User:CompactStar/Ed10/3: Difference between revisions

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=== Division of a thirteenth (e. g. 10/3) into n equal parts ===
{{Editable user page}}
Division of e. g. the 10:3 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 10:3 or another thirteenth as a base though, is apparent by being the top of the upper structure of jazz voicings. Most importantly, a minor thirteenth is the quadruple of a fourth while a major thirteenth is the triple of a fifth, so diatonic scales will not generate prime edXIIIs though these have 1-3-5-7-10 pentads rather than the tone clusters of an equal division of a (perfect) fourth or fifth. Although they no longer count as equivalent, 2-3 and 4-3 are still as valid suspensions of normal root position pentads as 9-10 and 11-10. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy.  
The '''equal division of 10/3''' ('''ed10/3''') is a [[tuning]] obtained by dividing the [[10/3|just major thirteenth (10/3)]] into a number of [[equal]] steps.  


Incidentally, one way to treat 10/3 as an equivalence is the use of the 4:5:6:7:10 chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Just as in meantone it takes four 3/2 to get to 5/1, tempering out the comma 81/80. So, doing this yields 9, 12, 21 and 33 tone 3MOS. While the notes are rather farther apart, the scheme is uncannily similar to [[Augmented family|augmented]] temperament. "Macro-augmented" might be a practically perfect term for it if it hasn't been named yet.
== Properties ==
Division of 10/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed10/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.


The branches of the Bijou family are named thus:
The structural significance of 10/3 or another thirteenth is apparent by being the the top of the upper structure of jazz voicings, as well as a fairly trivial point to split the difference between the [[3/1|tritave]] and the [[4/1|double octave]]. 10/3 is also the complete ambitus of three, later five, of the church modes{{clarify}}.


2&10: Macro-Injera and Macro-Shrutar and Macro-[[Diaschismic family|srutal/pajara]] (Quadrifold Symmetric and Hexachordal Major)
One approach to ed10/3 tunings is the use of the 2:3:6 chord as the fundamental complete sonority in a very similar way to the 4:5:6 chord in meantone. Whereas in meantone it takes four [[3/2]] to get to [[5/4]], here it takes eight [[3/1]] to get to [[3/2]] (tempering out the comma 5000000/4782969 in the 5-limit). This [[regular temperament]] yields monolarge mos with 1–12 notes, followed by a 13-note [[12L 1s (10/3-equivalent)|12L 1s⟨10/3⟩]] mos.


3&9: Macro-augmented (Trifold Symmetric and Pentachordal Major)
[[Category:Ed10/3| ]] <!-- main article -->
 
[[Category:Edonoi]]
4&8: Macro-diminished (Bifold Symmetric and Tetrachordal Major)
[[Category:Lists of scales]]
 
5&7: (Contra-alto) Chromatic Major
 
6&6: Macro-Hexe
 
10/3 being a major thirteenth, any way to treat it as an equivalence is a member of the Kiriage Mangan family:
 
(Tetrad and Pentatonic - Mangan Temperament
 
Hexa- and Heptatonic - Haneman Temperament
 
Enneatonic plus or minus one - Baiman Temperament
 
Hen- and dodecatonic - Sanbaiman Temperament)
 
Triskaidekatonic - Yakuman Temperament List
 
(1L 12s and 12L 1s - Kazoe Yakuman)
 
7L 6s and 6L 7s - Daichīsei and Daisharin
 
9L 4s and 4L 9s - Shōsūshī and Daisūshī
 
10L 3s and 3L 10s - Shōsangen and Daisangen
 
5L 8s and 8L 5s - Ryūīsō
 
2L 11s and 11L 2s - Kokushimusō

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The equal division of 10/3 (ed10/3) is a tuning obtained by dividing the just major thirteenth (10/3) into a number of equal steps.

Properties

Division of 10/3 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed10/3 scales have a perceptually important false octave, with various degrees of accuracy.

The structural significance of 10/3 or another thirteenth is apparent by being the the top of the upper structure of jazz voicings, as well as a fairly trivial point to split the difference between the tritave and the double octave. 10/3 is also the complete ambitus of three, later five, of the church modes[clarification needed].

One approach to ed10/3 tunings is the use of the 2:3:6 chord as the fundamental complete sonority in a very similar way to the 4:5:6 chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/4, here it takes eight 3/1 to get to 3/2 (tempering out the comma 5000000/4782969 in the 5-limit). This regular temperament yields monolarge mos with 1–12 notes, followed by a 13-note 12L 1s⟨10/3⟩ mos.

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