explorations, since it can capture a fleeting idea before it evaporates, and also it is difficult to start writing down one's ideas in a new scale before learning the added signs for a quartertone notation. Too many problems presented at once that way; better to deal first with listening to the new sounds and trying them out, before choosing one of the many systems for writing quartertones down. Many people never even get to performing or listening to new scales, simply because of this artificial notation hurdle; I speak from 50 years' experience and interviewing musicians.
Now that could explain the numerous trials of two pianos tuned a quartertone apart, since each of the two pianists does not have to deal with more than 12 pitch-classes so there is no need for writing new signs when it is understood at the beginning that one pair of staves is to be taken as .25 tone higher or lower in pitch than the other. (Caution: do not attempt ever to tune a piano a quartertone HIGH! Always the other piano in the quartertone duo should be tuned a quartertone LOW--at A = 440 Hz this will put the A-semiflat at 427.5 Hz.) Harpsichords with two keyboards also should have the quartertones lower than standard pitch. With electronic organs and various synthesizers and electronic polyphonic keyboards, of course there is no objection to the second keyboard being a quartertone higher.
The disadvantage of this ordinary-notation-for-two-keyboards scheme, as already suggested, is that it forces the composer to think in terms of a duel rather than integrated duo; to keep the two sets of twelve pitches each apart from each other and not pass freely from one to the other; this impoverishes the results. Also it stresses chords and harmony too much for a system which is primarily melodic and does not differ from 12 as much as truly non-12 systems such as 19, 22, 31 and some others do, since 24 = 2 x 12 it continually sounds out the familiar 12-tone-tempered harmonies. The two-piano scheme seems to tell you to Keep Both Players Busy Enough or Else! And discourage you from presenting the simply unadorned single melodic line which is one of the greatest values of quartertones.
Indeed, one might recommend composing some quartertone melodies accompanied by a drone: this will help both composer and listener to measure the new intervals from the drone as zero-point and get used to them. From this bare-bones beginning, one can then start moving the drone every so often, and then go on to the usual constantly moving the patterns and to fuller harmony. Counterpoint should not be neglected--two-part counterpoint at first.
Going from 12 to 24 tones/octave does not improve the harmoniousness of the thirds and sixths, either major or minor. Both the 12- and 24-tone versions of these intervals have the same errors, as compared with the untempered or just-intonation forms of these intervals with exact integer ratios of their vibrations. This must be understood at the outset; this is a non-gain from choosing quartertones. However, this does not undercut the melodic gains and values mentioned above. Since we not longer have to make financial and other sacrifices to go beyond 12-tone, by the same token if you use quartertones that does not keep you from using other non-12 systems at all. With modern electronic instruments and modern tuning-devices and the new ease of electronic instruments and modern tuning-devices and the new ease of refretting guitars and similar instruments, one can have several non-12 tunings equally available and go from one to the other at leisure.
This new-found ease simply was out of the question 50 years ago. That is why you are going to encounter all manner of negative statements in books and magazine articles, and why so many Human Roadblocks and Spoilsports are going to try to frighten you off. Don't heed them! You are now free.
Let's get down to figures. The major third in both 12 and 24 is about 14 cents too sharp. (Cents are an interval measurements invented by Alexander J. Ellis about a century ago. Hundredths of a conventional semitone, 1/1200 octave.) The 14 cents (more accurately 13.686 but you could never hear such tiny discrepancies) of sharpness as in relation to the just-intonation major third with exact 4:5 ratio of the frequencies of the vibrations of the two tones producing that interval, just 386 cents and 12 or 24 tones 400 cents. For melodic playing, many performers will use the Pythagorean major third, of 408 cents, especially in such situations a violinist playing an E going to F in the key of C major, or a B going to C, reckoning that B from G to make the above-mentioned Pythagorean major third of ratio 64:81.
That is to say, melody and harmony are continually at odds, and the standard 12-tone equal temperament is biased against harmony and toward melody, thus having a mood restlessness and brilliance. So long as you realize that 12 and 24 afford excellent imitations of the Pythagorean intervals and only poor imitations of just thirds and sixths, there is really no problem, since other tunings are now feasible when you want the calm, serene, harmonious major thirds of just or 31 or some other tunings. (Pythagorean intervals are those reckoned only by fifths and fourths, or ratios involving the number 3.)
The major third and its inversion the minor sixth have the error in 12 and 24 of 14 cents as mentioned; this compares with 0.7 cents in 31-tone and 7 cents in 19. This means a big change in mood when using 19 or 31, especially for harmony.
Quartertones have an advantage over semitones when it comes to the "harmonic" or "natural" or subminor seventh. This interval is represented in the quartertone system by 950 cents, or 19 quartertones, instead of its just intonation value of 969 cents for the ratio 4:7. This is then a quartertone-system error of 19 cents. That is better than what 19-tone provides, 21.5 cents, but not as good as 22 with 13 cents and 31 with an amazing closeness of 1 cent. But looking at the gross seventh-harmonic error in ordinary 12 of 31 cents, 19 cents for 24-tone is surely an improvement, especially when realizing that now there is a difference between the subminor seventh and the ordinary minor seventh, which difference can be exploited in actual musical performance. Furthermore, violinists, cellists, etc. can bend pitches toward just and improve all the errors of the quartertone system. This interval is either ignored or condemned by most orthodox harmony textbooks, but actual performance practice an unaccompanied chorus singing prove that the interval is acceptable and is used despite the various objections made to it.
The eleventh harmonic is a more remote member of the harmonic series, but quartertones (11 quartertones, by coincidence!) represent it with about a 1-cent error, so this is an advantage of the system. The thirteenth harmonic is represented by 17 quartertones with an error of some 9.5 cents too sharp. Considering the remoteness of the thirteenth harmonic this error is quite small and not worth objecting to.
How important in actual worthwhile music the eleventh and thirteenth harmonics and the intervals derived from them are going to be, depends on a number of special considerations ordinarily never considered in orthodox music. Primarily it depends on the tempo, having the music [slow] and sustained enough to appreciate those intervals; and on the timbres of the instruments used to perform it. The piano is much too dull in tone to provide any clarity or definition for such intervals. So more harmonically-rich timbres are called for, and it helps if these tones are sustained. This is such a personal and individual matter of taste that it wouldn't be proper to make blanket recommendations. As for the more important seventh harmonic and the intervals derived from it, there will be more uses for these, and more chances of having a suitable timbre to exploit this expansion of the tonal range.
Now we come to another difference between 24 and the really non-twelve group of just intonation, 19, 22, and 31 tones/octave. This concerns existing music, such as Bach, beethoven, and Brahms, Inc., or folksongs and jazz and blues, or Schumann, Schubert, Liszt, Tchaikovsky, Rimsky-Korsakoff, et al., or the avant-garde music of the 20th century. Everything written in the Classical and Romantic periods and after, that belongs to "tonality" or the key-system, has a 90% chance or better of being playable in the 19, 22 and 31-tone systems with comparatively slight alterations. And even more chance than that of being played in just intonation, perhaps with considerable improvements for some styles of existing music. Since even classical players "bend" pitches even when they call it something else out of snobbery, these non-twelve systems often systematize those bends and cause improvement and an increase in general order. It's only with a limited range of modern atonal and serial music that we get a conscious and strict and deliberate attempt to make the music "twelvular" or "dodecaphonic" or strictly non-key.
Quartertones are set apart from the other non-12 or xenharmonic systems by the above considerations. Only a small part of existing Romantic and Classical period music lends itself to the addition of quartertones or the adaptation or rearrangement and modification that would be involved in inserting any appreciable amount of quartertones. Perhaps there would be enough instances where the quartertone representation of the seventh harmonic and its derivative intervals such as the subminor seventh and subminor third could be used to alter chords of the dominant seventh and the like. This would require discretion and be awfully controversial because of all the people who would argue about it.
The higher harmonics such as the 11th and 13th and more are simply not implied in ordinary 19th-century music save in exceptional instances. Blues and some kinds of jazz involve tone-bending and so quartertones might be used there to imitate these relatively free, rule-less bends.
If your main interest happens to be making old music sound better, or change to new moods, then quartertones will not be your principal xenharmonic system. You may still use them, but your order of priorities will be different from other people's. Your personal priorities and system of relative system-evaluations will be a matter of taste, and there is no reason to submit to self-appointed Authorities. Your opinions are as good as theirs. The worst interference and constraint has come from non-composers and very often non-performers who have no imagination. So-called Rules of Music cannot have the precision and force of laws of physics or chemistry of mathematics. Customs change with time and there has been a profound change in lifestyles and attitudes in the last 30 years and amazing technological developments in new instruments, such that quartertones as well as the rest of xenharmonics are in a favorable position.
As already stated, there are far too many quartertone notation-signs to list in this kind of monograph, so no disapproval of unlisted systems is intended.