GENERALIZED PYTHAGOREAN COMMA
Zvonimir Šikić
University of Zagreb, FSB, Lučićeva 5, [email protected]
Abstract:
Pythagorean tuning generates the notorious wolf fifth and two semitones, the apotome and the limma. The two semitones, and the two fifths (the wolf and the just), differ by the same Pythagorean comma. It is also the difference between 12 just fifths and 7 octaves. We prove that this is not a coincidence but a consequence of the (generalized) Pythagorean procedure.
Key words: Pythagorean tuning, Pythagorean comma, wolf fifth, apotome and limma
The ratios of the frequencies of tones that make up the perfect Pythagorean intervals are:
2/1 = Octave = 1200 cents
3/2 = Fifth = 701.96… cents 702 cents
4/3 = Fourth = 498.13… cents 498 cents.
(Cents are logarithms of ratios, with base b = 21/1200. They transform multiplicative structure of ratios into additive structure of cents. For example, the sum of a fifth and a fourth should be an octave. If we multiply the ratios of the fifth and fourth, (3/2)(4/3) = 2/1, the result is indeed an octave. If we add their amounts in cents, 702+698=1200, the result is again an octave.)
All the other intervals in the 12-tone scale (the chromatic scale) can be generated by moving successively up and down by these intervals. Moving up/down by a fourth is equivalent to moving down/up by a fifth (after moving up/down by an octave). Hence, fifths and octaves are enough. It is easy to see that moving up by fifths and transposing down by octaves will do the job. This procedure is the celebrated circle of fifths.
Beginning with C at frequency 1, or at 0 cents, we move up by fifths. It means we multiply by 3/2 or add 702 cents. When we get an interval which is outside of the basic octave, we transpose it down by an octave, i.e. we divide by 2 or subtract 1200 cents.)
C G D A E B F# C# G# D# A# F c
( ) ( ) ( ) ( ) ( ) ( )
1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
C G D A E B F# C# G# D# A# F c 1404 1608 1812 1314 1518 1722 0 702 204 906 408 1110 612 114 816 318 1020 522 1224
The problem is that 12 fifths, (3/2)12, do not line up with 7 octaves, 27. Namely, (3/2)12/27 1. (Expressed in cents, 12702 – 71200 = 24 0.) In fact (3/2)m/2n 1 for all integers m and n, because the unique factorization guarantees that 3m 2m+n. So, c is not at 1200 cents but at 1224 cents. It misses the perfect octave by 24 cents. Of course, in musical theory (and practice) we use c with 1200 cents. The result is that the final fifth, from F to c, has 678 cents. This imperfect fifth is called the wolf fifth because it sounds unpleasantly like wolves howling.
If we arrange the intervals in ascending order of magnitude (with c at 1200 cents) we get chromatic Pythagorean scale:
C C# D D# E F F# G G# A A# B c 0 114 204 318 408 522 612 702 816 906 1020 1110 1200
Hence, a closed cyclic scale system based on multiple fifths and octaves is impossible. But a closed cyclic system is required in order to allow music to be transposed to any key and still sounds in tune. The only possibility is a tempered scale, in which some intervals are flatted.
In the Pythagorean chromatic scale this is the extremely distorted wolf interval. In the equal- tempered scale all the intervals are tempered moderately:
C C# D D# E F F# G G# A A# B c 0 100 200 300 400 500 600 700 800 900 1000 1100 1200
The semitones of the equal-tempered scale are all the same (they all have 100 cents).
To determine the semitones of the Pythagorean scale we subtract the lower interval in the scale from its upper neighbor:
C C# D D# E F F# G G# A A# B c 114 90 114 90 114 90 90 114 90 114 90 90
There are two semitones, a smaller semitone with 90 cents called the Pythagorean diatonic semitone, or limma, and a larger semitone with 114 cents called Pythagorean chromatic semitone, or apotome. Their difference has 24 cents. It is called Pythagorean comma. It is also the difference between a perfect fifth with 702 cents and a wolf fifth with 678 cents. It is also the amount by which the interval of 12 perfect fifths differs from 7 octaves.
Is this threefold appearance of the Pythagorean comma coincidental, as is often concluded (e.g. [1] p. 54.)? No, it is a consequence of the Pythagorean cyclic procedure. We prove that bellow.
The equal-tempered scale could be tuned by moving successively up by 100 cents = 1 unus:
0 1 2 3 4 5 6 7 8 9 10 11 12
But it is not easy to tune 1 unus exactly. It is much easier to tune a fifth, i.e. 7 uni, and move successively up by 700 cents = 7 uni (and transpose down by an octave, when necessary):
14 16 18 13 15 17 0 7 2 9 4 11 6 1 8 3 10 5 12
In fact, what is really easy to tune is a perfect fifth of 7.0196… uni (easier then an equal-tempered fifth of 7 uni). Hence, the real Pythagorean tuning looks like this (with x = 0.0196… uni):
14+2x 16+4x 18+6x 13+7x 0 7+x 2+2x 9+3x 4+4x 11+5x 6+6x 1+7x
15+9x 17+11x 8+8x 3+9x 10+10x 5+11x 12
If we arrange the intervals in ascending order of magnitude we get:
0 1+7x 2+2x 3+9x 4+4x 5+11x 6+6x 1+7x 1–5x 1+7x 1–5x 1+7x 1–5x
6+6x 7+x 8+8x 9+3x 10+10x 11+5x 12 1–5x 1+7x 1-5x 1+7x 1–5x 1–5x
There are only two semitones, apotome with 1+7x uni and limma with 1–5x uni. Their difference has 12x uni:
1 + 7x – (1–5x) = 12x.
This is also the difference between a perfect fifth with 7+x uni and the final wolf fifth with 12–(5+11x) uni, i.e. 7–11x uni:
7 + x – (7–11x) = 12x.
Of course, it is also the amount by which the interval of 12 perfect fifths differs from 7 octaves: 12 (7+x) – 7 12 = 12x.
But why the Pythagorean tuning generates only two semitones (the apotome and the limma)?
Let us consider a general case in which an octave is divided into N equal semitones. The corresponding semitone tuning is:
0 1 2 . . . N–2 N–1 N
If N and n are relatively prime (and n < N) then the same scale is generated with the following n-step tuning:
0 n 2n . . . (N–2)n (N–1)n N.
The generalized Pythagorean (n+x)-step tuning (where Nx<1) is defined as:
0 n+x 2n+2x . . . (N–2)n + (N–2)x (N–1)n + (N–1)x N.
If we transpose the intervals to the basic octave and arrange them in ascending order of magnitude, we get the semitones of this generalized Pythagorean scale, as the differences of the neighboring intervals.
Let us suppose that pn and qn are neighbors in the equal-tempered scale (x=0), and that qn is the upper neighbor. Their difference is 1, i.e. qn–pn 1 (mod N).
The difference of the corresponding Pythagorean intervals is I+ = 1 + (q–p)x qn + qx – (pn + px) (mod N) if q > p, or I– = 1 – (p – q)x qn + qx – (pn + px) (mod N) if p > q (cf. Fig. 1.).
Fig 1. Pythagorean intervals I+ and I–
Now, from qn – pn = (q–p)n 1 (mod N) it follows that
q – p n–1 (mod N).
Similarly, from pn – qn = (p–q)n -1 (mod N) it follows that p – q -n–1 (mod N).
In both cases n–1 is the multiplicative inverse of n (mod N). (It exists because N and n are relatively prime.) Hence,
I+ = 1+ (n–1)x, I– = 1– (-n–1)x.
Notice that 1+ (n–1)x 1– (-n–1)x because the long – is from ordinary arithmetic and the short - is from arithmetic mod N (e.g. (n–1) + (-n–1) = N).
I+ and I- do not depend on p and q, so we have proved that I+ and I- are the only semitones of the scale. It also follows that:
I+ – I– = 1+ (n–1)x – (1– (-n–1)x) = ((n–1)+ (-n–1))x = Nx, which means that the difference between this two semitones is Nx. This is the generalized Pythagorean comma, defined as the difference between the tuning step n+x and the final “wolf” step N–[(N–1)n+(N–1)x]. Namely,
(n+x) – (N–[(N–1)n + (N–1)x]) = n + x – N + Nn – n + Nx – x = –N + Nn + Nx Nx (mod N).
Of course, it is also the difference between N Pythagorean n+x-steps and N equal-tempered n-steps:
N(n+x) – Nn = Nx.
In our special, chromatic, case N = 12, n=7, x 0.2. It follows that
n–1 = 7 because 77 1 (mod 12),
-n–1 = 5 because 7 + 5 0 (mod 12),
I+ = 1+ (n–1)x = 1 + 7 0.02 = 1.14,
I– = 1– (-n–1)x = 1 – 5 0.02 = 0.9.
These are the well-known apotome of 114 cents and limma of 90 cents. Of course, their difference is Nx = 120.02 = 0.24, which is the Pythagorean comma of 24 cents.
[1] Loy, Gareth: Musimathics, Vol. 1, MIT Press, 2006.