Editing Equal temperament (section)

== Related tuning systems ==
Equal temperament systems can be thought of in terms of the spacing of three intervals found in [[just intonation]], ''most'' of whose chords are harmonically perfectly in tune—a good property not quite achieved between almost all pitches in almost all equal temperaments. Most just chords sound amazingly consonant, and most equal-tempered chords sound at least slightly dissonant. In C&nbsp;major those three intervals are:<ref name=Milne-Sethares-Plamdon-2007/>
* the [[whole tone|greater tone]] {{nobr|{{math| ''T'' {{=}} {{sfrac| 9 | 8 }}  {{=}} }}}} the interval from C:D, F:G, and A:B;
* the [[whole tone|lesser tone]] {{nobr|{{math| ''t'' {{=}} {{sfrac| 10 | 9 }}  {{=}} }}}} the interval from D:E and G:A;
* the [[semitone|diatonic semitone]] {{nobr|{{math| ''s'' {{=}} {{sfrac| 16 | 15 }}  {{=}} }}}} the interval from E:F and B:C.

Analyzing an equal temperament in terms of how it modifies or adapts these three intervals provides a quick way to evaluate how consonant various chords can possibly be in that temperament, based on how distorted these intervals are.<ref name=Milne-Sethares-Plamdon-2007/>{{efn|
For 12&nbsp;pitch systems, either for a whole 12&nbsp;note scale, for or 12&nbsp;note subsequences embedded inside some larger scale,<ref name=Milne-Sethares-Plamdon-2007 /> use this analysis as a way to program software to microtune an electronic keyboard dynamically, or 'on the fly', while a musician is playing. The object is to fine tune the notes momentarily in use, and any likely subsequent notes involving consonant chords, to always produce pitches that are harmonically in-tune, inspired by how orchestras and choruses constantly re-tune their overall pitch on long-duration chords for greater consonance than possible with strict 12&nbsp;TET.<ref name=Milne-Sethares-Plamdon-2007/>
}}

=== Regular diatonic tunings ===
{{anchor|Figure 1}}[[File:Rank-2 temperaments with the generator close to a fifth and period an octave.jpg|right|250px|thumb|Figure 1: The [[regular diatonic tuning]]s continuum, which includes many notable "equal temperament" tunings<ref name=Milne-Sethares-Plamdon-2007>
{{cite journal
 |last1=Milne |first1=A.
 |last2=Sethares |first2=W.A. |author2-link=William Sethares
 |last3=Plamondon |first3=J.
 |title=Isomorphic controllers and dynamic tuning: Invariant fingerings across a tuning continuum
 |journal=[[Computer Music Journal]]
 |date=Winter 2007
 |volume=31 |issue=4 |pages=15–32
 |issn=0148-9267 |doi=10.1162/comj.2007.31.4.15 |doi-access=free
 |df=dmy-all
}} Online: {{ISSN|1531-5169}}</ref>]]
The diatonic tuning in ''12&nbsp;tone equal temperament'' {{nobr|(12 {{sc|TET}})}} can be generalized to any regular diatonic tuning dividing the octave as a sequence of steps {{nobr| {{mvar|T t s T t T s}} }} (or some [[circular shift]] or "rotation" of it). To be called a ''regular'' diatonic tuning, each of the two semitones ({{mvar| s }}) must be smaller than either of the tones ([[major tone|greater tone]], {{mvar| T }}, and [[minor tone|lesser tone]], {{mvar| t }}). The comma {{mvar|κ}} is implicit as the size ratio between the greater and lesser tones: Expressed as frequencies {{nobr|{{math|''κ'' {{=}} {{sfrac| ''T'' | ''t'' }}}} ,}} or as [[cent (music)|cents]] {{nobr|{{math| ''κ'' {{=}} ''T''  − ''t'' }} }}.

The notes in a regular diatonic tuning are connected in a "spiral of fifths" that does ''not'' close (unlike the [[circle of fifths]] in {{nobr|12 {{sc|TET}}).}} Starting on the subdominant {{sc|'''F'''}} (in the [[C major|key of C]]) there are three [[perfect fifth]]s in a row—{{sc|'''F'''}}–{{sc|'''C'''}}, {{sc|'''C'''}}–{{sc|'''G'''}}, and {{sc|'''G'''}}–{{sc|'''D'''}}—each a composite of some [[permutation]] of the smaller intervals {{nobr| {{mvar|T T t s}} .}} The three in-tune fifths are interrupted by the [[List of pitch intervals|grave fifth]] {{sc|'''D'''}}–{{sc|'''A'''}} {{=}} {{nobr| {{mvar|T t t s}} }}([[List of pitch intervals|''grave'']] means "flat by a [[comma (music)|comma]]"), followed by another perfect fifth, {{sc|'''E'''}}–{{sc|'''B'''}}, and another grave fifth, {{sc|'''B'''}}–{{sc|'''F'''}}{{music|#}}, and then restarting in the sharps with {{sc|'''F'''}}{{music|#}}–{{sc|'''C'''}}{{music|#}}; the same pattern repeats through the sharp notes, then the double-sharps, and so on, indefinitely. But each octave of all-natural or all-sharp or all-double-sharp notes flattens by two commas with every transition from naturals to sharps, or single sharps to double sharps, etc. The pattern is also reverse-symmetric in the flats: Descending by [[perfect fourth|fourths]] the pattern reciprocally sharpens notes by two commas with every transition from natural notes to flattened notes, or flats to double flats, etc. If left unmodified, the two grave fifths in each block of all-natural notes, or all-sharps, or all-flat notes, are [[wolf interval|"wolf" intervals]]: Each of the grave fifths out of tune by a [[diatonic comma]].

Since the comma, {{mvar|κ}}, expands the [[minor tone|lesser tone]] {{nobr|&thinsp;{{mvar|t {{=}} s c}} ,}} into the [[major tone|greater tone]], {{nobr|&thinsp;{{mvar|T {{=}} s c κ}} ,}} a [[just intonation|just]] octave {{nobr| {{mvar|T t s T t T s}} }} can be broken up into a sequence {{nobr|&thinsp;{{mvar|s c κ &thinsp; s c &thinsp; s &thinsp; s c κ &thinsp; s c &thinsp; s c κ &thinsp; s}} ,}} (or a [[circular shift]] of it) of 7&nbsp;diatonic semitones {{mvar|s}}, 5&nbsp;chromatic semitones {{mvar|c}}, and 3&nbsp;[[syntonic comma|commas]] {{nobr|&thinsp;{{mvar|κ}} .}} Various equal temperaments alter the interval sizes, usually breaking apart the three commas and then redistributing their parts into the seven diatonic semitones {{mvar|s}}, or into the five chromatic semitones {{mvar|c}}, or into both {{mvar|s}} and {{mvar|c}}, with some fixed proportion for each type of semitone.

The sequence of intervals {{mvar|s}}, {{mvar|c}}, and {{mvar|κ}} can be repeatedly appended to itself into a greater [[circle of fifths|spiral of 12&nbsp;fifths]], and made to connect at its far ends by slight adjustments to the size of one or several of the intervals, or left unmodified with occasional less-than-perfect fifths, flat by a comma.

=== Morphing diatonic tunings into EDO ===
Various equal temperaments can be understood and analyzed as having made adjustments to the sizes of and subdividing the three intervals—{{mvar| T }}, {{mvar| t }}, and {{mvar| s }}, or at finer resolution, their constituents {{mvar| s }}, {{mvar| c }}, and {{mvar| κ }}. An equal temperament can be created by making the sizes of the [[major tone|major]] and [[minor tone]]s ({{mvar|T}}, {{mvar|t}}) the same (say, by setting {{nobr|{{math|''κ'' {{=}} 0}}}}, with the others expanded to still fill out the octave), and both semitones ([[diatonic semitone|{{mvar|s}}]] and {{mvar|c}}) the same, then 12 equal semitones, two per tone, result. In {{nobr|12 {{sc|TET}}}}, the semitone, {{mvar|s}}, is exactly half the size of the same-size whole tones {{mvar|T}} = {{mvar|t}}.

Some of the intermediate sizes of tones and semitones can also be generated in equal temperament systems, by modifying the sizes of the comma and semitones. One obtains {{nobr|[[7 equal temperament|7 {{sc|TET}}]]}} in the limit as the size of {{mvar|c}} and {{mvar|κ}} tend to zero, with the octave kept fixed, and {{nobr|5 {{sc|TET}}}} in the limit as {{mvar|s}} and {{mvar|κ}} tend to zero; {{nobr|12 {{sc|TET}}}} is of course, the case {{nobr|&thinsp;{{mvar|s {{=}} c}}&thinsp;}} and {{nobr|&thinsp;{{math|''κ'' {{=}} 0}} .}} For instance:

;{{nobr|[[5 equal temperament|5 {{sc|tet}}]]}} and {{nobr|[[7 equal temperament|7 {{sc|tet}}]]}}: There are two extreme cases that bracket this framework: When {{mvar|s}} and {{mvar|κ}} reduce to zero with the octave size kept fixed, the result is {{nobr|{{mvar|t t t t t}} ,}} a 5&nbsp;tone equal temperament. As the {{mvar|s}} gets larger (and absorbs the space formerly used for the comma {{mvar|κ}}), eventually the steps are all the same size, {{nobr|{{mvar|t t t t t t t}} ,}} and the result is seven-tone equal temperament. These two extremes are not included as "regular" diatonic tunings.

;{{nobr|[[19 equal temperament|19 {{sc|tet}}]]}}: If the diatonic semitone is set double the size of the chromatic semitone, i.e. {{nobr|&thinsp;{{mvar|s {{=}} 2 c}} }} (in cents) and {{nobr|&thinsp;{{math|''κ'' {{=}} 0}} ,}} the result is {{nobr|[[19 equal temperament|19 {{sc|tet}}]],}} with one step for the chromatic semitone {{mvar|c}}, two steps for the diatonic semitone {{mvar|s}}, three steps for the tones {{mvar|T}} =  {{mvar|t}}, and the total number of steps {{math|{{nobr|&thinsp;3 ''T'' + 2 ''t'' + 2 ''s''}} {{=}} {{nobr| 9 + 6 + 4}} {{=}}&thinsp;}} 19&nbsp;steps. The imbedded 12&nbsp;tone sub-system closely approximates the historically important {{nobr|{{sfrac| 1 | 3 }} comma}} [[meantone temperament|meantone system]].

;{{nobr|[[31 equal temperament|31 {{sc|tet}}]]}}: If the chromatic semitone is two-thirds the size of the diatonic semitone, i.e. {{nobr|{{math|''c'' {{=}} {{sfrac| 2 | 3 }} ''s''}} ,}} with {{nobr|{{math|''κ'' {{=}} 0}} ,}} the result is [[31 equal temperament|31 {{sc|tet}}]], with two steps for the chromatic semitone, three steps for the diatonic semitone, and five steps for the tone, where {{math|{{nobr|&thinsp;3 ''T'' + 2 ''t'' + 2 ''s''}} {{=}} {{nobr| 15 + 10 + 6}} {{=}}&thinsp;}} [[31 equal temperament|31&nbsp;steps]]. The imbedded 12&nbsp;tone sub-system closely approximates the historically important [[quarter comma meantone|{{nobr|{{sfrac| 1 | 4 }} comma}} meantone]].

;{{nobr|[[43 equal temperament|43 {{sc|tet}}]]}}: If the chromatic semitone is three-fourths the size of the diatonic semitone, i.e. {{nobr|{{math|''c'' {{=}} {{sfrac| 3 | 4 }} ''s''}} ,}} with {{nobr|{{math|''κ'' {{=}} 0}} ,}} the result is [[43 equal temperament|43 {{sc|tet}}]], with three steps for the chromatic semitone, four steps for the diatonic semitone, and seven steps for the tone, where {{math|{{nobr|&thinsp;3 ''T'' + 2 ''t'' + 2 ''s''}} {{=}} {{nobr| 21 + 14 + 8}} {{=}}&thinsp;}} 43. The imbedded 12&nbsp;tone sub-system closely approximates {{nobr|{{sfrac| 1 | 5 }} comma}} meantone.

;{{nobr|[[53 equal temperament|53 {{sc|tet}}]]}}: If the chromatic semitone is made the same size as three commas, {{nobr|&thinsp;{{math|''c'' {{=}} 3 ''κ''}}&thinsp;}} (in cents, in frequency {{nobr|&thinsp;{{math|''c'' {{=}} ''κ''³}}&thinsp;}}) the diatonic the same as five commas, {{nobr|&thinsp;{{math|''s'' {{=}} 5 ''κ''}} ,}} that makes the lesser tone eight commas {{nobr|{{math|''t'' {{=}} ''s'' + ''c'' {{=}} 8 ''κ''}} ,}} and the greater tone nine, {{nobr|&thinsp;{{math|''T'' {{=}} ''s'' + ''c'' + ''κ'' {{=}} 9 ''κ''}} .}} Hence {{math|{{nobr|&thinsp;3 ''T'' + 2 ''t'' + 2 ''s''}} {{=}} {{nobr| 27 ''κ'' + 16 ''κ'' + 10 ''κ''}} {{=}} 53 ''κ''}} for [[53 equal temperament|53&nbsp;steps]] of one comma each. The comma size / step size is {{nobr|&thinsp;{{math|''κ'' {{=}} {{sfrac| 1 200 | 53 }} }}&thinsp;¢}} exactly, or {{nobr|&thinsp;{{math|''κ'' {{=}} 22.642}} ¢}} {{nobr|&thinsp;{{math|≈ 21.506}} ¢ ,}} the [[syntonic comma]]. It is an exceedingly close approximation to 5-limit [[just intonation]] and Pythagorean tuning, and is the basis for [[Turkish makam|Turkish music theory]].