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Thirty-one tone equal temperament was studied by Christiaan Huygens, a prominent Dutch mathematician and scientist who lived in The Netherlands from 1629 to 1695. This temperament was rediscovered ...

In the mid-1970s, Alain Louvier worked out microtonal scales called modes of progressive transposition and used them in many musical works. These modes have similar properties to major modes and are related to diatonicism. Some of them were known by Ivan Wyschnegradsky and Georgy Rimsky–Korsakov, the grand-son of Nikolai. Deep scales are well known in diatonic theory, and are special cases of these modes. However, their algebraic structure is not known. Although the diatonic theories have been developed by many musicologists, such as Agmon, Balzano, Carey, Clampitt, Noll, Zweifel and others, many questions remain open. In this paper, we describe some studies on microtonality, published over the last century, and we review what is known and what remains to understand in this field, in both theoretical and compositional aspects. In the first section, we study the modes called by Alain Louvier “imperfect modes”, a special case of modes of progressive transposition. He used them in several important works as Le Clavecin non tempere (1973), Canto di Natale (1976) and Anneaux de lumiere (1983), written in the 24 tone equal temperament. The mathematical study would be to find a criterion to easily determine all the modes of progressive transposition in any equal temperament, and in particular to determine all deep scales. In the second section, we study the enumeration of the modes of limited transposition, also known as Messiaen modes in any equal temperament. In the last section, we present another classification of the modes related to the plactic monoid. Along this paper, we question what, if they exist, the microtonal diatonicism and the microtonal modality could be.

A finite subset A of integers tiles the discrete line ℤ if the integers can be written as a disjoint union of translates of A. In some cases, necessary and sufficient conditions for A to tile the integers are known. We extend this result to a large class of nonperiodic tilings and give a new formulation of the Coven–Meyerowitz reciprocity conjecture which is equivalent to Fuglede's conjecture in one dimension.

The article studies the problem of generalizing the concept of ‘diatonic scale’ for a given ambient chromatic of N tones: ‘Which subset A⊂ℤ N shall be considered as a generalized diatonic scale?’ Each generic type of well-formed scale has exactly two specific manifestations in chromatic universes, which are large ME*-scales, i.e. which are maximally even non-degenerate well-formed scales, whose cardinality exceeds half of the chromatic cardinality. A qualitative distinction between these two large ME*-scales of the same type can be comfortably made on the basis of the shuffled Stern–Brocot tree, which is introduced in Section 2. The shuffled Stern–Brocot tree represents the same abstract binary tree as the traditional Stern–Brocot tree, but has a different planar arrangement. Candidates and final choices for generalized diatonic scales are studied in Section 3. Candidates are those large ME*-scales A⊂ℤ N which are tightly generated by a prime residue class m mod N. According to this property there is no n...

L'article présente deux applications de la théorie des nceuds et des entrelacs à l'univers musical. La première application est une classification des séries de douze sons selon des structures particulières appelées diagrammes de cordes. Cette taxinomie conduit à une nouvelle hiérarchie des structures profondes du lexique dodécaphonique. La chiralité et les degrés de symétrie des formes sérlelles sont directement accessibles sur ces diagrammes qui rŕsument les propriétés intrinsèques de la série, La forme schématique nodale remplit ainsi la double fonction d'integration méronymique et de contrôle combinatoire. La deuxlème application est une approche des problèmes du tempérament musical et de l'analyse des systèmes acoustiques contemporains par coloration des entrelacs. A chaque arc de la représentation planaire d'un nœud est associée une fréquence (appelée aussi couleur) du système acoustique envisagé. Lors du passage d'un arc à un autre par croisement des brins du nœud, la fréquence évolue selon les règles d'une grammaire prédéfinie, L'entrelacs ainsi étiqueté — appelé nœud polychrome — décnt complètement l'accord du système acoustique. Des questions théoriques comme la transposition des échelles tonales dans des univers nontempérés ou les variations micro-intervalliques dans Ie champ compositionnel s'etudient de manière naturelle à l'aide de ces nouvelles catégories musicales.

I found a surprising number of new musical patterns in formations as simple as the permutations, sums and subsets already discussed, and in the case of my “counting music”, even simpler ones, but I was always interested in finding new directions in all this. One new direction presented itself quite unexpectedly in 2003, when I heard a piece by a young Dutch composer, Samuel Vriezen. Using a scale of only 11 notes, Vriezen constructed 11 five-note chords in such a way that each chord had exactly two notes in common with each other chord. I asked the composer how he had ever found such a group of chords, and he told me it was not too complicated. He thought I could construct such a system myself, if I thought about it a bit.