Title: Musical intervals under 12-note equal temperament: a geometrical interpretation,
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.