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<span style="font-size: 19.5px;">Division of a tenth (e. g. 7/3) into n equal parts</span>
'''EdX''' short for '''Equal Division of the Tenth''' divide the septimal minor tenth ([[7/3]]) into n equal parts.


Division of e. g. the [[7/3|7:3]] into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of [[equivalence|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 [https://en.wikipedia.org/wiki/Dastg%C4%81h-e_M%C4%81hur 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 [[Pseudo-traditional_harmonic_functions_of_enneatonic_scale_degrees|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.
Division of [[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 [https://en.wikipedia.org/wiki/Dastg%C4%81h-e_M%C4%81hur 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 [[Pseudo-traditional harmonic functions of enneatonic scale_degrees|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.
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.
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The branches of the Middletown family are named thus:
The branches of the Middletown family are named thus:


3&amp;6: Tritetrachordal
* 3&amp;6: Tritetrachordal
 
* 4&amp;5: Montrose (between 5/4edo and 4/3edo in particular, MOS generated by [pseudo] octaves belong to this branch)
4&amp;5: Montrose (between 5/4edo and 4/3edo in particular, MOS generated by [pseudo] octaves belong to this branch)
* 2&amp;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).
2&amp;7: Terra Rubra
* The temperaments neighboring Middletown proper are named thus:
 
* 5&amp;6: Rosablanca
The family of interlaced octatonic scale based temperaments in the "Middletown valley" is called Vesuvius (i. e. the volcano east of Naples).
* 4&amp;7: Saptimpun (10 1/2)
 
* 5&amp;7: 8bittone
The temperaments neighboring Middletown proper are named thus:
* [[8edX]]
 
* [[9edX]]
5&amp;6: Rosablanca
* [[15edX]]
 
* [[16edX]]
4&amp;7: Saptimpun (10 1/2)
* [[17edX]]
 
* [[19edX]]
5&amp;7: 8bittone
 
[[8edX|8edX]]
 
[[9edX|9edX]]
 
[[15edX|15edX]]
 
[[16edX|16edX]]
 
[[17edX|17edX]]
 
[[19edX|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.
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.
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