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An Alternative Approach to Generalized Pythagorean Scales

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An Alternative Approach to Generalized Pythagorean Scales April 1, 2020 Journal of Mathematics and Music generalized_pythagorean_author_rev2 The Version of Record of this manuscript has been published and is available in Journal of Mathematics and Music, March 2020 https://doi.org/10.1080/17459737.2020.1726690 An alternative approach to generalized Pythagorean scales. Generation and properties derived in the frequency domain Rafael Cubarsi Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain Abstract scales are formalized as a cyclic group of classes of projection functions related to iterations of the scale generator. Their representatives in the frequency domain are used to built cyclic sequences of tone iterates satisfying the closure condition. The renement of cyclic sequences with regard to the best closure provides a constructive algorithm that allows to determine cyclic scales avoiding continued fractions. New proofs of the main properties are obtained as a consequence of the generating procedure. When the scale tones are generated from the two elementary factors associated with the generic widths of the step intervals we get the partition of the octave leading to the fundamental Bézout's identity relating several characteristic scale indices. This relationship is generalized to prove a new relationship expressing the partition that the frequency ratios associated with the two sizes composing the dierent step-intervals induce to a specic set of octaves. Keywords: well-formed scales; equal temperament; Pythagorean tuning; Pythagorean comma; Bézout's identity; continued fractions 2010 Mathematics Subject Classication: 05C12; 11A05; 54E35 1. Introduction An n-tone equal temperament (TET) scale does not include the pitch classes (PCs) of the har- monics and subharmonics of a fundamental frequency, not even the lowest orders, but in its favour its PCs remain equally spaced within the circle of the octave. The scale intervals have an algebraic structure similar to the group of integers modulo n, (Zn; +), and any PC may assume the role of the fundamental. On the other hand, a just intonation scale composed of a nite family of harmonics and subharmonics, apart from having non-equal temperament, is not closed with regard to the composition of its intervals. A compromise is reached in Pythagorean tuning, where the scale is tuned to the class of the third harmonic, the lowest harmonic not belonging to the class of the fundamental. The standard 12-tone Pythagorean scale gives an acceptable approximation of the harmonics up to the tenth except the seventh that sounds very low in most instruments (von Helmholtz 1863) and is close to a 12-TET scale with an error smaller than one Pythagorean comma. For these reasons, Pythagorean tuning has become the standard pattern of Western music. By choosing conveniently the number n of notes, it is possible to get a Pythagorean scale containing notes closer to the lower harmonics. On the other hand, it is well known that the frequency ratios between the scale tones are aurally signicant since the harmonic structure of many musical instruments causes their partials to overlap. For this reason, in order to produce specic overtone series, spectrum, or timbre (Sethares 1998), Pythagorean tuning can be generalized to scales generated by intervals other than the fth. While trying to substantiate the process of building a Pythagorean scale, one nds that the properties inherent to the formation of the scale do not depend on the generator interval chosen, Email: [email protected] April 1, 2020 Journal of Mathematics and Music generalized_pythagorean_author_rev2 rather they are a consequence of the process itself. Hence, as already suggested by Regener(1967), the procedure can be extended to musical scales generated as a nite sequence of iterations of an arbitrary frequency ratio h (with regard to a fundamental ν0 = 1). Since the octave is associated with a frequency ratio 2, the reduction of the frequencies by octaves to the interval [1; 2) determines the representatives of the frequency classes (FCs), which will be referred to as scale tones. When one iterate is closer than the others to either end of the interval, the scale formed up to this iterate satises the closure condition (Carey and Clampitt 1989). Since the reference octave is usually represented as a circle in the pitch domain ( of the frequencies), we log2 will say the scale is cyclic, meaning a well-formed scale of one generator, which will be described as a cyclic group. In the general case, h is assumed to be dierent from a rational power of 2, otherwise the scale is equally tempered, which is considered as a degenerate Pythagorean system. The procedure can be generalized by changing the period of the octave to any positive value !. In the current paper, cyclic scales will be studied by following such a constructive procedure, by pointing out, especially in the footnotes, the links to other approaches, namely the continued fractions approximation of , combinatorics of words, Farey sums, or the Stern-Brocot tree log2 h (Carey and Clampitt 2012, 2017; Clampitt and Noll 2011; Noll 2006, 2007, 2008, 2015; Jedrzejew- ski 2009, 2008). After introducing the notation, an abstract scale is described from a canonical Generalized Interval System (GIS) (Lewin 1987) with an interval function, that we call projection function, related to iterations of the generator tone. Under an equivalence relation, the represen- tatives of the classes of projection functions dene the tones of a cyclic scale. Hereafter, we leave the abstract formulation and adopt a more analytical viewpoint. In a rst instance, we present an approach leading for eciently determining these tones. It is worked out in the frequency domain instead of the pitch domain, as usual. The reason is to show in what simple way a cyclic scale can be generated: (a) there is an innite hierarchy of cyclic sequences increasing the number of tones; (b) each sequence is characterized by the minimum and maximum tones νm and νM , ordered as increasing pitches in ; (c) since νM , then νM νM νm , from where (1; 2) 2 < 1 < νm 2 < 2 < νm we prove that a new extreme tone of the next cyclic sequence must be νm+M . By identifying this tone of the sequence with the fundamental we get the cyclic scale. In this way, we obtain alternative proofs to results usually derived from the continued fractions approach (e.g., Carey and Clampitt 1989; Hellegouarch 1999; Douthett and Krantz 2007). Later on, the generating al- gorithm is reformulated at a generic level depending exclusively on the indices of the scale tones, without the need to estimate the comma (e.g., Noll 2007). In this form, the algorithm allows to distinguish easily between optimal scales (associated with the convergents of the continued fractions approach) and non-optimal scales (associated with the semi-convergents). In a second part, new proofs of several scale properties are derived from the generating pro- cedure. The paper provides new and alternative viewpoints to obtain the main characterizations of well-formed scales, such as those related to the best comma (Douthett and Krantz 2007), the chromatic length of the generator, Myhill's property (Clough and Myerson 1985, 1986), the spec- trum width, etc. As a new result we prove that, for a n-tone cyclic scale, the dierence between the chromatic length of the generating ratio N and the index of octave n (octave windings in n) tells whether the scale is or is not optimal. J K Finally, we study the partition of the octave. After providing a new demonstration of how the frequency ratios associated with the extreme tones (i.e., those corresponding to the two possible sizes of the step-interval j = 1) generate the partition of the octave into elementary factors such M 2 m 2 that ν ( ) = 2, we generalize it. Thus, if the pair νl, , with l + L = n, is the spectrum of m νM νL the j-th step-interval, then we get νL( 2 )l = 2j, which provides the partition of j octaves and l νL yields the generic relationship between the indices j; l; L. Generating the cyclic scale from such a pair of factors with indices coprime with , which in the domain is equivalent to the pairs n log2 generator/co-generator, gives rise to self-similar pitch structures representing the scale (Carey and Clampitt 1996; Clampitt and Noll 2011). In addition. it provides the deviation of the scale tones with regard to the n-TET scale. 2 April 1, 2020 Journal of Mathematics and Music generalized_pythagorean_author_rev2 2. Preliminaries 2.1. Well-formed scales Well-formed scales satisfy the symmetry condition, consisting of displaying several degrees of rotational symmetry, which Carey and Clampitt(1989) proved to be equivalent to the closure condition. abka(2009, 2010) generalized this result and proved the equivalence of the closure and symmetry conditions also for scales generated by two intervals. In the current paper, cyclic scale refers to a well-formed scale with one generator. If one wishes a particular frequency ratio h to be the generator of a non-degenerate cyclic scale, i.e., reproducing the closure and symme- try conditions, the cardinal k of its pitch class (PC) within the scale must be coprime with n. For generalized Pythagorean systems, well-formed scales are also known as moment of symme- try (MOS) (Wilson 1975) and natural scales (Hellegouarch 1999). We highlight the three main approaches leading to cyclic scales. The rst one is the MOS (Wilson 1974, 1975), which results from iterating a non-degenerate generalized Pythagorean system when the partition of the octave induced by the scale notes has exactly two sizes of scale steps and each number of generic intervals occurs also in two dierent sizes1. This condition is known as Myhill's property (Clough and Myerson 1985, 1986).
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