Ed7/3

The equal division of 7/3 (ed7/3) is a tuning obtained by dividing the septimal minor tenth (7/3) in a certain number of equal steps.

Properties

Division of 7/3 into equal parts does not necessarily imply directly using this interval as an equivalence. 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 ⁠^{[citation needed ]} (and even the range of a dastgah⁠ ⁠^{[citation needed ]}).

It also provides a fairly trivial point to split the difference between the octave and the tritave, which is why Joseph Ruhf has 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) and, as is the twelfth (tritave), an alternative interval where invertible counterpoint has classically occurred.

Chords and harmonies

Main article: Pseudo-traditional harmonic functions of enneatonic scale_degrees

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 structurally 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:(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 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 Middletown family

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 Middlebury temperament falls in the "Middletown valley", but its enneatonic scales are “generator-remainder".

The temperaments neighboring Middletown proper are named thus:

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

The pyrite tuning of edXs will turn out to divide a barely mistuned 5/2 of almost exactly 45\34edo.

Individual pages for ed7/3's

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