NOTATION FOR NON-TWELVE

by IVOR DARREG - PART I

"Have you written anything lately?" -- I am a composer, and that is the stock question. Since music is the art of sound-patterns in time, one would expect questions like "What does your latest piece sound like?" but they never get asked. Well, hardly ever.

In the 20th century musical notation has gained entirely too much power. Compositions and composers are evaluated mainly by eye. Inventors who should be giving us new instruments or new sounds by Whatever means, instead concentrate on devising new musical notations. Almost always, these new notations are piano and for the 12 sounds per octave of the organ, and in most cases they erect new obstacles for anyone who uses any other tuning system than 12-tone equal temperament. Frankly, they usually step backward.

One of the commonest such reforms is the keyboard staff. Alternate groups of two and three lines stand for the black keys of the organ/piano keyboard, in complete disregard of the fact that violins andcellos don't have keyboards and don't have frets either. Sometimes the lines are made vertical, so that the performer reads down instead of across -- this conforms to the perforations on a player-piano roll. Arnold Schoenberg once described a system which is based on the whole-tone scale (6 tones per octave), the other six pitches being symbolized by some kind of modification. Since then, a number of variations on this whole-tone-scale idea have appeared. Indeed, there are 6-6 keyboards that can be fitted to organs and pianos or attached on top of regular keyboards in some way. There is a group promoting this idea, which publishes a magazine with the Esperanto title Muziko Ses-Ses.

This sort of thing would have represented real progress and reform in Europe during the 1890's. Today, it would thwart and block the implementation of any really new music.

Another obvious variation of this idea is to use graph paper. Each horizontal line in the graph can represent one4welfrh octave. Such paper ruled in squares, twelve to the inch, is readily available. Joseph Schrninger tried to popularize this idea in the 1930's.

Now here is something much less objectionable. If we go to a temperament which is not twelve-tone, we can simply use some other convention -- let each horizontal line be 1/19 or 1/31 octave, or when you are working in a system like 34-tone or 28-tone which is not very harmonic, this method becomes a positive advantage, because it avoids false implications and suggestions of familiar intervals such as fifths which the system may not really possess. For unequal 19-tone and such variants, this method can still be useful. From graph paper to a perforated or specially-inked roll is not a big step.

Special staves with some lines thickened for certain intervals have been invented for the 31-tone system. That principle is adaptable to a number of other tunings as well.

It so happens that some of the new systems are suitable for performing a large portiom of existing music. This is especially true of 19- and 31-tone... There wrn be thousands of pieces where the matter of new notation can be completely evaded. If you have a guitar fretted to 19-tone, go right ahead and use your regular books of standard selections as is!

Sure, it will sound diffent, but very few pieces in the guitar repertoire would be impossible in 19. Concerts and recordings of 31-tone performances have already proved beyond the shadow of a doubt that a large portion of existing musical literature can be directly pedonned in that system.

The commonest suggestion for going beyond 12 tones has been merely to cut the semitones in half, getting the 24 or quartertone system. There are at present many rival systems of notation for quarter-tones. I recommend Mildred Couper's modification of the Vyshnegradski system, as follows:

George Secor and others have suggested using the above Couper system for some other temperaments, such as 48-tone.

Ervin Wilson has invented a long series of notations for various tunings, co-ordinated with his layouts for new keyboards and metallophones. These include 17, 22, 31, 41, and larger numbers of tones, as well as many just-intonation systems.

Perhaps this is where to discuss just notation for a moment: The conventional staff-notation was designed for a system such as Pythagorean (infinite chain of perfectly-tuned fifths) and/or meantone (infinite chain of specially-tempered fifths, such that four of them produce a perfectly-tuned major third). Not designed deliberately, but perhaps subconsciously. Conventional staff-notation evolved at first to meet the needs of singers, and this before the lush harmonics of the 19th century had been dreamt of. The conventional thing for me to do would be to bore you with endless speculations about what the musicians of the Middle Ages might have thought. I think all that should be in a separate article or monograph-it is important to an understanding of non-12 and the work of the Ancient Greeks, but would seriously distract us here.)

Standard staff-notation thus suits meantone or Pythagorean tuning far better than it does 1/4-tone equal temperament, since it distinguishes between such pitches as C-sharp and D-flat. It follows logically enough that it works fine for imitation Pythagorean such as the 31-tone equal temperament, or for the 29-tone, or indeed for 41- and 53-tone if these are treated as faithful imitations of Pythagorean intonation.

Equally logical, it follows that Imitation Meantone, such as the 19, 31, and 50 systems, or the Meantone Variants such as the 1/3 and 1/5-comma temperaments, will be well served by conventional staff notation.

But with just intonation we run into complications. There is an interval whose ratio is 81:80, known as the comma (sometimes called Ptolemy or Didymus to distinguish it from the Pythagorean comma of nearly the same size). Most temperaments have been devised to get rid of it, but the 41- and 53-tone temperaments flaunt it. If you want a multiple of 12, try the 72- and 84-tone systems.

The comma we are discussing here is the difference between a pure major third of ratio 4:5 and a melodic or Pythagorean major third obtained by tuning fifths upward, such as the C G D A E heard when violins, violas, and cellos are tuning up. The 12-tone major third is a fair imitation of the Pythagorean, not the just. The Pythagorean comma is the difference between twelve fifths and seven octaves, or between C and the B-sharp obtained by tuning twelve perfect fifths in succession, and the conventional notation does provide for this comma, but the relation 531441:524288 is so very remote that it has no musical meaning and so let's give it back to the organ and piano tuners to play with.

I wish I didn't have to bring it up at all, but if I don't some Human Roadblock will try to demolish this article. Some people refuse to admit the existence of the comma 81:80. Yet the writing of just compositions in ordinary staff-notation demands that it be indicated. Moritz Hauptmann, early in the 19th century, devised a notation for commas by capital and small letters and lines above and below the letters for further shifts of a comma. When translating Hermarm Helmholtz's Sensations of Tone, Alexander J. Ellis, about a century ago, devised a convenient modification of this, with super- and subscript numerals for notes a comma above and below the Infinite Chain of Pythagorean Fifths.

The just scale of C major would then be: C D E1 F G A1 B1 C. Frequently an extra note, D1, would be needed, and then when going to G major, A1 would be replaced by A a comma higher. The unmarked Bb for use in F major would not be the same pitch as the B1b in the descending scale of C minor. Neither of these would do for the septimal interval introduced by the harmonic form of the seventh chord of ratio 4:5:6:7. So some other sign for the 7-based intervals has to be invented. (Practically, we may use any one of the signs used for 31-tone temperament, for this septimal alteration.) Going from C major to A1 minor, we meet such new pitches as F2# and G2# which involve two major-third removes from C.

Now this seems terribly complicated. No wonder that many writers have either thrown just intonation out the window, condemned without a hearing, literally! For they have scrapped conventional notation entirely, using some private cryptic scheme, or like Partch have used some conventional notation without any comma-marhings, supplementing it with enough ratio-fractions to look like a high-school algebra problem. This is practical enough for instructions on how to tune the instruments, but very cumbersome for use throughout a musical composition. Worse yet, it is repelling and discouraging to most newcomers to non-twelve, who think despairingly "I could never learn all that stuff!"

For much just music, Ellis had a simple answer: the Duodene. This is a group of 12 pitchclasses selected from the Web of Fifths and Thirds, being the notes closely related to the tonic of the moment, and the tonic of the Duodene is placed in brackets above the staff, remaining in effect till another Duodene is needed. This gets rid of all comma-markings on the staff save those which are not members of the Duodene in force. Septimal intervals or eleven-based intervals might be indicated by special accidentals-why not use 31-tone accidentals for septimal alterations and quartertone accidentals for eleventh-harmonic-related alterations? This makes so much sense that I do not know of a single composer, theorist, or author who uses it:

Of course the above will not take care of all uses for just intonation, but it is an excellent way to start, and avoid8 marking up existing music too heavily. Actually, if you want real precision tuning and complicated music performed in such accurate tuning, it is best to have it played by a computer, and therefore to adapt some one of the many coding systems for controlling computers, to this new task. I for one am not going to torure any musicians by asking them to study higher mathematics and go against all their lifetime of piano programming in order to perform one of my new pieces. If what I must express myself with is too difficult to notate, then I should turn it over to the computer as a matter of common human decency!

Partch and his emulators use ratio-fractions for some scores, but for other scores and parts they use TABLATURE. So let us discuss that here. Conventional staff-notation and its modifications are for the listener as much as the performer -- one can learn to read it silently much as this article is being read silently although the letters of the English alphabet stand for speech-sounds in a very rough sort of way.

Even those who do not play instruments can use a score to follow along a live performance or a record disk or tape. Generally, this is next to impossible with tablature. This subject is made much more difficult and ambiguous because intermediate kinds of notation between standard notation and tablatures exist. So we must take those intermediary forms into account.

Today the commonest tablature is the chord diagram for guitar. This does not symbolize the pitches that will be heard when the music is played, but what the guitarist must do to get the desired pitches: put the fingers on such-and-such frets on such-and-such strings, or leave the strings "open" (not stopped at a fret). The usual diagram looks like a small piece cut out from graph paper, thus:

This is a legitimate descendent of the lute tablature which may be found in history books. Unless the tuning of the fretted instrument is specified in advance (a standard tuning is of course understood if not contradicted) this tablature means nothing. Now, if this is used for a non-12 guitar, banjo, or whatever, the frets and therefore the horizontal lines on the diagrams will denote a different fraction of an octave above the open string than 1/12 octave, and the non-12 tablature will look like 12-tone, so that would have to be specified clearly in advance.

The non-guitarist cannot imagine the sounds from such a diagram, but might be able to form some idea of the sounds from the equivalent staff-notation of the same chord. Tablature, then, is for performer's only. And for one kind of instrument only. Each instrument and modification or variant of said instrument must have a separate tablature. And in this now-very-important case of the guitar, there are several tablatures for the same instruments. Instead of using a graph of little squares as a map of the six strings and several frets, one may use a six-line staff for the six strings and number the frets (the open string being numbered zero):

It can readily be seen that going to non-12 is extremely simple -- just use more numbers for the additional frets. And a guitar with a different tuning merely has to have the tuning specified in advance. The Hawaiian or Steel guitar, while it has no frets, has fret-lines and these can be numbered in the same way as real frets. So the tablature does not look any different even though it may produce entirely different sounds in performance. The pedal steel has more strings, so the staff may have ten or more lines.

Transposing the music to another key is also extremely simple -- retune the guitar or put a capo over such-and-such a fret. In the case of a steel guitar a moveable bridge could be made and installed at the desired place. For wind instruments there are systems of circles and dots denoting keys and finger holes. For slide trombone the 7 semitone positions can be numbered, even though there are no physical guide-lines on the trombone to show where those positions are. These tablatures are not used primarily for performance, but are used in instruction books.

So many organ and piano tablatures have been invented over the centuries that there simply is no space here to describe them. Enough music history books and dictionaries exist that you may consult at leisure.

NOTATION FOR NON-TWELVE

by IVOR DARREG - PART II

Far from being obsolescent, keyboard are alive and well in many self-study instruction books, and now in the sophisticated form of Synthesizer scores with timbres and "patches" (connections among the electronic apparatus) specified in the score. We might mention here, that in order to put certain synthesizers into non-12 tunings, it is necessary only to turn a dial, and then use the conventional keyboard of the instrument for other pitches than the conventional pattern of white and black keys would seem to represent. No special new keyboards are required.

HOWEVER: Please do not write the usual pitches of the 12-tone keyboard when the pitches of the 19- or 224one system will be sounded. Write the notation for the 19, 22, or other non-12 scale for the sounds. If you must have tablature, number the keys on the keyboard, or some such scheme.

Partch invented a separate and different tablature for each kind of instrument in his ensemble, which means that the listener would not derive much if any information from seeing the score, and the player of one instrument would not have much idea of what sounds come out of another instrument with a different tablature until or unless he heard those sounds. The details of this cryptographic scheme may be found in his book, Genesis of a Music. For the Chromelodeon which had a conventional-looking 12-tone keyboard, but produced the just scale of his system, he wrote the 12-tone notation for the keys to be pressed, with the result that all uninitiated readers of such a tablature will imagine the 12-tone system sounds and be seriously misled. I am quite sure the 12-tone composition resulting from playing that notation on a piano has been heard a number of times and the just sounds intended by it have not been heard on many occasions where they should have been. The temptation is irresistible.

In many cases, both tablature and staff-notation are used together, often helping the beginner on some instrument to learn staff notation gradually and palnlessly.

We might just mention the Tonic Sol-Fa System, which was originally a 12-tone system but modified successfully for just intonation singing. In effect the one-letter abbreviations for the syllables, d r m f l t, were a kind of tablature, and there was a scheme for denoting rhythms, which many other tablatures ignore, or compel the use of some kind of external rhythm indications.

From the above described tablatures to typewritable code for input to computers is a very small step. Since new synthesizers and other electronic instruments will mostly be digital, and digital recording is coming into use now, the adoption of new coding systems to cause digital equipment to perform music will be of ever-increasing value. I can't very well describe something which is not yet fully worked out, but I can encourage you to heed this development and take active part in it if you are at all willing. We must not let the 12-tone people take over the entire computer and synthesizer music world and fence us out of it!

Now let's look at two excellent ideas from Mexico of the last 50 years or so. They are nowhere near as well-known as they deserve to be. Perhaps this is because they were both developed for new scale-systems, even though both are perfectly compatible with the ordinary 12-tone scale. Perhaps also, because the two ideas were developed independently and do not have much in common. So much the better, since they have different and equally valuable fields of application.

As the much-less-known of the two, I shall consider Augusto Novaro's ideas first. Over a period of years, Novaro experimented with new instruments and refretted instruments and wrote books, one of which set forth tables of calculations, and some proposed notations for various scales, with musical examples composed to put them through their paces.

One very important reform of Novaro's could be applied to every definite-pitched instrument: the clef-signs are abolished, and all ordinary five-line staves are considered to be governed by the G clef. A Roman numeral is placed where the clef would normally go, and this numeral denotes the octave range of the staff. By setting Roman numeral V as the normal pitch of the treble staff, Roman numerals from I to VIII take care of the most extreme pitches of organs or synthesizers.

The one-clef proposal is not new, but the need for it was not as great until now, and as long as we are introducing so many changes with new instruments and new tuning-systems, this is a most suitable time to get rid of clef nuisances, those dreadful 8va--------------- lines, and excessive ledger lines, so hard to write and read. The saxophones, the clarinet family, and some other band instruments have been on G-clef for a long time, and the tenor and alto voices have not been written in the C-clefs for ages, so there is some precedent for this move even among the most orthodox.

By tying this reform and some others in with the move to non-12, we can de-fuse the hecklers who feel threatened by the possible effect of this reform upon the staggering investment in existing printed sheet music. Nobody would ever have the time to rewrite older music anyhow. Conversely, the lovers of the past have no right to compel us to continue laboring under the burden of the excess baggage and old barnacles they have accumulated during their voyage through time.

Novaro tried out a number of scales, such as 14, 15, 19, 22, 81, and 53, and proposed notations for several of them. Since his 534one notation embodies interesting features, it will be shown here:

Actually, the above idea, of using numerals for Mercator degrees above and below the "natural" pitches of the notes, would work just as well in some other systems, such as 41-tone or 72-tone. Or as a comma-indicator in just intonation. The right-hand parenthesis for degrees downward is made necessary by the confusion that would have been caused by putting minus-signs on a staff with too many horizontal lines in it already. Novaro was clever to have thought of that! Note also that it unclutters the page. no sharps or flats or newly-invented accidentals such as are used for quarter-tones or 31-tone. You might even try it for 31.

Now for the proposals of Julian Carrillo, who was much better known and who also wrote books on his ideas and a considerable body of compositions beyond 12-tone. In his work Sistema General de Escritura Musical, he I)resents a formidable barrage of arguments for reform, and is much more radical about it than Novaro. Carrillo wanted nothing less than the abolition of the five-line staff and of noteheads. The flags and stems of notes are retained, but there are no open or blacked-in note-heads atop these stems. Instead, the "staff" becomes a single line, which could even be the lines on ordinary ruled notepaper.

One degree of this staff-the spaces and ledger-lines above or below-corresponds to a whole octave. Instead of noteheads, there are numbers. For ordinary 12-tone music these numbers run from 0 for C to 11 for B. For quartertones, they would run from 0 to 23. For eighth-tones, 0 to 47. A specially-bent stem has to be used for half-notes, now that there can be no distinction of open and blacked-in noteheads. Notes in the middle octave are on the one-line staff, so that line runs right through the middles of the numerals, like this:

which might cause problems in some methods of printing, and requires special skill in typewriting. Most of the conventional expression marks, braces, etc. remain unchanged. At the back of Carrillo's book are numerous examples of eyestraining complex compositions, and how his system would save space, make them much neater, and do away with clouds of key-signatures, clefs, and accidentals.

Carrillo's system represents a logical intermediate point between conventional notation and a completely typewritable code for input to computers or other digital music-performance devices for the composer. A slight extension of his system would make most if not all of it capable of being written on an ordinary typewriter keyboard.

Carrillo's numbering of all the pitches in an octave is theoretically suitable for any number of such pitches, but in practice he seems to have confined himself to those systems (12 18 24 30..96) which were multiples of 6 tones per octave and thus compatible with the 6-tone scale of Debussy) et al. That should not deter anybody from using his system for 17 or 31 or whatever else, since he did mention the possibility. For inharmonic scales especially (such as 13 or 23) that have no fourths or fifths and do not conform to the usual scheme of tonality, numbering all the tones as Carrillo does is the best possible evasion. For that matter, there are experimental scales that don't even have octaves! One might just go on numbering every tone to the top of the audible range.

Carrillo did not make any provisions for just intonation, so let me do it: instead of numbering the degrees of an equal temperament, let the note-head-number be the number of cents above the starting-tone, assumed to be C, in the octave in question. That will solve the problem for unequal temperaments as well as for just systems, including Partch's. Another wilder possibility is to have these note-head-numbers stand for the number of hertz (cycles per second) of the tone. Then you can even do away with the one-line staff, since that takes care of octaves. Why, you could even use number of mels (a psychologist's non-harmonic pitch scale used in some scientific work) for the note-head-numbers. A further possibility would be the use of letters or symbols such as asterisks instead of numbers to denote various kinds of noises.

In closing, there are some very important loose ends to take care of: musical notation is an abstraction -- much of what happens and is heard in actual performance has to be left out. Therefore ordinary sheet music is "unphonetically spelled" -- not as badly as English is spelled, but surely as badly as French.

There are very irritating details of musical orthography, such as putting a big base clef at the beginning of a cello piece and immediately canceling it with a teenyweeny treble clef you can hardly see, or using key-signatures with 5 or 6 sharps and then almost every note has a double-sharp or a natural in front of it; or the truly horrid practice of tying a note for 7 or 8 measures, and its prefixed accidental sharp or flat is given only at the first measure, and you are expected to remember that accidental for two or three lines!

Most important of all: I am fed up as a composer with the insane custom of "transposing instruments" which may have had some reason for existing in the time of Bach and Beethoven, but in this age of precision and logic and commonsense, and the almost universal abandonment of key-signatures and the key-system, has become an intolerable nuisance. Let the Beethoven Symphonies keep it, but for heaven's sake stop its use for non-twelve-tone new compositions. There will be enough mistakes with GUARANTEEING myriads of wrong notes. I grieved for days when I saw a recommendation in a book on the latest synthesizers to put them in B-flat when playing like a clarinet. Surely everybody knows better than to waste time writing orchestra scores in microtones for the Podunk Philharmonic! Or the Dizzyland Jazz Band either. Our kind of new music will demand new instruments and some of these already exist and here is a golden opportunity to get rid of such Organized Nonsense as writing C or F in order to hear somebody play B-flat. If you still insist on doing that silly sort of stuff, you are a crazy masochist and I hate you to pieces.