Syntonic temperament

A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size (TTTS or a permutation of that) which makes it a Linear temperament with the tempered fifth as a generator.

In the ordinary diatonic scales the T's here are tones and the S's are semitones which are half, or approximately half the size of the tone. But in the more general regular diatonic tunings, the two steps can be of any relation within the range between T=171.43 (S=T) and T=240 (S=0) cents (fifth between 685.71 and 720). Note that regular diatonic tunings are not limited to the notes of the diatonic scale which defines them.

One may determine the corresponding cents of S, T, and the fifth, given one of the values:

When the S's reduce to zero (T=240 cents) the result is TTTTT or a five tone equal temperament. As the semitones get larger, eventually the steps are all the same size, and the result is in seven tone equal temperament (S=T=171.43). These two end points are not included as regular diatonic tunings, because to be regular the pattern of large and small steps has to be preserved, but everything in between is included, however small the semitones are, or however similar they are to the whole tones.

"Regular" here is understood in the sense of a mapping from Pythagorean diatonic such that all the interval relationships are preserved. For instance, in all regular diatonic tunings, just as for the pythagorean diatonic:

and so on; in all those examples the result is reduced to the octave.

If one continues to increase the size of the S further, so that it is larger than the T, one gets scales with two large steps and five small steps, and eventually, when all the T's vanish the result is SS, so a tritone division of the octave. These scales however are not included as regular diatonic tunings.

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