Editing Equal temperament (section)

=== Proportions between semitone and whole tone ===
{{More citations needed section|date=August 2017}}
In this section, ''semitone'' and ''whole tone'' may not have their usual 12&nbsp;EDO meanings, as it discusses how they may be tempered in different ways from their just versions to produce desired relationships. Let the number of steps in a semitone be {{mvar|s}}, and the number of steps in a tone be {{mvar|t}}.

There is exactly one family of equal temperaments that fixes the semitone to any [[proper fraction]] of a whole tone, while keeping the notes in the right order (meaning that, for example, {{sc|'''C'''}}, {{sc|'''D'''}}, {{sc|'''E'''}}, {{sc|'''F'''}}, and {{sc|'''F'''}}{{music|#}} are in ascending order if they preserve their usual relationships to {{sc|'''C'''}}). That is, fixing {{mvar|q}} to a proper fraction in the relationship {{nobr|{{math|''q t'' {{=}} ''s''}} }} also defines a unique family of one equal temperament and its multiples that fulfil this relationship.

For example, where {{mvar|k}} is an integer, {{nobr|12{{mvar|k}} {{sc|EDO}} }} sets {{nobr|{{math|''q'' {{=}} {{sfrac|1|2}}}},}} {{nobr|19 {{mvar|k}} {{sc|EDO}} }} sets {{nobr|{{math|''q'' {{=}} {{sfrac|1|3}}}},}} and {{nobr|31 {{mvar|k}} {{sc|EDO}} }} sets {{nobr|{{math|''q'' {{=}} {{sfrac| 2 | 5 }} }}.}} The smallest multiples in these families (e.g. 12, 19 and 31 above) has the additional property of having no notes outside the [[circle of fifths]]. (This is not true in general; in 24&nbsp;{{sc|EDO}}, the half-sharps and half-flats are not in the circle of fifths generated starting from {{sc|'''C'''}}.) The extreme cases are {{nobr|5 {{mvar|k}} {{sc|EDO}},}} where {{nobr|{{math|''q'' {{=}} 0}} }} and the semitone becomes a unison, and {{nobr|7 {{mvar|k}} {{sc|EDO}} }}, where {{nobr|{{math|''q'' {{=}} 1}} }} and the semitone and tone are the same interval.

Once one knows how many steps a semitone and a tone are in this equal temperament, one can find the number of steps it has in the octave. An equal temperament with the above properties (including having no notes outside the circle of fifths) divides the octave into {{nobr|{{math| 7 ''t'' − 2 ''s''}} steps}} and the perfect fifth into {{nobr|{{math| 4 ''t'' − ''s'' }} steps.}} If there are notes outside the circle of fifths, one must then multiply these results by {{mvar|n}}, the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24&nbsp;{{sc|EDO}}, six in 72&nbsp;{{sc|EDO}}). (One must take the small semitone for this purpose: 19&nbsp;{{sc|EDO}} has two semitones, one being {{sfrac| 1 | 3 }} tone and the other being {{sfrac| 2 | 3 }}. Similarly, 31&nbsp;{{sc|EDO}} has two semitones, one being {{sfrac| 2 | 5 }} tone and the other being {{sfrac| 3 | 5 }}).

The smallest of these families is {{nobr|12 {{mvar|k}} {{sc|EDO}},}} and in particular, 12&nbsp;{{sc|EDO}} is the smallest equal temperament with the above properties. Additionally, it makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons 12&nbsp;{{sc|EDO}} has become the most commonly used equal temperament. (Another reason is that 12&nbsp;EDO is the smallest equal temperament to closely approximate 5&nbsp;limit harmony, the next-smallest being 19&nbsp;EDO.)

Each choice of fraction {{mvar|q}} for the relationship results in exactly one equal temperament family, but the converse is not true: 47&nbsp;{{sc|EDO}} has two different semitones, where one is {{sfrac| 1 | 7 }} tone and the other is {{sfrac| 8 | 9 }}, which are not complements of each other like in 19&nbsp;{{sc|EDO}} ({{sfrac| 1 | 3 }} and {{sfrac| 2 | 3 }}). Taking each semitone results in a different choice of perfect fifth.