Abstract
Symmetry forms the foundation of combinatorial theories and algorithms of enumeration such as Möbius inversion, Euler totient functions, and the celebrated Pólya’s theory of enumeration under the symmetric group action. As machine learning and artificial intelligence techniques play increasingly important roles in the machine perception of music to image processing that are central to many disciplines, combinatorics, graph theory, and symmetry act as powerful bridges to the developments of algorithms for such varied applications. In this review, we bring together the confluence of music theory and spectroscopy as two primary disciplines to outline several interconnections of combinatorial and symmetry techniques in the development of algorithms for machine generation of musical patterns of the east and west and a variety of spectroscopic signatures of molecules. Combinatorial techniques in conjunction with group theory can be harnessed to generate the musical scales, intensity patterns in ESR spectra, multiple quantum NMR spectra, nuclear spin statistics of both fermions and bosons, colorings of hyperplanes of hypercubes, enumeration of chiral isomers, and vibrational modes of complex systems including supergiant fullerenes, as exemplified by our work on the golden fullerene C150,000. Combinatorial techniques are shown to yield algorithms for the enumeration and construction of musical chords and scales called ragas in music theory, as we exemplify by the machine construction of ragas and machine perception of musical patterns. We also outline the applications of Hadamard matrices and magic squares in the development of algorithms for the generation of balanced-pitch chords. Machine perception of musical, spectroscopic, and symmetry patterns are considered.
Highlights
We laid the combinatorial foundations of music theory and spectroscopy and pointed out a number of similarities between spectroscopy and music theory
Symmetry is shown to play a critical and integral part in both of these disciplines as the concepts of rhythm, proportion, and harmony, when expressed in mathematical group theoretical and combinatorial structures, find direct applications to both music theory and spectroscopy, as established in this review. Spectroscopic concepts such as the blue and red shifts can be directly applied to musical scales in order to transform one musical scale into another through such tonal shifts. Combinatorial techniques such as the inclusion–exclusion principle, Möbius inversion, Pólya’s theory of counting and its generalization to encompass all irreducible representations of the symmetry group, Latin squares, Hadamard matrices, magic squares, ordered partitions, and many more combinatorial structures are shown to find extensive applications both in music theory and spectroscopy
In the context of spectroscopy, we demonstrated the applications of such combinatorial and group theoretical techniques to giant golden fullerene domes by way of providing elegant solutions to the complex problems of machine construction of their MQ-NMR, ESR, and vibrational modes with applications up to supergiant fullerene C150,000
Summary
The above enumeration scheme can be generalized to scales of other lengths such as the pentatonic, hexatonic, and those that are symmetrical as well as asymmetrical. Analogous to the hexatonic enumeration, by replacing all binomials by 1 in the above expression, we obtain the total number of pentatonic symmetric scales as 236. The total generating function for all of the non-kinky musical scales is given by the product of the ascent and descent inventories: RIa × RId = 1 + x + 11x2 + 54 x3 + 142 x 4 + 236 x5 + 204 x6 + 72 x7 × 1 + 11y2 + 54y3 + 142y4 + 236y5 + 204y6 + 72y7 , (14). The coefficient of xm yn in the generating function RIa × RId enumerates the number of musical scales with m-tonic notes in the ascent and n-tonic notes in the descent. The common pentatonic scale of the Asian music system, known as Bhoopali in the north Indian music and Mohanam in the south Indian music system, is covered by the term MN in Equation (9)
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