Comma 32 805 / 32 768
by Jan Haluska
published in
International Journal of Uncertainty,
Fuzziness and Knowledge-Based Systems
vol 6, no 3 (1998), p 295-305
© 1998 World Scientific Publishing Company
edited for the web 1999 by Joseph L. Monzo
[ABSTRACT:] We find a fifth approximation of the Just Intonation which generalizes Equal Temperament. The intervals causing a dilemna are the 2nd and the minor 7th and the tritone because they are unambiguous in Just Intonation (the relative frequencies 10/9, 9/8, 8/7 and 7/4, 16/9, 18/10 and 45/62, 64/45, respectively). If we do not consider the 2nd and the 7th with the relative frequencies 8/7 and 7/4, respectively, all music intervals in this approximation either coincide with the Just Intonation interval values (the 8ve, 5th, 4th, 2nd (9/8) and the minor 7th (16/9)) or are exactly the one comma distant from the corresponding Just intonation intervals. This comma is 32 805/32 768
~ 1.00112915, which is less than the ratio of frequencies of the perfect and the equal tempered 5ths (~ 1.00112989). *
1. Introduction
Mathematicians have solved problems connected with tuning since antiquity. It seems that in these considerations vearious mathematical structures on real line can be applied. Copnenections between mathematics and music are mutually enriching, depending on the progress in mathjematics or music. Therefore problemss having their origin in music attract the interest of scientists and musicians thru-out the history up to present days.
Pythagorean Tuning c.f. e.g. [1], [2], was created as a sequence of ratios, i.e., products 2p3q, where p and q are integers , cf. Table 1.
C(P)
2030
1.0
Db(P)
C#(P)
283-5
2-1137
2187/2048
1.053497942
1.067871094
D(P)
2-332
1.125
Eb(P)
D#(P)
253-3
2-1439
19683/16384
1.185185185
1.201354981
E(P)
2-634
1.265625
F(P)
223-1
1.333333333
Gb(P)
F#(P)
2103-6
2-936
729/512
1.404663923
1.423828125
G(P)
2-131
1.5
Ab(P)
G#(P)
273-4
2-1238
6561/4096
1.580246914
1.601806641
A(P)
2-433
1.6875
Bb(P)
A#(P)
243-2
2-15310
59049/32768
1.777777777
1.802032473
B(P)
2-735
1.898437528
C' (P)
2
2.0
This tuning was established about 500 years BC and used in Western music up to the 14th century.
The gradual development of polyphony led to the introduction of 3rds (5/4 and 6/5) and 6ths (8/5 and 5/3). After that the problem why musical ratios of integers are considered attempts to provide a strict mathematical definition of consonance. The practical result of this period was the creation of the Just Intonation set (abbreviation: JI). JI is based on the intervals which are integer exponents of numbers 2, 3, 5, and 7. The key figures in these studies were Gioseffo Zarlino (1517-1590), Simon Stevin (1548-1620), Johann Kepler (1571-1630). The present microtonalists assert that "Just Intonation is any system of tuning in which all of the intervals can be represented by ratios of whole numbers, with a strongly implied preference for the smallest numbers compatible with a given musical purpose" (David B. Doty). We will deal with JI given by Table 2 (see [3]).
C(JI) | 203050 | 1.0 | |
C#(JI), Db(JI) | 243-15-1 | 1.066666666 | |
D(JI) | 213-251
2-332 237-1 | 9/8 8/7 | 1.1111111111
1.125 1.14285714 |
D#(JI), Eb(JI) | 21315-1 | 1.2 | |
E(JI) | 2-251 | 1.25 | |
F(JI) | 223-1 | 1.333333333 | |
F#(JI), Gb(JI) | 2-53251
263-251 | 64/45 | 1.40625
1.422222222 |
G(JI) | 2-131 | 1.5 | |
G#(JI), Ab(JI) | 235-1 | 1.6 | |
A(JI) | 3-151 | 1.666666666 | |
A#(JI), Bb(JI) | 2-271
243-2 325-1 | 16/9 9/5 | 1.75
1.777777777 1.8 |
B(JI) | 2-33151 | 1.875 | |
C' (JI) | 21 | 2.0 |
The further progress in tuning continued by so-called temperaments which did not avoid the inharmonic music intervals given by irrational numbers. Rather the most popular today, [12-tone] Equal Temperament, was already known to Andreas Werckmeister in 1698. It is represented by the sequence
W = {1, , (
)2,..., (
)12=2}
= {C(W),C#(W),...,
B(W), C' (W)}.
Then there have been many theoretical and experimental works concerned with the assumed universality of diatonic scales - both pro and contra, cf. [4], [5]. However, our aim is not to judge such non-mathematical problems.
In the present paper, we find a fifth approximation of the JI, , cf. Table 3, which generalizes Equal Temperament.
Table 3. The JI Approximation A, x = 28/35, y = 37/211
C(A) | x0y0 | 2030 | 1.0 | |
C#(A), Db(A) | x0y1 | 2-1137 | 1.067871094 | |
D(A) | x2y0
x1y1 x0y2 | 2163-10
2-332 2-22314 | 9/8 4782969/4194304 | 1.10985791
1.125 1.140348673 |
D#(A), Eb(A) | x1y2 | 2-1439 | 1.201354981 | |
E(A) | x3y1 | 2133-8 | 1.248590154 | |
F(A) | x3y2 | 223-1 | 1.333333333 | |
F#(A)
Gb(A) | x4y2
x3y3 | 2103-6
2-936 | 729/512 | 1.404663923
1.423828125 |
G(A) | x4y3 | 2-131 | 1.5 | |
G#(A), Ab(A) | x4y4 | 2-1238 | 1.601806641 | |
A(A) | x6y3 | 2153-9 | 1.664786873 | |
A#(A), Bb(A) | x7y3
x6y4 x5y5 | 2233-14
243-2 2-15310 | 16/9 59049/32768 | 1.753849545
1.777777777 1.802032471 |
B(A) | x7y4 | 2123-7 | 1.872885232 | |
C' (A) | x7y5 | 2130 | 2.0 |
The intervals causing a dilemna are the 2nd
and the minor 7th and the tritone because they are
unambiguous (the relative frequencies [10/9, 9/8, 8/7]
and [7/4, 16/9, 18/10] and [45/62, 64/45], respectively) in JI. If we do not
consider the 2nd and the 7th with the relative frequencies 8/7 and 7/4,
respectively, all music intervals in this approximation either coincide
with the JI interval values (the 8ve, 5th, 4th, 2nd (9/8)
and the minor 7th (16/9)) or are exactly the one
comma
[= schisma]
distant from
the corresponding JI intervals.
This comma is 32 805/32 768 (~ 1.00112915), which is
less than the ratio of frequencies of the perfect [= JI]
and the equal tempered 5ths (~ 1.00112989).
The author is indebted [to] the referee for pointing out that related ideas are treated in a more general setting in several papers of Yves Hellegouarch, in particular, in "Gammes naturelles", Publications de l'APMER, no 53, 1983; so, some other papers of Yves Hellegouarch are quoted in the references below, [6], [7].
The proof of the following lemma is easy.
Lemma 1. Let q = const ELEMENT OF (0, infinity).
MUCH MORE STILL TO COME!!
I would appreciate help with the HTML math equations
which still need to be done.
1J. Girbau, "Mathematics and musical scales", Butl. Sec. Math. 18 (1985), 3-27. (back to text)
2J. Haluska, "On fuzzy coding of information in music", BUSEFAL 69 (1997), 37-42. (back to text)
3V. A. Lefebvre, "A rational equation for attractive proportions", Math. Psychology 36 (1992), 100-128. (back to text)
4H. Helmholtz, Die Lehre von den Tonemphindungen als phychologische Grundlage fur die Theories der Musik (Braunschweig, 1863). (back to text)
5J Haluska, "Diatonic Scales Summary", Proc. 7th IFSA World Congress, Prague, June 25-29, 1997 (Academia Prague, 1997), vol 4, p 320-322. (back to text)
6Y. Hellegouarch, "Scales". C. R. Math. Rep. Acad. Sci. Canada 4 (1982), 277-281. (back to text)
7Y. Hellegouarch, "A la recherche de l'arithmetique qui si cache dans la musique", Gaz. Math. 33 (1987), 71-80. (back to text)
*Haluska is here calling a 'comma' that interval which is usually referred to as a 'schisma', defined as this ratio by Alexander Ellis. See Ellis's English translation of Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music. (London, 1875) (Dover reprint, 1954, based on 2nd English edition of 1885). (back to text)
Jan Haluska
Mathematical Institute, Slovak Academy of Science
Gresakova 6, 040 01 Kosice, Slovakia
email: Jan Haluska
or try some definitions. |
I welcome
feedback about this webpage:
![]() ![]() |