Editing Equal temperament (section)

=== 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]].