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On Rabbits, Mathematics and Musical Scales

John S. Allen

"...vast numbers of tribes and cultures...developed music independently, and in the most varied surroundings...They exhibit enormous differences in their language, customs, clothes, modes of life and so forth, but all who have advanced beyond homophonic music have, if not precisely the same musical scale, at least scales which are built on the same principle.

"The main differences are found in the numbers of notes which form the scale. By stopping at differing places in the sequence. F-C-G-D-A-...,* we obtain the various scales which have figured in the musics of practically all those races which have advanced beyond the one-part music of primitive man."

* There is no theoretical reason for starting with F rather than with any other note of the scale. F has been selected merely in order to keep off the black notes of the piano for as long as possible.

Sir James Jeans, Science and Music, pp. 163-164

Let's discuss music as it relates to rabbits.

No, not the Bunny Hop, or Here Comes Peter Cottontail -- or Jefferson Airplane's White Rabbit, or I'm Late, I'm Late (for a very important date) from the Disney movie, or del Tredeci's Final Alice (hmmm, that rabbit gets around...).

We're going to discuss rabbits as they relate to musical scales.

in his Liber Abaci, the 13th Century Italian mathematician Fibonacci set out to determine how fast rabbits would multiply, given certain mathematical assumptions: a rabbit couple could reproduce once a month beginning at the age of two months, bearing a litter of one male and one female, and no rabbits died. Let's say that you start with one pair of baby rabbits in January. Then, from month to month, the number of rabbit couples will be:

month month no. calculation rabbit couples
January 1 1 + 0 = 1
February 2 1 + 0 = 1
March 3 1 + 1 = 2
April 4 2 + 1 = 3
May 5 3 + 2 = 5
June 6 5 + 3 = 8
July 7 8 + 5 = 13
August 8 13 + 8 = 21

and so forth...

For any month's total number of rabbits, you add the number of rabbits existing one month earlier to the number of rabbits born since then -- the same as the number of rabbits existing two months earlier. Generalizing this to a mathematical formula, we can say that

xn = xn-1 + xn-2 (x0 = 1, x1 = 1).

Where

x represents the number of rabbit couples,

n represents the number of the month, 0, 1, 2 etc., and so,

xn is the number of rabbits existing in month n.

Application to music

But what does this rabbit formula have to do with music? Lots! If we only substitute x0= 2 and x1 = 5 for the first two terms, we get the series 2, 5, 7, 12, 19, 31, 50... which bears an important relationship to the structure of musical scales. Any musician will recognize 5, 7 and 12 as the numbers of pitches per octave in pentatonic, diatonic and chromatic scales; and also as the number of black keys, white keys and keys per octave of the traditional musical keyboard.

Now let's see how these numbers build musical structures. The simplest musical interval is the octave, a 2/1 frequency ratio. The next simplest are the fifth, a 3/2 ratio (more or less, in various tunings) and its inversion, the fourth (4/3, more or less). The numbers in the musical Fibonacci series 2, 5, 7, 12, 19 ... all are generated by increasingly long series of musical fourths and fifths, as Sir James Jeans described and shown in the table below.

scale no. calculation number of intervals Series of musical fifths
1 2 + 0 = 2 F-C (G)
2 5 + 0 = 5 F-C-G-D-A- (E)
3 5 + 2 = 7 F-C-G-D-A-E-B-(F#)
4 7 + 5 = 12 F-C-G-D...G#-D#-A#-(E#)
5 12 +7 = 19 F-C-G-D...Gx-Dx-Ax-(Ex)
6 19 + 12 = 31 Gbb-Dbb...Gx-Dx-Ax-(Ex)

and so forth...

Two-tone scale

Our first division of the octave (scale no. 1 in the table) is into a scale of two tones a fifth apart, say F and C, with a frequency ratio of 3/2. The next fifth after our two tones is a G, and the resulting sequence of two fifths, 3/2 x 3/2, takes us to 9/4 (2.25), which brings us close to an F a musical octave (2/1) higher than our first F. We think of the G as the same as the F, and stop with a two-tone scale.

With two tones per octave, it is already possible to play music which distinguishes a tonic and dominant -- like a typical tympani part of the early Classical era.

We may also lower the fifths to produce a 2-tone equal temperament. In fact, we can play this temperament on any conventional musical instrument by lowering the C of our two-tone scale by a half tone to B. But assigning different roles such as "tonic" and "dominant" to each of the two pitches of our scale is difficult. The intervals between them are equal, and so we can't tell which pitch has which role.

Five-tone scale

Now if we prolong the series of fifths, F-C-G-D-A-(E), the E at the end is close to F again. If we leave the fifths mathematically exact, we can now play pentatonic music ("Mama makes shortnin', shortnin bread ...") in one key signature, the way we do on the black keys of any conventional keyboard instrument. As you can see by looking at the black keys, the pentatonic scale consists of groups of three and two pitches separated by two larger steps. This pattern repeats in every octave, allowing us to identify the role of the tones of a pentatonic scale.

But suppose that we raise all of the fifths a little, until the final E is at the same pitch as F. In this way, we construct a 5-tone equal temperament. We can't play this temperament on a regular keyboard, but we can play it on any synthesizer which lets us set nonstandard intervals: set the semitone or octave interval for 2.4 times the standard one (or 1.2 times and play on alternate keys). The resulting scale will sound somewhat out of tune to our ears which are used to standard 12-tone equal temperament, but its melodic and harmonic properties are clear enough. It is not only a theoretical temperament; it is the standard Javanese tuning called slendro (see for example Helmholtz, p. 518.)

In the 5-tone temperament, we can distinguish tonic and dominant of two-tone scales and modulate freely among five key signatures - but pentatonic melodies do not work. Even though there are five pitches per octave in this scale, ordinary pentatonic music sounds oddly rootless -- we can't determine the keynote. With five equal intervals, there is no differentiation between wide and narrow intervals to identify the pentatonic key signature. This is the same problem we had earlier in distinguishing the tonic and dominant in our 2-tone temperament.

And onwards...

Now that we have established the first two terms of our Fibonacci series, succeeding terms follow logically. Each new, extended series of fifths is the sum of the previous two. The reason for this not the same as for the numbers of Fibonacci's rabbits, but is just as simple: since sequences of two or five fifths approximate whole numbers of octaves, sums of these numbers of fifths also approximate whole numbers of octaves and allow us to close a circle of fifths by filling in the wider steps in the scale which the earlier series of fifths built. So now let's increase the number of tones per octave, just as Fibonacci did with his rabbit population.

Seven-tone scale

Continuing from 5 on to 7, the series F-C-G-D-A-E-B-(F#) once again brings us close to F. If we leave the fifths exact, we can now play seven-tone diatonic music (like playing on the white keys of the piano), in one key. We can also play two-and five-tone music.

If, on the other hand, we lower all of the fifths of our seven-tone scale a little, we arrive at a seven-tone equal temperament. Set the semitone or octave on a synthesizer to 1.7142 (that's 12/7) times standard to play in this temperament. In a seven-tone temperament, we can play in the 2 and 5 tone scales and also play any ordinary pentatonic music with full freedom of modulation in seven keys.

The seven-tone temperament is not just a theoretical construction. It is the standard tuning used in traditional Siamese music (see for example Helmholtz, p. 556). You might think that, with 7 tones, you could also play standard diatonic music in this temperament.

But if you try, it sounds oddly rootless. Though we have the seven pitches needed to play diatonic music, the intervals between them are all the same. There is no distinction between large and small intervals to tell us what step of the musical scale we are on -- the same problem we had playing pentatonic music in the five-tone temperament in the previous step of our experiment, or two-tone music in the step before that.

12-tone scale

Next, we proceed to the series of 12 fifths, F-C-G-D-A-E-B-F#-C#-G#-D#-A#-(E#). If we lower all of the fifths very slightly, we arrive at our familiar 12-tone equally-tempered tuning. Now we can play all of the music we played in the earlier 2, 5 and 7-tone scales, and we can also play diatonic music using 7-tone scales with full freedom of modulation through 12 keys.

The progression from 7 to 12 tones occurred in European music between the Middle Ages and the Baroque era. Composers have explored the potential of the 12-tone scale over several centuries, and these explorations have culminated in the highly chromatic music of the later Romantic era and "atonal" music of the 20th century.

The term "atonal" reflects the oddly rootless quality of music that moves freely among all 12 of the tones of the scale. Does this begin to sound familiar?

19 tones: Joseph Yasser's proposal

Each new temperament generated by extending the series of fifths encompasses the melodic and harmonic possibilities of all of the earlier ones. The mathematical logic of this progression of equal temperaments, and the way it generates new musical possibilities at each step, lead us to ask "what if we went on to the next step of this progression, a 19-tone temperament?"

In the 1930's, music theorist Joseph Yasser published a book, Theory of Evolving Tonality, exploring that question. Yasser suggested that certain composers of his time, notably Scriabin, were already composing in a 12-tone "supradiatonic" scale with distinct tonalities identifiable by the wide and narrow intervals a 12-tone scale would have in a 19-tone temperament. Yasser also suggested that a logical next step in musical instrument design would be to adopt a 19-tone temperament, which he contended would open up possibilities in music similar to those which followed the transition from the 7-tone to the 12-tone scale.

Why don't we use Yasser's system?

The advance to a 19-tone scale which Yasser proposed has not yet occurred. Why? I think that there are a number of reasons.

Ingrained habits

One is that conventional musical instruments, music theory and practice are so deeply ingrained in our musical culture that breaking away from them would be difficult. The perception is grounded in the traditional design of musical instruments, and particularly of keyboard instruments, that D# and Eb (for example) are the same pitch, even though we spell them differently in musical notation.

Keyboard difficulties

Another problem is that a practical keyboard to play in a 19-tone temperament does not evolve simply from the traditional keyboard.

A 19-tone keyboard based on the traditional keyboard would have pairs of black keys (Db and C#, Eb and D#, etc.), and an additional black key between the semitone intervals E-F and B-C. Every diatonic semitone would, then, become a two-step interval, and every diatonic whole tone a three-step interval. The octaves would have to be lengthened to make room for the fingers, with all of the additional keys.

An alternative, favored by Yasser, would be exchange the positions of the black and white keys, front to rear. This keyboard might be somewhat easier to play, but it would still have long octaves and require difficult relearning.

Poor intonation of diatonic intervals

Another fundamental reason that the 19-tone system has not developed naturally out of the 12-tone system is that the 12-tone system works rather well. The 2, 5, and 7-tone systems which lead up to the 12-tone system have very impure fifths. The fifths in the 12-tone system are quite good.

But going from 12 to 19 tones requires that we make the fifths worse rather than better.

In 19-tone, the minor third is very pure but the major third, though more harmonious than in the 12-tone temperament, is slightly flat, compromising the melodic role of the leading tone.

Poor intonation of additional intervals

We should also hope that a system with more than 12 tones would provide good approximations to additional harmonic intervals used in non-western and experimental music (and discussed by Yasser), but the 19-tone system does poorly at this. The 19-tone scale is, for example, not capable of distinguishing between the 7th and 11th harmonics, or of distinguishing the 11th from the augmented fourth.

The rule of seven plus or minus two

The seven-tone scales in the twelve-tone system approach the limit of what the human mind can assimilate. The rule of "seven plus or minus two" in sensory psychology states that for any sensory continuum, humans describe between five and nine different categories: to give an example, we describe the gray scale using the categories white, off-white, light gray, medium gray, dark gray, near black and black. Though we can discriminate more shades of gray when they are placed side by side for comparison, we do not give names to them, or use the discrimination between them as part of a conceptual structure based on unaided observation and memory.

The musical octave is regarded by sensory psychologists as a continuum like the gray scale, and it follows that 12 distinct pitch classes may be more than we can easily assimilate as elements of a musical structure.

It's not as if 19-tone wasn't tried...

Historically, the evolution toward a 19-tone scale would in fact have succeeded if not for the problems discussed here. The mean-tone temperament used from the late Renaissance through the early Classical era had unequal semitones; it is possible to play Yasser's theoretical 12-tone scale in one key signature on any keyboard instrument tuned in the mean-tone temperament. Not only this, many keyboard instruments, especially organs, had extra keys -- for example, separate keys for Eb and D# (see for example Jeans, p. 173). The extra keys let the instruments play in several more key signatures than a standard 12-tone keyboard and still avoid the discordant "wolf fifth" which resulted from the mean tone series of fifths' not closing at 12. Handel, among others, had an organ with such a keyboard. The logical culmination of such keyboards with doubled sharps and flats would have been a 19-tone system.

But unlike the transition from the 7-tone to the 12-tone keyboard, the transition to 19 tones never occurred. Instead, 12-tone equal temperament gradually replaced mean-tone temperament beginning around 1700 AD and culminating around 1850 AD.

What if we leap over 19 to 31?

Suppose, however, that we leap past the 19-tone temperament and examine the next step of our Fibonacci series, with 31 tones.

This temperament turns out to be extremely close to the historical mean-tone temperament, with major thirds that are nearly pure. In addition, it gives a nearly perfect natural seventh, and a natural 11th which is slightly flat, but distinct from the augmented fourth.

If we put aside the idea of a 12 or 19-tone diatonic scale and regard the 31-tone tuning primarily as a tool to improve the intonation of traditional scales and intervals, then it looks like a sensible idea. I have tried it on a guitar, which sounds very pleasant, although it is difficult to play. The question of how to construct keyboard instruments which use this temperament has been addressed by Fokker. In forthcoming articles, I will address this tuning, among others, as they relate to modern, computer-controlled instruments. The next article in this series describes musical tunings as mathematical series.

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1997 John S. Allen, except for quotations as indicated.

Last modified 24 December 1998