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Musical intervals and special linear transformations

Musical intervals and special linear transformations,10.1080/17459730701375026,Journal of Mathematics and Music,Thomas Noll

Musical intervals and special linear transformations   (Citations: 8)
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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 B3 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 ℝ 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.
Journal: Journal of Mathematics and Music - J MATH MUSIC , vol. 1, no. 2, pp. 121-137, 2007
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    • ...In 16, the two dimensional special linear transformations are used to explore the well-formed scales which are shown to be related to secondary convergents...

    Hermann Heßling. On the Euler scale and the μEuclidean integer relation algorithm

    • ...The present article implements an announcement, which I made in an earlier transformational account to scale theory [7, footnote on p...

    Thomas Noll. Ionian theorem

    • ...Thomas Noll 16 linked symplectic transformations of integer pairs with continued fractions and developed in 17 a transformational model for scale states based on the Stern–Brocot tree (see also 12 13)...

    Franck Jedrzejewski. Generalized diatonic scales

    • ...I thank Franck Jedrzejewski for an initial reference in word theory and Val´erie Berth´e for helpful comments on [34]...

    Thomas Noll. STURMIAN SEQUENCES AND MORPHISMS A MUSIC-THEORETICAL APPLICATION

    • ...T. Noll introduced this term in [10] and established a recursive construction of these words, which represent the cyclic step-interval patterns of maximally even sets...
    • ...that transforms paths in the Christoffel tree into matrices of the monoid generated byL andR, which is isomorphic toSL(2;N); (See the proof given by T. Noll in [10])...
    • ...Observe that the applicationSL(2;N) ! Q+ given by a c b d = a+d c+d (called mediant ratio by T. Noll in [10]) transforms each matrix Aw into juj jvj ;which is the slope of the dual word w . Denoting by A w the matrix d c b a (flipped the main diagonal), then (A w)=...
    • ...3. The transformational theory for well-formed scales as proposed in [10] is mainly covered by the theory of Sturmian morphisms...
    • ...It is therefore challenging to review the music-theoretical interpretations in [10] within the full algebraic context of Sturmian morphisms...

    Manuel Domínguezet al. WF Scales, ME Sets, and Christoffel Words

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