Glossary

A concise list of essential terms in xenharmonic music theory

hertz - A unit for frequency; counts per second.

pitch - The property of some sounds that allows them to be ordered on a one-dimensional 'pitch axis'. often measured by the frequency (in hertz) of a sine wave having the same pitch.

cents - A logarithmic units of measure. For a ratio R, 1200 * log2(R).

interval - The distance between two pitches, expressed as a ratio (of their associated frequencies in hertz) or in cents.

scale - An ordered list of intervals.

harmonic distance - A property of intervals other than size; a second type of distance, such that pitches separated by 'short' intervals are more likely to be confused (as by novice singers) than those separated by 'long' intervals. e.g. an octave is 'shorter' than a tritone. related to consonance. e.g. Tenney height.

equave or interval of equivalence - An interval considered to be equivalent to the unison, such as an octave (pitch ratio 2/1 or 1200 cents).

periodic scale - A scale that repeats at an interval of equivalence. Exactly what can be described by a Scala file.

mode - A particular rotation of a scale, e.g. ionian, dorian etc. for the diatonic scale.

just intonation - The set of intervals expressible as rational numbers.

generators - A set of intervals that generate a larger set through linear combination. e.g. the primes {2, 3} generate the Pythogorean scale.

temperament - A mapping from a just intonation subgroup to a set of intervals with fewer generators. expressible as a mapping matrix whose columns are generators of the just intonation and whose rows are generators of the temperament. In particular, the row vectors are called vals.

comma - A rational number that maps to 1/1 in a given temperament.

rank - The number of generators of a set of intervals. e.g. 12-tone equal temperament is rank-1 because it can be generated by the semitone; the Pythagorean scale is rank-2 because it can be generated by the primes {2, 3}

equal temperament - A rank-1 temperament. The temperament-agnostic term is equal-step tuning.

tuning of a temperament - The concrete realization of a temperament, with concrete cent values (for each basis element of the temperament). A temperament's character does not depend on the exact choice of tuning, e.g. a semitone of 100.1 cents produces a scale similar to that produced by a semitone of 100.0 cents.

optimal tuning - A tuning that optimizes some desired property (usually the accuracy of the temperament's approximations to just intonation). e.g. POTE tuning.

subtend - Of an interval size, to occur as a certain number of steps in a given scale.

In the diatonic scale, the perfect fifth always subtends 4 steps.