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]).

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.

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, n^o 53, 1983; so, some other papers of Yves Hellegouarch are quoted in the references below, [6], [7].

2. Preliminaries

Denote by N, Z, Q, R, C the sets of all natural, integer, rational, real, and complex numbers, respectively. Denote by L = ((0, infinity),*,1,=<) the usual multiplicative group with the natural order on R. So if a=b with a, b ELEMENTS OF (0, infinity), then b/a is an L-length of the interval (a, b). Since this terminology is not obvious, we borrow the usual musical terminology, i.e. we simply say that b/a is an interval. This inaccuracy does not lead to any misunderstanding because in this paper the term "interval" is used only in this sense.

The proof of the following lemma is easy.

Lemma 1. Let q = const ELEMENT OF (0, infinity).

Then p_q(u,v) = |log₂

REFERENCES

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)