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Taking Music Seriously: on the Dynamics of 'Mathemusical' Research with a Focus on Hexachordal Theorems

Moreno Andreatta, Corentin Guichaoua, Nicolas Juillet

Abstract

After presenting the general framework of 'mathemusical' dynamics, we focus on one music-theoretical problem concerning a special case of homometry theory applied to music composition, namely Milton Babbitt's hexachordal theorem. We briefly discuss some historical aspects of homometric structures and their ramifications in crystallography, spectral analysis and music composition via the construction of rhythmic canons tiling the integer line. We then present the probabilistic generalization of Babbitt's result we recently introduced in a paper entitled ''New hexachordal theorems in metric spaces with probability measure'' and illustrate the new approach with original constructions and examples.

Taking Music Seriously: on the Dynamics of 'Mathemusical' Research with a Focus on Hexachordal Theorems

Abstract

After presenting the general framework of 'mathemusical' dynamics, we focus on one music-theoretical problem concerning a special case of homometry theory applied to music composition, namely Milton Babbitt's hexachordal theorem. We briefly discuss some historical aspects of homometric structures and their ramifications in crystallography, spectral analysis and music composition via the construction of rhythmic canons tiling the integer line. We then present the probabilistic generalization of Babbitt's result we recently introduced in a paper entitled ''New hexachordal theorems in metric spaces with probability measure'' and illustrate the new approach with original constructions and examples.
Paper Structure (17 sections, 5 theorems, 20 equations, 3 figures, 2 tables)

This paper contains 17 sections, 5 theorems, 20 equations, 3 figures, 2 tables.

Table of Contents

  1. An introduction to "mathemusical" research
  2. Around the domain of structural music information research
  3. Approaching spectral theory and homometry musically
  4. A digression on Fuglede's spectral conjecture and its application to the discrete integer line
  5. A special case of homometric structures: Babbitt's theorem
  6. Generalizing Babbitt's hexachordal theorem
  7. Naive illustration of the hexachordal phenomenon and heuristic proof in the case of the sphere $\boldsymbol{\mathbb{S}^2}$
  8. Statement of three generalized hexachordal theorems
  9. Metric probability spaces satisfying (CVC)
  10. Discrete and continuous CVC spaces
  11. Constructions with CVC spaces
  12. Remarks on the theorems
  13. Remark on weakening $\boldsymbol{\mu(A)=1/2}$ in Theorem \ref{['thm1']}
  14. Remarks on $\boldsymbol{({\rm Ind})}$ in Theorem \ref{['them3']}
  15. Remark on $\boldsymbol{({\rm Ind})\Rightarrow({\rm Hex}')}$ in Theorem \ref{['them3']}
  16. ...and 2 more sections

Key Result

Proposition 1.2

Let $A$, $B$ be two subsets of $\mathbb{Z}_n$. Then $\mathbb{Z}_n =A\oplus B$ if and only if $Z_A \cup Z_B = \mathbb{Z}_n \backslash \{0\}$ and $\#A \times \#B=n.$

Figures (3)

  • Figure 1: A diagram showing the underlying "mathemusical" dynamics between music and mathematics through computer-science, also including some epistemological and cognitive aspects.
  • Figure 2: A more detailed perspective on the "mathemusical" research diagram as presented in Figure \ref{['fig:diagram1']}, with the indication of the three main ingredients of the dynamics (namely "formalization", "generalization" and "application").
  • Figure 3: Two points randomly picked in the bright region of the sphere have distance distributed equally as the one between points picked in the dark region (made of two caps).

Theorems & Definitions (16)

  • Conjecture 1.1: Fuglede
  • Definition 1.1
  • Proposition 1.2
  • Definition 2.1: constant volume condition
  • Theorem 2.2: hexachordal theorem for metric probability spaces
  • Theorem 2.3: characterization for metric probability spaces
  • Theorem 2.4: characterization for abstract probability spaces
  • Definition 3.1: transitive space
  • Remark 3.2: nontransitive CVC spaces
  • Example 3.3: products
  • ...and 6 more