Editing Equal temperament (section)

=== 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.