Musical intervals and special linear transformations

Abstract

This paper presents a transformational approach to musical intervals with particular focus on their constitutive role for well-formed scales. These scales have the property that their binary step-interval pattern is maximally even. Transposition classes of well-formed scales are therefore characterized by two step intervals and their characteristic binary pattern, or, more abstractly, by four numbers: two step intervals and two associated multiplicities. The proposed transformational approach therefore studies group actions (1) on interval pairs, (2) on multiplicity pairs, such that the two intervals with their associated multiplicities form two pairs of canonically conjugated variables, and (3) on binary cycle words. The group is the same in all three cases: the modular group Γ=SL(2, ℤ). For any free commutative interval group G we have a faithful action of Γ on G×G through transvections. This action has a refinement in terms of an action of the braid group B ₃ on the product F×F of any non-commutative free group F with itself. The non-commutative interval group considers intervals as pathways rather than sums. The action of the modular group on ℝ⁴ through canonical transformations is given in terms of a representation of this group through symplectic 4×4-matrices. This left action can be comfortably rewritten in terms of a right action on 2×2-matrices. The submonoid SL(2, ℕ) exemplifies the Stern–Brocot tree and provides a link to the classical theory of well-formed scales. We recapitulate how the processes of approximating a scale generator g through its semi-convergents and of generating smaller and smaller step interval sizes are transformationally interconnected. The action of SL(2, ℤ) on cycle words with directed letter is based on parallel rewriting rules. The maximally even patterns form exactly one orbit of this group action: the orbit generated by the one-letter-word . Looking to the future, the author indicates how the extension of this theory can be musically explored by the technology of Sethares spectra and how new questions arise from an inspection of intrinsic properties of the modular group Γ.

Keywords:

• Transvections
• Scale states
• Canonical transformations
• Well-formed scales
• Clough words
• Congruence subgroups

Acknowledgements

The author thanks Emmanuel Amiot, Vittorio Cafagna, Norman Carey, David Clampitt, Patrick Dehornoy, Noam Elkies, Franck Jedrzejewski, Guerino Mazzola, Andreas Nestke, Robert Peck, William Sethares, Domenico Vicinanza and the reviewers for helpful information and critique, and numerous helpful comments.

Notes

^*Franck Jedrzejewski (cf. 12–14) was the first to interpret braids in music-theoretical terms. In a section entitled Tuning Braids, Jedrzejewski [14, pp. 121–124] describes binary step interval patterns of scales in terms of words in the Artin group B ₃. In 13 he investigates the infinite sequence of upward fifths and downward fourths—representing the octave classes of the infinite pythagorean system within the fundamental domain of one octave register. With each point in this sequence he associates an element of the Artin group B ₃. One may roughly characterize these braid-interpretations as being inherited from the action B ₃×B ₃→B ₃ of the braid group on itself. This entails the difficulty of interpreting the Artin relation music-theoretically, as this relation is imposed from outside on the musical structures. A somewhat similar critique applies to the study of interval apperception processes directly in the group SL(2, ℤ) in a paper by Andreas Nestke and myself 5. Although that paper is deliberately written for the purpose of music-theoretically interpreting the defining relations of SL(2, ℤ), there remain lots of open questions. It is insightful to compare directly the pythagorean braids in [13] with those in the present paper. If one constructs the fundamental domain of the infinite pythagorean system with respect to a fixed tone C of reference, Jedrejewski associates a braid with each finite subsequence of upwards-fifths/downwards-fourths departing from C. In the context of the present paper braids are only associated with those subsequences that are related to well-formed pythagorean tone systems. And these braids are different from those in Jedrejewski's approach, as they are built upon another group action of B ₃.

^*And generalized musical spaces.

^*These symbols follow a common notation: m2=minor second, M2=major second, …, P4=perfect fourth, A4=augmented fourth.

^*This observation is featured by the ‘pitch-nominalist’ approaches.

^*This space is equipped with the symplectic form with , where

^*Canonical (or symplectic) transformations have to satisfy the condition .

^*This meaning does not coincide with the concepts of primary and secondary intervals in [9, p. 193].

^*It was independently discovered by Moriz Stern (1858) and Achille Brocot (1860).

^*The term act is chosen on mathematical grounds of monoid- and group-actions as well as with reference to transformational approaches to music theory.

^*The content of the present paper has several close relations to the subfield of Algebraic Combinatorics of Words which studies Christoffel words and Sturmian words—cf. 20. While revising this paper it became clear to me that the right mathematical context for the rewriting rules are Sturmian morphisms and morphisms of Christoffel words.

^*John Clough substantially contributed to the investigation and music-theoretical interpretation of these patterns.

^*The discussion here only touches upon the music-theoretic interpretation of the central application. For example, the concept of Carey–Clampitt Duality, introduced in 10, corresponds to an inner anti-automorphism of the modular group Γ and is investigated in a forthcoming joint article by David Clampitt and myself. This investigation further widens the limited scope of this central list of examples and interprets matters of musical harmony such as diatonic triads and seventh chords in terms of scale theory and thus touches the domain of harmonic tonality.

^*I am grateful to William Sethares for kindly providing his 10TET-library for such musical experiments and to Daniel Figols Cuevas for his kind permission to add his illustrative example to this article.

^*It is Carey–Clampitt-dual to the fifth-generated diatonic.

^*In favor of this assumption we may argue that a capability like the focusing of our eyes could be related to an action of the group SL(2, ℝ)$ on ℝ×ℝ which in linear optics describes lens systems as transformations. The discrete group SL(2, ℤ) sits nicely within SL(2, ℝ) and so does B ₃ within the universal covering group SL(2, ℝ)∼ of SL(2, ℝ).

^* SL(2, ℤ₁₂)≅SL(2, ℤ₃)×SL(2, ℤ₄) with abelianization $\mathbb ℤ₃×ℤ₄. More generally, for any k≥1 the abelianizations of SL(2, ℤ_3^k ) and SL(2, ℤ_2^k+1 ) are isomorphic to ℤ₃ and ℤ₄, respectively. Instead, SL(2, ℤ _N ) is simple whenever N is not divisible by 2 or 3. Thus the abelianization of SL(2, ℤ _N ) is ℤ₁₂ and hence [Γ, Γ] is a congruence subgroup of level N, if and only if N is divisible by $12.

^*The converse is not true. SL(2, ℤ₁₂) has 1152 elements and 96 of them are still commutators.