Glossary

This page lists various terms conventionally used in xenharmony (or in some cases, general music theory as it applies to xen) that can be briefly described.

Don't put idiosyncratic terms here. When using personal terminology in an article, either explain it there or link to an article about your theory that explains the term.

Cent

A cent (abbreviated to c or ¢) is the conventional measurement unit of the logarithmic (perceptual) distance between frequencies; in other words, the size of the interval between them. A cent is defined as a frequency ratio of 2^(1/1200), or a factor of about 1.0005778, such that the octave (2/1) spans exactly 1200 cents, and therefore that each step of 12edo spans exactly 100.

Constant structure

A constant structure (CS; Erv Wilson's term) is a scale such that no two of its interval classes share a common interval.

Pyth[7] is a constant structure, but 12edo diatonic is not.

Detempering

Detempering a tempered scale results in a scale that has pitches in JI (or a temperament that tempers less). Each tempered pitch corresponds to one or more pitches in the detempered scale, which map to the tempered pitch under the temperament.

The Zarlino scale in 5-limit JI is a detempering of Meantone diatonic. Pental blackdye is another detempering of Meantone diatonic, but with some cases of multiple detempered pitches corresponding to a tempered pitch.

Equave

An equave or interval of equivalence is an interval that separates notes that are considered equivalent. Most commonly the octave (2/1), but 3/1, 3/2, and other intervals are sometimes used.

Interval class

An interval class is the set of all intervals that occur as a given number of steps in a given scale. For example, the interval class of 4-step intervals in 12edo diatonic is {700c, 600c}.

Limit

In just intonation, limit has two distinct senses:

• The p-prime limit is the lattice consisting of all JI ratios with primes up to p in their prime factorization. 3/2, 5/3, 7/4, and 49/36 are all in the 7-prime limit, but 11/7 is not.
• The n-odd limit is the set of all JI ratios with both the numerator and the denominator at most n, after removing powers of 2. Equivalently, it is the set of all intervals that appear in the harmonic series scale k:(k+1):...:2k (and all their octave equivalents), where k = n/2 + 1/2. For example, the 15-odd limit is the set of intervals that occur in the harmonic series scale 8:9:10:11:12:13:14:15:16.

The term "limit" without qualification more commonly means prime limit.

Signature

A signature is a list of numbers giving useful but incomplete information about an object. Usually refers to one of:

• a step signature, a list of how many of each step size a scale has; e.g. 4L3m2s.
• a delta signature, a list of frequency increases between adjacent notes measured relative to a reference frequency increase, e.g. +1+1+2 for the chord 6.465:7.465:8.465:10.465.

Superparticular

A superparticular or Delta-1 ratio is a ratio between two whole numbers which differ by 1: e.g. 2/1, 3/2, 4/3, 5/4, etc, representing intervals between consecutive members of the harmonic series. These are distinguished from superpartient ratios (all other rational ratios), which can be classified as Delta-2, Delta-3, etc. by the difference between their numerator and denominator. Note that the ratio between consecutive superparticulars is itself superparticular.

Ternary

A ternary scale is a scale with exactly three step sizes (usually denoted L, m, and s).

Union

The union of two scales/chords is a scale/chord with all pitches that occur in either scale/chord. In other words, it's a shorter way of saying "superimposition".

Examples:

• A cross-set is a union of copies of the same scale placed on different offsets.
• Polysystemic tuning uses a union of multiple systems, for example 5edo and 7edo.

Variety

Variety (or interval variety) refers to how many interval sizes an interval class comes in. We often refer to maximum variety (MV) or strict variety (SV). For example, MOS scales can be defined as scales that are MV2.