Xenharmonic Reference - User contributions [en]

14edo

2026-02-03T17:46:50Z

Calvera: fix

'''14edo''', or 14 equal divisions of the octave, is the equal tuning featuring steps of (1200/14) ~≃ 85.714 cents, 14 of which stack to the perfect octave 2/1. While it approximates the 5:7:9:11:17:19 harmony relatively well for its size, it lacks a convincing realization of other low-complexity just intervals. Consequently, [[Delta-rational chord|DR]]-based approaches may be more practically useful.

As a superset of the popular 7edo scale, it offers recognizable triadic harmonies built on subminor, neutral, and supermajor thirds; however, its poor approximation of perfect fourths and fifths gives it a distinctly xenharmonic character.

14edo

2026-02-03T17:45:42Z

Calvera: Created page with "'''14edo''', or 14 equal divisions of the octave, is the equal tuning featuring steps of (1200/14) ~≃ 85.714 cents, 14 of which stack to the perfect octave 2/1. While it approximates the 5:7:9:11:17:19 harmony relatively well for its size, it lacks a convincing realization of other low-complexity just intervals. Consequently, Delta-rational chord-based approaches may be more practically useful. As a superset of the popular 7edo scale, it offers recognizable tri..."

'''14edo''', or 14 equal divisions of the octave, is the equal tuning featuring steps of (1200/14) ~≃ 85.714 cents, 14 of which stack to the perfect octave 2/1. While it approximates the 5:7:9:11:17:19 harmony relatively well for its size, it lacks a convincing realization of other low-complexity just intervals. Consequently, [[DR|Delta-rational chord]]-based approaches may be more practically useful.

As a superset of the popular 7edo scale, it offers recognizable triadic harmonies built on subminor, neutral, and supermajor thirds; however, its poor approximation of perfect fourths and fifths gives it a distinctly xenharmonic character.

Temperament

2025-12-13T02:20:39Z

Calvera: stub for temperament article

'''Temperament''' is a method of tuning musical instruments based on approximating certain [[just intonation|just intervals]] with others, in order to maintain the desired harmony between sounds. Process of application of temperament is called '''tempering''', and it is performed in the '''tempering zone''', usually twelve notes within one octave, which is subsequently repeated in remaining octaves.

Temperaments can be defined as sequences of operations or more formally as mathematical functions. Most commonly used can be divided into [[EDO|equal temperaments]], [[well temperament|well temperaments]] and [[regular temperament|regular temperaments]].

== History ==

In Medieval Western Europe [[Pythagorean tuning]] was the most widely used, in which fifths and fourths are just. At the time, thirds were omitted and considered a dissonance which followed from characteristics of Pythagorean tuning. Instrumental music was based on two-part voice leading - a majestic bourdon and melodic descant. To overcome these limitations, a compromise solution was sought that would allow the use of just thirds without sacrificing the purity of fifths.

That lead to adoption of [[meantone temperament]] (meantone tuning) in the end of the 15th century. Fifths began to be narrowed, which led to the complete disappearance of just fifth, instead impression of purity of harmony was achieved by just thirds; differently than before when the pure fifths could not compensate for wide major thirds. This enabled introduction of third-based chords and development of [[tertian harmony]].

The downside of meantone was that not every diatonic scale known in European music in that time could be played in tune on tempered instruments. In practice, meantone was being modified in a various way by tuners to reduce dissonance created by [[wolf interval|wolf intervals]]. Some theoreticians since 16th century proposed to resolve the problem by use of equal temperaments like 12, 19 or 31 edo.

Adoption of 12-tone equal temperament became widely accepted practice already in the beginnings of 18th century.

[[Category:Core knowledge]]

Comma

2025-12-13T00:56:52Z

Calvera: fixd

'''Comma''' is typically a small interval that, in a [[temperament]], defines the logarithmic difference (division) between two other intervals which are assigned to the same approximate interval. For example: in meantone temperament, the [[syntonic comma]], 81/80, is result of dividing four justly tuned perfect fifths (3/2)<sup>4</sup> and two octaves plus a justly tuned major third 4/1 × 5/4.

[[Category:Core knowledge]]

Comma

2025-12-13T00:51:39Z

Calvera: stub of a comma article

'''Comma''' is typically a small interval that, in a temperament, defines the logarithmic difference (division) between two other intervals which are assigned to the same approximate interval. For example: in meantone temperament, the [[syntonic comma]], 81/80, is result of dividing four justly tuned perfect fifths (3/2)<sup>4</sup> and two octaves plus a justly tuned major third 4/1 × 5/4.

[[Category:Core knowledge]]

Regular temperament

2025-12-13T00:14:45Z

Calvera: comma link

A '''regular temperament''' is a [[temperament]] (an approximation of [[just intonation]]) that is [[consistent]]. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, if you stack the tempered version of 9/8, it must always produce the tempered versions of 81/64, 729/512, 6561/4096, etc. It follows from this that unlimited free modulation must be possible - any interval can be stacked as many times as you like, like in just intonation. In contrast, temperaments that do not meet this requirement (such as [[well temperament]]s) are considered ''irregular''.

[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], meanwhile any just intonation subgroup itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]).

In order to simplify JI, [[comma|commas]] are tempered out - small (or not-so-small) differences between intervals that the regular temperament represents with the unison. For example, in [[meantone]], [[5/4]] (the classical major third) and [[Major third|81/64]] (the pythagorean major third) are equated. Their difference in just intonation is the ratio [[81/80]], so we say that meantone '''tempers out''' 81/80. As a consequence, due to the rules of a regular temperament, 6561/6400, 531441/512000, etc (ratios obtained by stacking 81/80) are also tempered out, but knowing that meantone tempers out 81/80 is enough to determine this information, so it is usually left unstated.

It turns out that tempering out a single comma reduces the dimensionality of an [[interval space]] by 1 (as long as that comma or any multiple of it can't be reached by stacking the other intervals you temper out) - that is, if a JI subgroup can be reached by stacking any multiples of a minimum of three distinct intervals (such as 2, 3, and 5 in the 5-limit), tempering out 1 comma leads to a system built out of two distinct intervals. This means that a [[rank-2 temperament]] (one with two generators, or a period and a generator) must always temper out ''p''-2 commas in a subgroup with ''p'' primes.

See also: [[List of regular temperaments]]