[Top: John S. Allen's Home Page]
[Up: John Allen's Music Site]

[contact John S. Allen by e-mail]

Site Logo (Track bike) (1.3 KB GIF)Bikexprt.com Web Site

Notation for microtonal scales, part 1

1997, John S. Allen

One of the great virtues of traditional musical notation is its consistent, logical and easily readable representation of pitch. Before discussing microtonal notation, it is useful for us to examine some of the features of conventional notation. This article briefly describes the features of standard notation and begins an exploration of how it may be expanded to include microtones.

Spatial and symbolic representation of musical pitch

A system of notation may represent musical pitch spatially, by the location of symbols on a surface -- or symbolically, by the appearance of the symbols.

An entirely spatial representation of pitch, as in a piano roll, assigns a different spatial location to every defined pitch. Some experimental systems of notation have represented pitch in this way, but they have not been widely accepted. They are not as compact as conventional notation, and the larger staff they require makes reading difficult. The only place where the piano roll representation is commonly used today is in music software for computers, where it is necessary to provide a mathematically accurate representation of pitch and timing on a computer screen.

Pitch may be represented non-spatially, purely symbolically, with letter symbols such as A3 or c#'''. Such symbols are commonly used in printed text such as this article. While symbolic notation is satisfactory to identify single musical pitches, it becomes difficult to read when applied to entire compositions.

Conventional music notation

Conventional musical notation uses both spatial and symbolic representation of pitch, -- and the result is greater than the sum of the parts. The symbolic representation came first, in the Middle Ages. With the addition of a single staff line, and then additional lines, the spatial reference was added to represent increasingly complex musical structures while keeping the notation readable. The great staff, with 11 lines, represented the high point of this evolution -- but the great staff was difficult to read. The system of treble and bass clef used today for keyboard music is, however, simply a great staff split apart in the middle, and covers exactly the same range. Besides its easier readability, the system with two five-line staffs allows for clearer notation of overlapping figures in the two hands.

The several different symbols -- naturals, sharps and flats -- at each degree of the staff make possible a compact and easily readable presentation. The compactness of the presentation is assisted further by placing notes in the spaces as well as on the lines of the staff. Conventional notation uses a similarly two-pronged approach in representing timing. The start of a note is represented spatially, but its duration is represented symbolically.

Notational Spelling

The sharp and flat signs let conventional western notation distinguish 35 pitches per octave, if we allow alterations to sharp, flat, double sharp and double flat. Such distinctions have existed in fact as well as in theory in western music. When mean-tone tuning was standard for keyboard instruments, the distinction between pitches such as G# and Ab was not just one of correct musical "spelling." It reflected an actual -- and not subtle -- difference in musical pitch.

The rules of musical spelling continue to be observed today, even in music written for today's equal-tempered keyboard instruments, on which different spellings no longer represent different pitches. Spelling continues to be important because it clarifies the role of pitches in tonal structures and makes the notation easier to read.

For example, we spell an E major triad E G# B, not E Ab B. The G# is a musical third above the E, as required in a triad. The Ab appears in notation as a flatted fourth; not only would this spelling lead to confusion as to whether the structure represented is a major triad, but more importantly, the Ab would occupy the position on the musical staff already occupied by the A natural of the E major scale. Could we solve this problem by calling the A a Bb? No, because our scale already includes a B. If we spell incorrectly, we are playing a game of musical chairs with musical notes. Two degrees of the scale are going to end up sitting in one line or space of the musical staff. This happens much less often, at least in tonal music, when spelling is correct.

Expanding the notational system to include microtones

The microtonal system which I favor is based on sequences of musical fifths. Such a system may use nearly conventional system of notation. In fact, we may conveniently use conventional notation unchanged to represent the pitches of a series of 17 fifths from Gb through A# -- the seven naturals, plus the five sharps and five flats.

      -5         0         5      
Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A#
Table 1: Representation of 17 distinct pitches using standard notation


However, if we carry this system onward into the double sharps and double flats, it becomes confusing. For example, a Cb is the same or nearly the same as a B, not a C; an Ebb is the same or nearly the same as a F, not an E. We avoid such difficulties as much as possible by using enharmonic transpositions. In twelve-tone equal temperament, the five sharps and five flats duplicate each other. There is enough overlap that double sharps and double flats are rarely needed.

Since this duplication does not occur in a microtonal system, we can not use enharmonic transpositions in the same way. Furthermore, even though 35 distinct pitch classes are possible using double sharps and double flats, more than 35 may be needed: some interesting microtonal systems are based on cycles of 41, 43, 50 and 53 fifths. And, as already stated, excessive use of double sharps and double flats makes notation clumsy and confusing. For all of these reasons, we do well to look for another approach.

Dr. Adriaan Fokker's microtonal notation

Dr. Adriaan Fokker, of Leyden, Holland, took up the study of music theory when prevented from working as a physicist during the Nazi occupation. He developed a theory of 31-tone music and built instruments to play in 31-tone equal-temperament. Dr. Fokker's work is a good point of departure in exploring our options for microtonal notation.

Other than that it continues on past 12 pitch classes, the 31-tone scale which Dr. Fokker used is almost identical to traditional mean-tone tuning. Its major thirds are nearly just and its fifths are slightly flatter than those of 12-tone equal temperament. As a result, a Db, for example, is somewhat sharper than a C#. If we continue into the double sharps and double flats, a Dbb is, in the same way, flatter than a Db and a Cx is sharper than a C#, and we get the sequence shown here:

C Dbb C# Db Cx D

Since the 31-tone temperament is an equal temperament, all of the steps here are equal.

Fokker eliminated the double sharp and double flat notation, using instead the notation:  

C C# C# Db Db D

or, with different symbols for the double sharps and double flats,

C C C# Db D| D

(The 1/2 sharp symbol used here is, fortuitously, just like Fokker's -- half of a sharp symbol. Fokker's half-flat symbol, along the same lines, replaces the closed loop of the conventional flat symbol with a small hook, so the symbol looks like a backwards J. The 1/2 flat symbol here, a vertical line, is the closest thing available for use in an HTML table. Think of it as a flat symbol without the loop. The keyboard diagram later in this article shows the actual appearance of Fokker's half-flat symbol.)

The complete scale of 31 tones is as shown below. Conventional notation is shown in the middle row of the table, with the naturals in white, the sharps and flats in black, and the double sharps and flats in gray. Fokker's notation is indicated in the lower row, with the half-sharps and half-flats in blue. Notice that the naturals are spaced essentially as on a standard keyboard; they provide useful points of reference when studying the table,

1         6         11         16         21         26         31
C Dbb C# Db Cx D Ebb D# Eb Dx E Fb E# F Gbb F# Gb Fx G Abb G# Ab Gx A Bbb A# Bb Ax B Cb B#
C C C# Db D| D D D# Eb E| E E F| F F F# Gb G| G G G# Ab A| A A A# Bb B| B B C|
Table 2: 31-tone scale in conventional notation (top row) and Fokker's notation (bottom row)

All of the naturals in this 31-tone scale are approximately the same as in the 12-tone system (and almost exactly as in mean-tone tuning). Fokker's "half-sharps" and "half-flats," replacing double sharps and double flats, make it possible for a musician to read Fokker's notation with almost no training. About all the musician needs to know is: "flats are sharper than the enharmonically equivalent sharps; half-sharps are a shade sharper than the corresponding naturals, and half-flats, a shade flatter." The purer harmonies of the 31-tone system provide a further guide to correct intonation in this system

Through the sequence of 17 musical fourths and fifths from Gb through A#, Fokker's notation is conventional. Following the A#, an additional series of fifths through seven "half-flats" from F1/2b to B1/2b is followed by seven "half-sharps" from F1/2# to B1/2# and the return to the conventional sequence at Gb. The 17 conventionally notated pitches, the seven "half-flats" and seven "half-sharps" conveniently add to a total of 31. As already described in the article on the Fibonacci series, this is not a mere mathematical accident: it results from the requirement that a whole number of octaves and a whole number of fifths nearly coincide.

Fokker designed and built 31-tone keyboards. In the terminology I use in this series of articles, they were horizontal row, sharps-forward general keyboards with alternating, interlaced columns of 6 and 5 keys and 65 keys per octave. Fokker's keyboard layout is shown in the two-octave section here:

Fig. 1. Fokker's 31-tone keyboard

Fokker's 31-tone keyboard layout (17K GIF)

John S. Allen, 1997

In any musical progression which goes into more and more sharps, an enharmonic transposition into flats is needed sooner or later -- and vice versa. In the 12-tone system, the transposition might, for example, be from F# major (6 sharps) to Gb major (6 flats). In the 31-tone system using conventional notation, the transposition might be from Dx major (5 double sharps and 2 triple sharps) to Fbb major (6 double flats and 1 triple flat). Fokker's notation, however, uses three enharmonic transpositions. Fokker's notation is easier to read than conventional notation, but is inconsistent in keys which use his half-sharps and half-flats. The chart below shows the sequence of fifths in conventional notation above that for Fokker's notation.. The numbers in the top row indicate the number of musical fifths away from the central D. Inconsistently notated fifths are indicated by red arrows.

-15         -10         -5         0         5         10         15
Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax
F C G D A E B Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# F| C| G| D| A| E| B|
-^           ^--- -^                               ^--- -^           ^---
Table 3: Inconsistencies of Fokker's notation

Another weakness of Fokker's notation, for all of its cleverness, is that it does not apply to scale systems which use more than 31 fifths.

A forthcoming article will explore solutions to these problems.  

Top: John S. Allen's Home Page]
[Up: John Allen's Music Site]

[contact John S. Allen by e-mail]

Contents 1997 John S. Allen

Last revised 8 April 1997