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Division of a tenth (e. g. 7/3) into n equal parts


Division of e. g. the 7:3 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of equivalence has not even been posed yet. The utility of 7:3 or another tenth as a base though, is apparent by being the absolute widest range most generally used in popular songs (and even the range of a dastgah) as well as a fairly trivial point to split the difference between the octave and the tritave (which is why I have named the region of intervals between 6 and 7 degrees of 5edo the "Middletown valley", the proper Middletown temperament family being based on an enneatonic scale generated by a third or a fifth optionally with a period of a [wolf] fourth at most 560 cents wide). Incidentally enneatonic scales, especially those equivalent at e. g. 7:3, can sensibly take tetrads as the fundamental complete sonorities of a pseudo-traditional functional harmony due to their seventh degree being as structrally important as it is. Many, though not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.

Incidentally, one way to treat 7/3 as an equivalence is the use of the 3:4:5:6:(7) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes two 28/15 to get to 7/2 (tempering out the comma 225/224). So, doing this yields 2 15, 19, and 34 note MOS 2/1 apart. While the notes are rather farther apart, the scheme is uncannily similar to meantone. "Macrobichromatic" might be a practically perfect term for it if it hasn't been named yet.

The branches of the Middletown family are named thus:

3&6: Tritetrachordal
4&5: Montrose (between 5/4edo and 4/3edo in particular, MOS generated by [pseudo] octaves belong to this branch)
2&7: Terra Rubra

The family of interlaced octatonic scale based temperaments in the "Middletown valley" is called Vesuvius (i. e. the volcano east of Naples).

The temperaments neighboring Middletown proper are named thus:

5&6: Rosablanca
4&7: Saptimpun (10 1/2)
5&7: 8bittone

8edX
9edX
15edX
16edX
17edX
19edX

Sort of unsurprisingly, though not so evidently, the golden tuning of edXs will turn out to divide a barely mistuned 5:2 of alomst exactly 45/34edo.