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 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 condition 1.
• [2, 3, 5, 15] is not a basis for the 5-limit, on account of not satisfying condition 2.

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

Categories: RTT, JI, Math terms

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.

Categories: Core knowledge

Chord

A chord is a finite set of (usually three or more) pitches, often implying a context when the pitches are played together. Two chords are usually considered the same chord if they only differ by transposition.

Categories: Core knowledge

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 which suffice to determine every comma that is tempered out or every pair of intervals that is equated. "Tempering out" means that all JI ratios/stacks that are separated by that comma are equated, e.g. tempering out 81/80 not only equates 81/64 and 5/4 but also equates 40/27 and 3/2. This follows from the principles of regular temperament.
3. An interval region of intervals around 20 cents, less than about 30 cents.

Categories: JI, RTT

Complexity

The complexity of a rank-2 temperament is fairly easy to intuit: it is how many stacked generators are needed to reach simple JI ratios. There is often a tradeoff between simplicity and accuracy in temperaments. For example, 5-limit Schismic is a more accurate but more complex temperament than 5-limit Meantone, since more generators are needed to reach 5/4 in the former.

Categories: RTT

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:

	1	2	3	4	5	6
1/1	9/8	81/64	4/3	3/2	27/16	243/128
9/8	9/8	32/27	4/3	3/2	27/16	16/9
81/64	256/243	32/27	4/3	3/2	128/81	16/9
4/3	9/8	81/64	729/512	3/2	27/16	243/128
3/2	9/8	81/64	4/3	3/2	27/16	16/9
27/16	9/8	32/27	4/3	3/2	128/81	16/9
243/128	256/243	32/27	4/3	1024/729	128/81	16/9

But 12edo diatonic is not, because 600c is both a 3-step interval and a 4-step one:

	1	2	3	4	5	6
0\12	200.0	400.0	500.0	700.0	900.0	1100.0
2\12	200.0	300.0	500.0	700.0	900.0	1000.0
4\12	100.0	300.0	500.0	700.0	800.0	1000.0
5\12	200.0	400.0	600.0	700.0	900.0	1100.0
7\12	200.0	400.0	500.0	700.0	900.0	1000.0
9\12	200.0	300.0	500.0	700.0	800.0	1000.0
11\12	100.0	300.0	500.0	600.0	800.0	1000.0

Some find CS a desirable property for JI scales, and some people find constant structure scales easier to navigate on keyboards.

A JI scale being a CS is not equivalent to it being a detempering of an equal temperament. The latter implies the former, but not vice versa.

Categories: Scales

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.

Categories: RTT, JI

Enharmonic

Sense 1

Two notes or intervals are enharmonic, or enharmonically equivalent, if they map to the same degree of the chromatic scale (the 12-note MOS scale generated by a perfect fifth). This can be generalized to pairs of notes separated by the difference between a chroma and a small step in a given scale, where enharmonic intervals are separated by a diesis, and can be equated by tempering out said diesis.

Sense 2

A 17- or 19-note MOS scale generated by a perfect fifth, which assigns enharmonically equivalent diatonic intervals their own scale degrees by making the diatonic diesis a small scale step. Schismic[17] is usable as a scale for Schismic temperament.

Sense 3

A Greek scale in which the lower two of the three intervals of a tetrachord are less than a semitone each.

Sense 4 (proscribed)

In 12edo, enharmonic notes in sense 1 are equated, which has led to a secondary use of "enharmonic" to refer to other equations between notes of a scaleform in some tuning system (such as B# = Cb in 19edo). This particular use is discouraged due to the potential for confusion with other meanings of this already overloaded term.

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.

Categories: Core knowledge

Extension

An extension of a temperament is a temperament that interprets the tempered intervals of the original temperament within a larger JI group. A weak extension introduces new tempered intervals in addition to those of the original temperament, whereas a strong extension uses the same set of intervals as the original temperament. The opposite of an extension is a restriction, which interprets a temperament as a subset of the original JI group, and strong and weak restrictions are defined similarly.

For instance, Meantone introduces 5-limit interpretations of intervals on a chain of tempered fifths by making the equivalence (3/2)⁴ = 2² × 5/4 (tempering out the comma 81/80 and finding 5 at 4 fifths up). But if the chain of fifths is continued further, 7-limit harmonies can be introduced: (3/2)² × (5/4)² = 2 × 7/4, which can be worked out to place 7 at 10 fifths up, a mapping of 7 known as septimal Meantone, which is a strong extension of 5-limit 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. As an example, Mothra is a temperament where the 3/2 Meantone generator is split into 3 parts, and then (3/2)^(1/3) is interpreted as 8/7. It is a weak extension of pental Meantone, as Meantone natively doesn't have something that is one-third of a 3/2, to the 7-limit. If you don't interpret the new intervals of a weak extension, the result is called contorsion.

Note that temperaments of different ranks are not considered extensions or restrictions of one another.

For this wiki's guidelines on what extensions a given temperament name refers to, see XenReference:Guidelines.

Categories: RTT, Somewhat technical

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.

Categories: JI

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.

Categories: JI

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.

Categories: JI, Core knowledge

Interval class

An interval class or generic interval is the set of all intervals that occur as a given number of steps in a given scale. For example, the interval class of fifths (4-step intervals) in 12edo diatonic is {700c, 600c}. Sometimes called an ordinal, because these are called ordinal numbers in conventional diatonic theory: "seconds", "thirds", etc. Other schemes such as TAMNAMS use a 0-indexing scheme: "1-step" for "seconds", "2-step" for "thirds", etc. See also #k-step.

Categories: Scales

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 of a group 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 (called basis elements or formal primes in this context) separated by full stops: for example, 2.3.5.7 denotes the 7-prime-limit. Usually, the first basis element is assumed to represent the equave: "3.2.5" would be a version of 2.3.5 that repeats on the tritave, though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group.

Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as 2.3.7), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.

Groups can be generalized to non-JI generators, for example 2.√6 (representing a chain of perfect hemififths), or 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.

Categories: JI, RTT, Math terms

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.

Categories: JI

Linearly independent

A set of vectors (such as a set of monzos or a set of vals) is linearly independent if no vector in the set is redundant: no nonzero multiple of a vector can be written as a sum of multiples of other vectors. In Xen Reference we will often shorten this to independent. In other sources the term co-unique may be used. This is technically $\mathbb{\mathbb{Z}}$-linear independence; $\mathbb{\mathbb{Z}}$-modules and abelian groups are the same concept.

Examples (for vals):

• ⟨12 19 28] and ⟨19 30 44] (12edo and 19edo patent vals in the 5-limit) are independent.
• ⟨12 19 28], ⟨19 30 44], and ⟨31 49 72] are not independent, since the 31edo val is a sum of the 12edo and 19edo patent vals. We say that three vectors are collinear if they taken together are not linearly independent though any two of them are.
• ⟨24 38 96] and ⟨36 57 84] are not independent, since they share a common multiple.

Examples of where this concept shows up in RTT:

• Basis elements for any applicable group must be independent.
• Two independent vals (equal temperaments) determine a rank-2 temperament, three independent vals determine a rank-3 one, ...

Categories: RTT, Math terms, Somewhat technical

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⟩.

Categories: RTT

Natave

The frequency ratio e/1. Sometimes called a neper. Mainly used in theoretical xen math.

Neji

A neji ("near-equal/equivalent JI") is a (possibly somewhat loose) JI approximation to a non-JI scale (often an edo), usually a subset of a chosen harmonic mode. The term was introduced by Zhea Erose.

Nejis are usually written as enumerated chords (i.e. written in the form a:b:...:z in ascending order): for example, the 12edo neji used in Zhea Erose's Eurybia is 22:23:25:26:28:30:31:33:35:37:39:42:44.

Categories: JI

Period

Period has the following related but different senses:

• The smallest unit at which a given scale repeats — a fraction of the equave but not necessarily the equave itself.
  • Example: Pentawood (5L5s, LsLsLsLsLs) has period 1\5 (240c).
• One of the generators of a regular temperament, specifically chosen to be a fraction of the equave (usually 2/1).
  • Example: The temperament Blackwood has period 1\5.

The two senses are related in that a multiperiod scale or equal division often supports a multiperiod temperament interpretation, and a multiperiod temperament requires the equal division that supports it to be divisible by some number (namely, the number of periods in the equave).

Categories: Scales, RTT

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.

Categories: Core knowledge

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. A temperament like 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. Any equal tuning is rank 1 because all intervals in an equal tuning are a stack of that tuning's step size.

Rank-2 temperaments deserve special mention as they can be described as stacking a single generator against a period. As a result, a very clear method for constructing scales from rank-2 temperaments exists, that being forming a MOS from the temperament's generator and period, which is quite nontrivial to generalize to systems of higher rank.

Categories: RTT, Math terms

Regular temperament

Main article: Regular temperament

A regular temperament (often just temperament) is a way of assigning JI interpretations (from a chosen JI group) to intervals in a non-JI tuning. We assign the interpretations so that the stack of two JI ratios gets assigned to the stack of the corresponding tempered versions of the two ratios. This is what makes a regular temperament "regular".

If you know what notes of a tempered tuning the basis generators of a chosen JI group get assigned to, that suffices to determine the interpretations assigned to any particular interval (provided that every interval is indeed interpreted, as in the overwhelming majority of practical cases). This is how vals and mappings for regular temperaments work — they specify what tempered notes correspond to the basis elements of the JI group.

The study of regular temperaments is called regular temperament theory (RTT).

Categories: RTT

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 XenReference, scales are periodic unless stated otherwise. A scale can be visualized as a set of points in the circle of equave-equivalent pitch classes.

Categories: Core knowledge, Scale

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.

Categories: Somewhat technical

k-step

An abbreviation for "k-step interval". For example, the fifth in the diatonic scale is a 4-step. See also #Interval class.

Categories: Scales

Square-superparticular

A square-superparticular or square-particular is a superparticular of the form

$\frac{k^2}{k^2-1}=\frac{k}{k-1}\frac{k}{k+1}=\frac{\frac{k}{k-1}}{\frac{k+1}{k}},$

denoted Sk in xen math.

A square-superparticular is the difference between consecutive suparparticulars. When a square-superparticular Sk is tempered out, it makes harmonics k - 1, k, and k + 1 equally spaced. For example, tempering out S9 = 81/80 makes harmonics 8, 9, and 10 equally spaced. Factoring a comma into a product of square-particulars, called an S-expression, is often helpful for understanding it.

The ratio between two consecutive square-superparticulars is called an ultraparticular, which has the form Sk/S(k + 1). Tempering out an ultraparticular equates the differences between three consecutive superparticulars.

Categories: RTT

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.

Categories: JI, Math terms

Ternary

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

Categories: Scales

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". If you change the offset between the two scales/chords, taking their union usually yields a different scale/chord.

Examples:

• A cross-set is a union of copies of the same scale placed on different offsets.
  • 15edo pentawood uses two copies of 5edo offset by 1\15; 20edo pentawood uses two copies of 5edo offset by 1\20.
• Polysystemic tuning uses a union of multiple systems, for example 5edo and 7edo.

Categories: Math terms

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.

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 (showing how the equal temperament maps the JI ratio) 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 operation is called 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.

Categories: RTT

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.

Categories: Scales