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'''EdX''' means '''Division of a tenth (e. g. 7/3) into n equal parts'''.
'''Ed7/3''' means '''Division of the septimal minor tenth ([[7/3]]) into n equal parts'''.


== Properties ==
== Properties ==
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) and, as is the twelfth, an alternative interval where [[wikipedia:Inversion_(music)#Counterpoint|invertible counterpoint]] has classically occurred. 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) and, as is the twelfth, an alternative interval where [[wikipedia:Inversion_(music)#Counterpoint|invertible counterpoint]] has classically occurred. 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:(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.
 
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 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:
The branches of the Middletown family are named thus:
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Sort of unsurprisingly, though not so evidently, the pyrite 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 pyrite tuning of edXs will turn out to divide a barely mistuned 5:2 of alomst exactly 45\34edo.


== Individual pages for EDXs ==
== Individual pages for ED7/3s ==
 
* 8 - [[8ed7/3|Eighth root of 7/3]] ([[8edX]])
* [[8edX]]
* 9 - [[9ed7/3|Ninth root of 7/3]] ([[9edX]])
* [[9edX]]
* 15 - [[15ed7/3|15th root of 7/3]] ([[15edX]])
* [[15edX]]
* 16 - [[16ed7/3|16th root of 7/3]] ([[16edX]])
* [[16edX]]
* 17 - [[17ed7/3|17th root of 7/3]] ([[17edX]])
* [[17edX]]
* 19 - [[19ed7/3|19th root of 7/3]] ([[19edX]])
* [[19edX]]
 
=== Equal Divisions of the Just Major Tenth (5/2) ===
 
* 16 - [[16ed5/2|Sixteenth root of 5/2]]
* 18 - [[18ed5/2|Eighteenth root of 5/2]]
* 25 - [[25ed5/2|25th root of 5/2]]
 
=== Equal Divisions of the Septimal Minor Tenth (7/3) ===
 
* 15 - [[15ed7/3|15th root of 7/3]]
* 19 - [[19ed7/3|19th root of 7/3]]
* 30 - [[30ed7/3|30th root of 7/3]]
* 30 - [[30ed7/3|30th root of 7/3]]
* 34 - [[34ed7/3|34th root of 7/3]]
* 34 - [[34ed7/3|34th root of 7/3]]
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* 106 - [[106ed7/3|106th root of 7/3]]
* 106 - [[106ed7/3|106th root of 7/3]]


[[Category:EdX| ]] <!-- main article -->
[[Category:EdX]]
[[Category:ed7/3]]
[[Category:Ed7/3| ]] <!-- main article -->
[[Category:Equal-step tuning]]
[[Category:Equal-step tuning]]
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