Glossary

A concise list of essential terms in xenharmonic music theory

Hertz (Hz)

A unit of frequency defined as one cycle 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 unit of measure of intervals. For a ratio R, 1200 · log₂(R).

interval

The distance between two pitches, expressed as a ratio (of their associated frequencies in hertz) or as a logarithmic measure (e.g. 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 (interval of equivalence)

An interval considered to be equivalent to the unison, such as an octave (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 (JI)

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 Pythagorean scale.

just intonation subgroup

The set of JI ratios obtainable by stacking (finitely many) copies of a finite set of JI generators up or down. For example, 7/6 and 49/32 are both in the 2.3.7 subgroup, the set of JI ratios obtained by stacking copies of 2/1, 3/1 and 7/1 up and down.

temperament

A mapping from a domain (such as a prime limit or 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 or maps.

comma

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

rank

The number of generators of a set of intervals, i.e. the set's dimensionality (not to be confused with the dimensionality of the temperament itself). For example, 12edo 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

A concrete realization of a(n abstract) 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 of a temperament that optimizes some desired property (usually the accuracy of the temperament's approximations to just intonation). e.g. CTE 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.

1, 2, 3...-step

A 1, 2, 3, ...-step interval in any scale.

delta-rational (DR)

Of a chord, having at least two dyads which represent frequency differences which have a rational ratio. This property is thought to result in synchronized interference beating between the dyads.

scale signature

An expression using the step sizes of a scale, where the coefficient in front of each step tells you how many of that step size the scale has. A mos scale has a scale signature of aL bs for step sizes L > s. Other scales may have more complex signatures such as 5L 2M 3s.

n-form

An n-note constant structure scale.

See also

• Acronyms