# Musical intervals under 12-note equal temperament: a geometrical interpretation

## TL;DR

This paper presents a geometric interpretation of musical intervals in 12-tone equal temperament, analyzing the combinatorics and symmetries of pitch-class sets using Euclidean spaces and lattice polytopes, and compares these with group theory results.

## Contribution

It introduces a novel geometric framework for understanding $n$-chords and their symmetries, extending combinatorial analysis with a geometric perspective and including palindrome considerations.

## Key findings

- Number of $n$-chords and set classes determined
- Geometric interpretation via inclined $n$-hedrons established
- Comparison with group theory results performed

## Abstract

Musical intervals in multiple of semitones under 12-note equal temperament, or more specifically pitch-class subsets of assigned cardinality ($n$-chords) are conceived as positive integer points within an Euclidean $n$-space. The number of distinct $n$-chords is inferred from combinatorics with the extension to $n=0$, involving an Euclidean 0-space. The number of repeating $n$-chords, or points which are turned into themselves during a circular permutation, $T_n$, of their coordinates, is inferred from algebraic considerations. Finally, the total number of $n$-chords and the number of $T_n$ set classes are determined. Palindrome and pseudo palindrome $n$-chords are defined and included among repeating $n$-chords, with regard to an equivalence relation, $T_n/T_nI$, where reflection is added to circular permutation. To this respect, the number of $T_n$ set classes is inferred concerning palindrome and pseudo palindrome $n$-chords and the remaining $n$-chords. The above results are reproduced within the framework of a geometrical interpretation, where positive integer points related to $n$-chords of cardinality, $n$, belong to a regular inclined $n$-hedron, $\Psi_{12}^n$, the vertexes lying on the coordinate axes of a Cartesian orthogonal reference frame at a distance, $x_i=12$, $1\le i\le n$, from the origin. Considering $\Psi_{12}^n$ as special cases of lattice polytopes, the number of related nonnegative integer points is also determined for completeness. A comparison is performed with the results inferred from group theory.