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A concise list of essential terms in xenharmonic music theory

Hertz - unit for frequency; counts per second

pitch - 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 - logarithmic units of measure. for a ratio R, 1200 * log2(R)

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

scale - an ordered list of intervals; exactly what can be described by a Scala file

harmonic distance - 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

interval of equivalence - an interval considered to have small or zero harmonic distance, such as an octave (pitch ratio 2/1 or 1200 cents)

periodic scale - a scale that repeats at an interval of equivalence

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

just intonation - a set of intervals considered consonant and 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 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

tuning - the generators of a 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