Regular diatonic tuning
One may determine the corresponding cents of s, T, and the fifth (p5), given one of the values: When the (diatonic) semitones, s, are reduced to zero (T = 240 ¢) the octave is T T T T T, 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 ¢).
"Regular" here is understood in the sense of a mapping from Pythagorean diatone such that all the interval relationships are preserved.
If one breaks the rule for "regular" that s must be smaller than T and continues to increase the size of s further, so that it becomes larger than the T, one gets irregular scales with two large steps and five small steps, and eventually, when all the T-s vanish the result is s s, so a division of the octave into tritones.
All regular diatonic tunings are also generated collections (also called moments of symmetry) and the chain of fifths can be continued in either direction to obtain a twelve tone system F C G D A E B F♯ C♯ G♯ D♯ A♯, where the interval F♯-G is the same as B♭-B, etc., another moment of symmetry with two interval sizes.
The diatonic semitone, d, called S above, is the change in pitch of a sequence of six notes spaced by fifths, e.g. from E to F or B to C. For any tuning, the diatonic semitone is the relative pitch difference on a standard keyboard between two white keys that have no black key between them.
When the fifths are slightly flatter than in just intonation, then we are in the region of the historical meantone tunings, which distribute or temper out the syntonic comma.
That leaves the two extremes: Diatonic scales constructed in equal temperaments can have fifths either wider or narrower than a just 3 / 2 .
Hence, it also defines an invariant mapping -- all across the tuning continuum -- between (a) the notes at these (pseudo-Just) generated tonal intervals, and (b) the corresponding partials of a similarly-generated pseudo-Harmonic timbre.
The combination of an isomorphic keyboard and continuously variable tuning supports Dynamic tonality as described above.
