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.

Basis

A basis (pl. bases) for a JI group or a similar group is a list of intervals called generators such that

1. anything in the group can be written as a stack of intervals of the basis or their inverses (possibly with repetition)
2. the list is non-redundant in the sense that there is only one way to write any particular interval in the group as a stack of generators.

Examples:

• [2, 3/2, 5/4] is a basis for the 5-limit; so is [2, 3, 5]
• [2, 5/3] and [2, 9, 5] are not bases for the 5-limit, on account of not satisfying 1
• [2, 3, 5, 15] is not a basis for the 5-limit, on account of not satisfying 2

JI groups are denoted using basis elements separated by full stops, for example 2.5.11/3.

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.

Comma

Comma may refer to:

1. a small JI interval. A fundamental fact about JI is that no stack of one prime is a stack of any other set of primes. Commas hence occur frequently in stacking-based JI.
2. The commas of a regular temperament are the intervals it tempers out, which can all be written as stacks of a certain number of commas known as the comma basis. "Tempering out" means that all JI ratios that are separated by that comma are equated, e.g. tempering out 81/80 equates 81/64 and 5/4, but it also equates 40/27 and 3/2.
3. An interval region of intervals around 20 cents, less than about 30 cents.

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.

Pythagorean diatonic 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.

Extension

An extension of a temperament is a temperament that interprets the tempered intervals of the original temperament as a larger JI group. The opposite of an extension is a restriction, which interprets a temperament as a subset of the original JI group. Temperaments of different ranks are not considered extensions or restrictions of one another.

A weak extension uses a larger set of tempered intervals (so that Mothra, a 2.3.5.7 temperament that tempers 1029/1024 in addition to 81/80, is a weak extension of pental Meantone, as pental Meantone natively doesn't have something that is one-third of a 3/2), whereas a strong extension uses the same set of tempered intervals (so that Septimal Meantone is a strong extension of pental Meantone). Weak extensions are often created by dividing the original period or (a choice of) generator into equal parts and then interpreting the split parts; in the Mothra example, the 3/2 Meantone generator is split into 3 parts, and then (3/2)^(1/3) is interpreted as 8/7. If you don't interpret the new intervals of a weak extension, the result is contorsion.

Harmonic mode

A harmonic segment of the form n::2n, considered as an octave-equivalent scale. For example, mode 7 of the harmonic series is 7:8:9:10:11:12:13:14.

Harmonic segment

Any finite set of consecutive harmonics in the harmonic series. Can be denoted m::n. For example, 5:6:7:8:9:10 is written 5::10.

Harmonic series

The infinite sequence of whole-number frequency multiples, called harmonics, above a fundamental frequency. The harmonics of 110 Hz are:

• 1st harmonic (fundamental): 110 Hz
• 2nd harmonic: 220 Hz
• 3rd harmonic: 330 Hz
• 4th harmonic: 440 Hz
• 5th harmonic: 550 Hz
• 6th harmonic: 660 Hz
• ...

Every JI interval occurs in the harmonic series as the pitch difference between some pair of harmonics.

Differences in relative loudnesses of various harmonics above a note, as well as deviations from mathematically exact harmonics (called inharmonicity), are perceived as different timbres of the same note.

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}.

JI group

A JI group is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called subgroups, as they can be seen as subgroups (subsets that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not prime-limits, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a JI lattice, though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

JI groups are denoted by generators separated by full stops: for example, 2.3.5.7 denotes the 7-prime-limit.

Prime-limits are JI groups. Some non-prime-limit JI groups are 2.3.7 or 2.5/3.11/3.

Groups can be generalized to non-JI generators, for example 2.φ.

A regular temperament starts with a JI group and maps the group to a tempered group. For example, Meantone maps 2.3.5 to the group generated by tempered 2 and tempered 3/2.

Mathematically, a group is a set with

• a binary operation * (for all group elements g and h, g * h is also an element of the group)
• the binary operation * is associative (thus no parentheses are needed when writing the group operation on more than two elements)
• an identity element: a unique element e such that g * e = e * g = g for all g in the group
• an inverse element for every element: every g corresponds to a unique element g^-1 such that g * g^-1 = g^-1 * g = e

A subgroup generated by a subset of a group is the group formed by iterating the binary operation on elements in the subset. Equivalently, it is the smallest subgroup of the larger group containing that subset.

Groups in xen theory are typically a much more specific type of groups, namely free abelian groups.

Limit

In just intonation, limit most commonly has two distinct senses:

• The p-prime-limit is the set 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 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; 21/16 is not in the 15-odd-limit.

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

Monzo

A monzo is a vector (list of coordinates) representing a JI ratio, whose coordinates are (usually) prime exponents. Also called an interval vector or a prime count vector.

Example: 81/80 = 3^4/(2^4 * 5^1) = 2^-4 * 3^4 * 5^-1 can be written in monzo form as [-4 4 -1⟩.

Pitch class

Assuming an equave, two pitches or two intervals belong to the same pitch class if they are separated by a multiple of the equave. Pitch class space is a circle, whereas pitch space is a line.

Lattice diagrams of JI or tempered scales show the pitches in a pitch-class lattice, a lattice one dimension lower than the original JI group, where equave differences are ignored.

Rank

The term rank just means "dimensionality". The rank of a temperament is the dimension of the group of tempered JI ratios under that temperament. Meantone has rank (dimension) 2 because any interval in Meantone can be written as a stack of some number of tempered octaves and some number of tempered fifths.

Scale

A scale is a collection of pitches; two scales are considered the same scale if they only differ by transposition. Unlike chords, scales are usually periodic, i.e. the same pattern of intervals repeats at some interval called the equave. On XenBase, scales are periodic unless stated otherwise. A scale can be visualized as a set of points in the circle of equave-equivalent pitch classes.

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.

Val

A val (short for "valuation") is a vector whose coordinates are step mappings of primes in an equal temperament. It can mathematically be called a "covector", since it is a kind of a vector "dual" (complementary) to interval vectors. Taking the dot product of a monzo (for a JI ratio) and a val (for an equal temperament) shows how the equal temperament maps the JI ratio.

Example: 12et maps 2/1 to 12 steps, 3/1 to 19 steps (reduced: 7 steps), and 5/1 to 28 steps (reduced: 4 steps). We write this in val form as ⟨12 19 28]. Vals can be evaluated at monzos by multiplying each pair of corresponding entries and summing the results together. This can be seen as, for a monzo with entries m and a val with entries v, "stepping" by each v m times for its corresponding m. In linear algebra, this is equivalent to taking the dot product. This is denoted by ⟨val][monzo⟩. Evaluating this val at [-4 4 -1⟩ (the monzo for 81/80) shows that 12et tempers out 81/80:

⟨12 19 28][-4 4 -1⟩ = 12 * -4 + 19 * 4 + 28 * -1 = -48 + 76 - 28 = 0.

Patent vals are the most common kinds of vals to consider. The "patent" means that the closest approximations in the edo tuning in question are used for the step mappings. The above val is the 12edo patent val in the 5-limit. An example of a non-patent val is ⟨12 19 27], since the closest approximation to 5/1 in 12edo is not 27 steps, but 28 steps.

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.