XENHARMONICS AND MUSICAL NOTATION
IVOR DARREG
(1979?)
There is a growing interest in scales and tunings outside the ordinary 12-tone equal temperament. For one things, accurate tuning devices are now available. For another, the ordinary 12-tone system has been mined to exhaustion by thousands of composers and using other non-12 scales is one way to escape the risk of merely repeating what one's predecessors have already composed.
Electronic musical instruments of various kinds can be retuned in some cases. They are not bound by the mechanical restrictions of the piano or pipe organ. New keyboard designs exist for systems with more than 12 tones per octave, but it is also quite practical to map a new scale onto a standard conventional keyboard.
Conventional musical notation is at least theoretically based upon the Pythagorean tuning, which consists of an infinite series of perfect fifths,the pitches of which never return to the starting point. This system works out in such a way that D-sharp is about one-eighth of a whole-step sharper than E-flat. If you compose in Pythagorean, there is really no notation problem. It fits hand-in-glove.
The very different-sounding meantone temperament also has an infinite series of fifths and therefore conventional notation and nomenclature is perfectly suited to it also. Contrary to the widely-published falsehood reprinted from book to book, meantone temperament has an endless supply of pitches, no just 12 tones per octave, and the so-called wolf is not necessary at all if your instrument can play more than 12 tones per octave. The wolf is the fierce dissonance occurring if G-sharp and E-flat are played together instead of either G-sharp and D-sharp or A-flat and E-flat. MORAL: Get an electronic whatever; don't tune meantone on a piano and then say it's defective. Ignore the misstatements in nineteenth-century books and those 20th century books and articles which merely parrot them. THe piano simply cannot have enough pitches to play meantone as it should be played because of its constructional constraints and the cold hard financial fact that providing more pitches would cost too much. With new electronic instruments this is no longer the case. The difference between such pairs of notes as G-sharp and A-flat is in the OPPPORITE DIRECTION from that in Pythagorean and is much greater. Instead of one-eighth of a whole-step it is one-fifth of a whole-step and is decidedly audible. G-sharp is FLATTER than A-flat by that amount. THis is for the so-called one-fourth-comma meantone, the standard variety, which has just major thirds. (There are variations of meantone such as one-fifth-comma with slightly different intervals, but conventional notation and nomenclature still fits them perfectly.)
It so happens that here is no audible difference between the standard one-fourth-comma meantone and the 31-tone equal temperament except under septic laboratory conditions, since the failure of meantone to close a circle at the thirty-first fifth up or down is only 6 cents or 1/200 octave, which discrepancy could not be heard in any conceivable musical performance or composition--nobody is going to sustain the tones long enough to hear the difference in real pieces.
There has been too much theory and not enough practice in this field under discussion, so we must emphasize that the theorist or acoustician in a laboratory surrounded by thousands of dollars' worth of test equipment and measuring devices is not in the same environment as the musicians and audience at a concert. In particular non-composers ought not to keep telling composers what not to do or set rules for them. New scales and tunings should not be rejected without a HEARING!
What was out of the question in 1900 or 1920 or 1940 is now affordable and feasible and accessible. It's that simple. Music has the same right to progress as the other arts and sciences.
When we say that 31-tone equal temperament is not audibly different from meantone, we mean in the Real World Out there, such as a music studio or a concert hall or a guitar fretted to 31 tones per octave or a synthesizer so tuned or some kind of electronic organ or a performance by a computer in 31-tone. We are assuming either existing music played at normal tempo or new compositions played at normal tempo, Adagio through Presto. The tiny discrepancies can only be heard by playing too slowly for listeners to tolerate, and sustaining them in isolation. Are you really that intent on showing everybody the difference between A-quadruple-sharp and C?
So for practical purposes, conventional notation will do for 31-tone temperament. However, Adriaan Fokker in the Netherlands revived some special accidentals used by Tartini for semi- and sesqui-sharps and flats and set up a system of equivalents to make 31-tone notation easier to read. These have been used in the printing of new compositions by a number of composers who write int he 31-tone system. THe fact still remains that an enormous amount of existing sheet musician can be read as is and played int he 31-tone equal temperament with a meantone effect. A number of keyboards for 31-tone already exist. It is no longer necessary to tune 31-tone by ear, since the tuning can be built into an instrument, or a guitar can be refretted. The crippling of meantone by using only 12 of its pitch-classes is no longer necessary since nobody has to tune even the twelve pitch-classes when they can be built in or done with an electronic tuning-device or programmed into a computer.
It is understandable that when a piano-tuner must practice for six months in order to be able to do the 12-tone equal temperament properly, no tuner would be willing nor have the patience to learn a tuning-by-ear routine for some other scale. Especially if it were a question of learning all the following scales which are available on actual instruments in this studio right now: 5 7 10 12 13 14 15 16 17 18 19 22 24 31 and 34. And Just Intonation. Thus a tremendous obstacle has been removed only lately. We no longer have to relegate new tunings to mere silent theory-on-paper.
Obviously the 5- and 7-tone scales can use ordinary notation without any accidentals whatever--no sharps or flats needed.
The ten-tone scale would best be dealt with by numbering the tones. (Julian Carrillo's notation will be discussed in a moment.) Similarly for the rarely-used 11-tone scale. The twelve-tone scale uses the standard notation but identifies such pairs of pitches as C-sharp and D-flat, D-sharp and E-flat, D-doubleflat B-sharp and C. THis has led to the invention of innumerable systems of notation and nomenclature which also ignore the difference between such pairs of pitches in non-twelve scales; and that of course is outside the scope of the present article. interested particles who want new ways of writing 12-tone equal temperament are referred to the Music Notation Modernization Association, P.O. Box 241, Kirksville MO 63501.
Near the beginning fo the 20th century, Julian Carrillo of Mexico wanted to notate both the ordinary 12-tone temperament and all the multiples of 6 tones per octave (divisions of the whole-step) up as far as sixteenth-tones, 96 tones per octave. To avoid inventing new accidentals and to avoid the existing clutter of sharps, flats, and naturals on the ordinary music-page, he abolished the 5-line staff and replaced it with a single line,s o that ordinary ruled writing-paper could be used. Note-heads were also eliminated. In place of note-heads,the tones of the scale being used are indicated by numbers, from zero through n-1. Thus the 12-tone system uses numbers 0 through 11; the quartertone system 0 through 23; the 72-tone scale 0 through 71; &c. Half-notes are shown by bending the stem, since there no longer are open vs. closed note heads. The degrees of the one-line staff plus ledger-lines indicate entire octaves instead of successive diatonic degrees. No clefs are needed at all, but most other standard music-signs are retained. This system is quite suitable for such scales as 13 16 18 20 21 22 23 etc., which are very difficult to notate or name with standard staff notation and nomenclature. A name like B or F-sharp is ridiculous int he 13-tone equal temperament. But numerals 0 through 12 are very convenient there. Carrillo never used 13 or 17 or 19, but why can't we use his system for these and for 22 or other things, since it works? He didn't say we shouldn't.
The 14-tone scale presents a new situation: while one might use ordinary sharps and flats in addition to conventional naturals for the notes of the 7-tone-equal temperament, it would be misleading and confusing to do so, because there is a 7-tone circle of fifths (admittedly quite distorted) already notatable and nameable as F C G D A E B in the usual manner. But there is no 14-tone circle of fifths. There is simply a second set of 7 fifths in a circle which does not intersect the with the first set. Thus is we think of B-flat and B, or B-natural and F-sharp, the 14-tone-system interval would NOT be a fifth of that system and would not sound like one, since B F would be the very same kind of distorted fifth that C G or A E happens to be in 7 or 14. Our suggestion is to call the new notes of 14, the second set of 7, F* C* G* D* A* E* B*, and use asterisks or arrows or whatever you please on the staff. Or just number the tones as for 13.
With Carrillo's numbers, we need not tarry on the 15- or 16-tone notations. Why spend time inventing new signs and names when we can number everything? This is the Computer Age.
17-tone is an exaggeration of Pythagorean tuning and so fits present notation perfectly without an additional signs at all. C-sharp is much sharper than D-flat. A-sharp is the same pitch as C-flat. So we don't have to invent new names to confuse people.
For 18-tone as well as 15 and 16, there is no need to invent new names and signs. However, besides a considerable number of 18-tone compositions by Carrillo, there are compositions by Alois Haba, who did devise a notation for it. Addition, Busoni recommended 18-tone but did not live long enough to exploit it. Busoni made proposals for a notation.
The 19-tone system is very important and often will be the first one for composers to try outside of 12. FOrtunately it needs to new accidentals or special staves or signs since the present notation is ideally suited to it. All that changes is the meaning of such designations as C-sharp and D-flat, and they are now different pitches, and the circle of 19 fifths closes at such a point that E-sharp and Fl-flat are the same pitch under two names, while B-sharp and C-flat are also the same pitch under two names. To get back to C in the sharpward direction we have to take fifths up to B-double-sharp, not B-single-sharp, to be identical with C. In the flatward direction it would be D-triple-flat rather than D-double-flat. If you really have to know! Joel Mandelbaum does not use equivalents in his compositions in 19, simply F-flat through A-sharp.
The only problem with 19-tone is convincing people that there is NO problem. 90% of printed music in existence is playable in 19-tone without marking it up in any way. You may or may not like the different sound, but there is no shortage of things to perform in the system even if you are not a composer. Why wait?
The 20-tone scale might as well be done with numbers.
The 21-tone scale could be done with two kinds of accidentals on a regular staff, taking the first of the three series of 7 fifths as normal and therefore the "naturals." Otherwise number the tones.
The 22-tone scale is important and useful, but has been sadly neglected simply because the present note-names break down and the staff-notation does not fit it. Numbering becomes almost a necessity. Note, please, that the system of 22 srutis used in India is NOT equal like 22-tone equal temperament, and has its own notation in India and does not sound very much like 22-tone equal temperament.
The 23-tone scale can be numbered.
The 24-tone or Quartertone scale already has a considerable body of compositions, and these have been written in almost as many different notations as there are composers! Too much has already been written for almost a century in quartertones, for anyone to undertake the impossible task of transcribing everything into one "standard" system. For new composers we suggest using the Couper modification of Ivan Vyshnegradsky's system of accidentals, in addition to the conventional staff-notation for 12-tone equal. Or follow Carrillo.
Beyond 24 tones per octave we already have taken care of 31. Beyond 31 it is unlikely anyone will notate sheet music for others to perform from so why not leave it open?
While such systems as 34 36 41 43 46 50 53 65 72 and more are worth composing in, no performer is going to learn to read all kinds of special notations for them, so let's do them on computers!
In order to have a computer perform music or control a synthesizer or other means to perform the music, one has to use a typewriter-keyboard-compatible code of some kind. That is the notation. So beyond 31, it is unrealistic and impractical to expect performers to learn all the new notations that would be required and practice and rehearse. Who has the time, and what composers have the money?
Now we come to Just Intonation. Theroretically, Just Intonation or an untempered tuning, requires an INFINITE number of pitches and would take all eternity to tune. Actually, some 20 to 30 pitches would do for most compositions, either existing or newly-created. Harry Partch set a nominal number of 43 pitches for his very special needs which are quite unlike those of other just-intonation composers. ACtually he often used less and once in a while needed more.
Alexander John Ellis, in his Appendix to Helmholtz' Sensations of Tone, and Ellis' Appendix in a book in its own right, solved the problem of notating just intonation for a majority of cases. No-one seems to want to acknowledge this fact nor give him any respect. THe scheme is as follow:: construct an array of tones related only by the prime factors 3 and 5 and combinatiosn thereof. Ignore 2 since octave duplications are obvious. I.e., consider first those pitches which like within one octave. Take the names of the tones in the Pythagorean system for granted. Place the series of fifths up and down from C in the center column with those names and do not add marks to them. First column to the right is a major third up, so to all those names of the major thirds or 5-related tones add a subscript 1 for one comma down from the Pythagorean tones of the same name. Construct a second column on the right with subscript 2 for 2 commas down, and so on. To the left of the center use a super-script 1 in the exponent position for one comma up which is produced by taking major thirds down, and so on. In the book, Ellis carries it out to 117 pitches but this is merely show that tiny intervals appear which would be too small to tune on instruments, and therefore in actual performance one does not need such niceties.
His legion of enemies, however, will not read his disclaimers, and accuse him of everything they can possibly think of. So his notation-help has been spurned for a century. To go on: Given an array of at least 30 of the notes in the array constructed as described, take 12 of them centered on the tonic of the place being notated at the time--say C. Call this array, which comprises 4 rows and 3 columns, a Duodene. Above the staff place [c] i.e., the tonic of the Duodene in square brackets. Then all ordinary notes are to be taken as the 12 members of the Duodene in power at the time, so it will not be necessary to clutter the staff with comma-markings for those notes which are one or more commas above or below those of the same name int he center column of the array. If you need notes which are outside, not members of the Duodene, you must work those and those only. If you modulate to another tonic, then put that tonic up in brackets and use its Duodene unmarked but mark only those notes which are not members of it. If you need 7- or 11-related notes outside what Partch called the 5-limit, they need special marks too. Even so, the number of special markings is cut down to a sensible minimum, and this system is so convenient that nobody will use it. Again, to get enough pitches for adequate just intonation, especially if you go to a 7 or 11 limit, and to keep those notes in tune, requires special instruments and in most cases the answer is going to be a computer. Thus notation is not too important really, since the typewritten code for instructing the computer what to play and how to play it is all the notation you need, and there are no performers involved, so why bother with clefs and staves and notes?
There are many rival systems of just-intonation notation out there but nobody has either the time or the patience to learn them; they are too complicated. Partch's notations, for example, were really tablatures differing for each of his instruments! Some of his instruments had to be retuned according to the composition to be performed on them. That was one good reason for tablature.
Happily, there is viable alternative: improvisation! One person can compose by overdubbing and re-recording layer upon layer. The instruments can be tuned just and no need to notate this since recordings can be copyrighted. Improvisation in just intonation is still practicable for two or three persons in a live performance. This reduces the contemporary need for special just notations.
There are other notations worthy of mention before we close. One is the chord symbols used in popular music--accompaniments and harmonic structures are denoted by letters and figures, such as C7 Bbm F+ etc. It would be very easy to extend this notation to many of the new scales or just intonation, in some cases without change.
There are Chord Diagrams showing where the fingers go on a fretted instrument. THis also is convertible to unusual frettings.
For fretless instruments such as steel guitars and pedal steels there are tablatures which have varying numbers of lines in the staff, according to how many strings there are; then the painted fret-line to place the steel on is written over the line denoting the string to be sounded. That is, the line runs through the numeral. It would be obviously easy to use this system for any system of fret-lines whatever.
For just intonation there is a system of ratio/fractions which has been used by some other composers since Partch wrote his book. While it is helpful for tuning purposes and learning how to construct just chords, it becomes very unwieldy when modulations are attempted. That is, it is valuable for studying harmony and not much use for melody.
Pitch-bending, or deviations from whatever the scale in use is assumed to be at the moment, is probably as old as scales themselves. However, at least in classical music, it has hardly ever been notated; it has been treated as "underground" or strictly a matter between teacher and pupil and the composer had no way of calling for it an asking that it not be done. Until recently: now that many synthesizers have pitch-bend levers or wheels, this is beginning to be notated.