Group theoretic description of just intonation

In this paper we present a group theoretic description of just intonation. All possible musical intervals belonging to 5-limit just intonation can be represented by the mathematical group {2 p 3 q 5 r |p,q,r ∈Z}. Considering only the intervals within one octave, an isomorphism with the 2-dimensional lattice Z 2 can be made. Plotting the intervals on the lattice according to the isomorphism, we can identify certain connected and convex sets of elements that represent the major and minor diatonic scale and the 12-tone chromatic scale. These sets of elements remain convex and the area each set spans remains invariant under lattice transformations. The fact that the major, minor and chromatic scale arise naturally from our representation of 5limit just intonation triggers discussion about the origin of scales and might suggest that those scales have a mathematical origin. With this, we challenge the framework of the 12-tone system. Our representation of 5-limit just intonation is compared with Balzano's `thirds-space ' in which he builds the intervals of the 12-tone scale from major and minor thirds also using group theory. This comparison gives a new insight on the compromises regarding intonation that are made when introducing a 12-tone temperament. 1

Pitch spelling: Investigating reductions of the search space

Abstract — Pitch spelling addresses the question of how to derive traditional score notation from pitch classes or MIDI numbers. In this paper, we motivate that the diatonic notes in a piece of music are easier to spell correctly than the non-diatonic notes. Then we investigate 1) whether the generally used method of calculating the proportion of correctly spelled notes to evaluate pitch spelling models can be replaced by a method that concentrates only on the nondiatonic pitches, and 2) if an extra evaluation measure to distinguish the incorrectly spelled diatonic notes from the incorrectly spelled non-diatonic notes would be useful. To this end, we calculate the typical percentage of pitch classes that correspond to diatonic notes and check whether those pitch classes do indeed refer to diatonic notes in a piece of music. We explore extensions of the diatonic set. Finally, a good performing pitch spelling algorithm is investigated to see what percentage of its incorrectly spelled notes are diatonic notes. It turns out that a substantial part of the incorrectly spelled notes consist of diatonic notes, which means that the standard evaluation measure of pitch spelling algorithms cannot be replaced by a measure that only concentrates on non-diatonic notes without losing important information. We propose instead that two evaluation measures could be added to the standard correctness rate to be able to give a more complete view of a pitch spelling model. I

Measures of consonance in a goodness-of-fit model for equal-tempered scales

In this paper a general model is described which measures the goodness of equal-tempered scales. To investigate the nature of this ’goodness’, the consonance measures developed by Euler and Helmholtz are discussed and applied to two different sets of intervals. Based on our model, the familiar 12-tone equal temperament does not have an extraordinary goodness. Others, such as the 19-tone equal temperament look as least as promising. A surprising outcome is that when intervals from the just minor scale are chosen to be approximated by an  -tone equal temperament system, good values for   are ¡£¢¥¤¦¤§¢¥¤© ¨ and �¦ � , rather than the commonly used  ��� � ¤.

Compactness in the Euler-lattice: A parsimonious pitch spelling model

  Compactness and convexity have been shown to represent important principles in music, reflecting a notion of consonance in scales and chords, and have been successfully applied to well-known problems from music research. In this paper, the notion of compactness is applied to the problem of pitch spelling. Pitch spelling addresses the question of how to derive traditional score notation from 12-tone pitch classes or MIDI. This paper proposes a pitch spelling algorithm that is based on only one principle: compactness in the Euler-lattice. Generally, the goodness of a pitch spelling model is measured in terms of its spelling accuracy. In this paper, we concentrate on the parsimony, cognitive plausibility and generalizability of the model as well. The spelling accuracy of the algorithm was evaluated on the first book of Bach s Well-tempered Clavier and had a success rate of 99.21%. A qualitative discussion of the model s cognitive plausibility, its parsimony and its generalizability is given.   

Convexity and the well-formedness of musical objects

It is well known that subsets of the two-dimensional space Z 2 can represent prominent musical and music-theoretical objects such as scales, chords and chord vocabularies. It has been noted that the major and minor diatonic scale form convex subsets in this space. This triggers the question whether convexity is a more widespread concept in music. The current paper systematically investigates the convexity for a number of musical phenomena among which scales, chords and (harmonic) reduction. We hypothesise that the notion of convexity may be a covering concept of musical phenomena and possibly reflects other mathematical properties of these musical structures. Furthermore, convexity can be used in a pitch spelling model. ∗ corresponding author