arXiv subjects
Publications and source records attributed to Emmanuel Amiot.
The import of the magnitude of fourier coefficients of a pitch class set is fairly well known. This paper deals with the angular component of these compelx numbers, the phase. It enables to shed new light on triads, the Tonnetz, and continuous gestures between diverse pc-sets, even those with different cardinalities.
Several musical scales, like the major scale, can be described as finite arithmetic sequences modulo octave, i.e. chunks of an arithmetic sequence in a cyclic group. Hence the question of how many different arithmetic sequences in a cyclic group will give the same support set. We prove that this number is always a totient number and characterize the different possible cases. In particular, there exists scales with an arbitrarily large number of different generators, but none with 14 generators. Some connex results and extensions are also given, for instance on characterization via a Discrete Fourier Transform, and about finite or infinite arithmetic sequences in the torus R/Z.
The problem of phase retrieval is a difficult one which remains far from solved. Two homometric sets are always connected by way of a convolution product by some spectral unit, though not necessarily in a unique way. Here we elucidate one small aspect, the subgroup of spectral units with finite order. Its elements are completely characterized by relations between their eigenvalues. This sheds some light on the beltway problem.
This paper, in french, gives a new approach to the concept of Maximally Even Sets based on discrete Fourier transform, with several elementary but interesting and previously unpublished results. Maximally Even Sets have been invented by musicologists but have been found to bear deep relationships to other areas of science, such as the Ising model in Physics. They describe economically and characterise many famous 'scales' or subsets of the cyclic group as modelised here.