On the Euler scale and the μEuclidean integer relation algorithm

Abstract

The equal-tempered 10-tone scale e ^n/10 (n=0,±1,±2, …), using the Euler number e=2.71828… as a pseudo-octave is shown to approximate well the prime number harmonics 2, 3, 5, and 11. Equal-tempered scales simultaneously approximating certain frequency ratios, shall be called tonal scales.1 Some of the properties of the Euler scale and its relation to other tonal scales are explored. The general mathematical problem of identifying tonal scales can be solved by investigating integer relations, using the μEuclidean algorithm, a modification of the PSLQ algorithm. If restricted to two numbers, the μEuclidean algorithm goes over identically into the ancient Euclidean algorithm, contrary to the PSLQ algorithm. The μEuclidean is able to solve a certain class of higher dimensional integer relations where the PSLQ (not the PPSLQ) algorithm breaks down. In general, the μEuclidean algorithm finds smaller integer relations than the (P)PSLQ algorithm. In an appendix, a simple alternative procedure is presented for determining tonal scales based on continued fractions.

Keywords:

• equal-tempered scales
• integer relations
• continued fractions
• harmonics
• intervals

Acknowledgements

Stimulating conversations with Arturo Raffaele Grolimund, Georg Hajdu, and Manfred Stahnke are gratefully acknowledged. Our investigation of the Euler scale started in March 2010 when A.R. Grolimund informed us about his concert on a Bohlen–Pierce pan flute at the Bohlen–Pierce symposium in Boston. M. Stahnke recognized the close relationship between traditional African and oceanic music and the Euler scale. We are grateful for his critical reading of the manuscript. G. Hajdu pointed our attention to John Chowning's composition Stria. It is a pleasure to thank Heinz Bohlen for valuable comments. We really appreciate the support from Thomas Noll for helpful suggestions during the publication process, in particular for hints on current work based on continued fractions.

Notes

In this particular meaning, the term is used among microtonal composers.

The Stahnke scale can be derived from the transcendental equations and . They are solved by the numbers

The solution μ is a transcendental number contrary to the number s which is simpler (algebraic number). The number μ characterizes the deviation of the intervals 4 and 10 of the Stahnke scale from the frequency ratios 5/4 and 7/4, respectively. The deviation corresponds to  cent. The smallness of μ is a kind of a magic.

The 6th harmonic corresponds nicely to the interval n=32 (Table 1).

The relevant equation 3=2 ^x can be solved by a simple ratio x if the continued fraction expansion is considered.

By doubling the number of subintervals (N=20) also the 7th harmonic can be well represented as .

Measures for estimating the approximation error of a tonal scale are also considered in Section 4 and in the appendix.

The 10-tone ET Euler scale and the 11-tone ET Bohlen–Pierce scale differ only weakly. The equation 3=e ^x can be solved by a ‘simple’ ratio x if the continued fraction expansion is considered. Thus, . The 11-tone ET Bohlen–Pierce scale was already suggested in [footnote 26].

Traditional Thai instruments are also tuned equiheptatonically, see [Chapter 15].

Some properties of continued fractions are collected in the appendix.

Dirac's bra–(c)ket notation is used. The scalar product between a bra vector and a ket vector reads . In our setting, a bra vector is the transpose of a ket vector, .

In the PSLQ algorithm, the relevant norm (max _i |H _{i, i} |) is reduced monotonically, but not strongly monotonic.

Let p≥1 be a real number. The p-norm of a k-dimensional vector is defined as . The case p=2 corresponds to the Euclidean distance measure, . In the limit the maximum norm is obtained, . Let q>p, then i.e. the topology induced by the p-norm is finer than the topology induced by the q-norm. The pq-norm of a (k×l)-matrix A can be defined as where |x⟩ is an l-dimensional vector. Clearly, . Let 1/p+1/q=1. Then where ⟨ a _i | is the ith row of the matrix A. The assertion follows from the Hölder inequality, if 1/p+1/q=1.

Without the rounding operation, i.e. for , the transformed matrix H′ would have been simply diagonal since then .

for and else, where is a tiny real number.

The inequality does not hold for i=k−1 although the matrices S, S′ can easily be modified to cover also this case. However, then the direct contact to the ancient Euclidean algorithm (k=2) is lost. But this detail is of no significant relevance as the Euclidean algorithm is converging fast, since as already mentioned.

The inequality (2) can be extended to all offdiagonals by generalizing the mirror transformation implicitly: in the definition of the reducing matrix D the definition of q needs to be replaced by ; ; else . Then, the contact to the ancient Euclidean algorithm is lost, however, this deficiency can be repaired by using the q-replacement not for i=k, j=k−1. Although in this case, the μEuclidean algorithm is (almost) equivalent to the PSLQ algorithm with respect to the convergence behaviour, the matrix norm of H is also not decreasing strongly monotonic, see footnote 11.

The phenomenon does not occur in the parallelized PSLQ (PPSLQ) algorithm 14 as its initial phase is modified essentially compared to the initial step of the PSLQ algorithm. We thank thank Bailey for pointing the difference out to us.

A breakdown occurs if .

There are infinitely many counter examples. Let, for example, α≥3 be any integer. Applied to the integer relation , the (P)SLQ algorithm terminates after the first iteration and finds and . On the other hand, the μEuclidean algorithm identifies the smallest integer relation . Since |m _c ⟩ is independent of α, both solutions |m _a ⟩, and |m _b ⟩ violate the upper bound the stronger the larger α.

For comparison, we also implemented the PSLQ algorithm in Mathematica and found that a working precision of 179 decimal digits is needed to find the solution (after 1426 iterations and with the termination condition that the absolute values of one of the entries of or is smaller than ). A working precision of 85 decimal digits as mentioned in 13 could be confirmed by an implementation the PSLQ algorithm in the software package Maple 15. The difference in the working precision should be due to the special error propagation handling within Mathematica.

All entries of a row are either positive of negative provided all input numbers have the same sign.

In practical applications, it is often convenient to represent errors in units of cent

An approximation is called reasonable if cents, good if cents, and impressive if cents.

The results of this section are obtained by an implementation of the μEuclidean algorithm in the programming language JavaScript (see http://www.hessling.net/mEuclid.html). JavaScript has a working precision of 16 decimal digits which is far more than sufficient for the calculations within this section (we set and ).

The tonal content of the Stahnke scale, i.e. the set of numbers (6, 2, 5), can be obtained without referring to the Stahnke number by choosing, for example, the harmonic seventh (7/4) as the basis and determining approximations for the major third (5/4) and the octave (2/1) with respect to this basis. Inserting the numbers we obtain,

The error associated with the first row of the matrix B ⁻¹ is quite large, roughly a quarter tone, since octaves are poorly approximated within the Stahnke scale. The numbers of the second line of the matrix (25, 10, 31) show a close connection between the Stahnke scale and the 31-tone ET system proposed by Huygens and others, see also footnote 28. The error is less than 2 cent. The solution (−13,−5,−16) in the last line of the matrix is not stable against a permutation of the input numbers.

The Editors pointed our attention to the authors of 20 which also explored 1–dimensional continued fractions for analysing non–standard tonal scales.

The right hand side of the equation is well–defined if the quantities are introduced.

The 3rd harmonic is nothing but a fifth (3/2) on top of an octave (2/1) since .

The basic interval Cent) is almost identical to the basic interval Cent) of the 31–tone ET scale considered already by Huygens (and others).

This observation motivated us to look for an alternative to 1–dimensional continued fractions and led us, eventually, to the μEuclidean algorithm.