User:CompactStar/Ed10/3: Difference between revisions

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Full cleanup of this page. Removed ancient references to the pre-"talk: Equal-step tuning" article name of "edXIII". 4:5:6:7 is a tetrachord so can't really be used in a similar fashion to 4:5:6 in 5-limit meantone. I suggest 2:3:6 since it is the basic chord of the 10/3.3.5 subgroup.
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=== Division of a thirteenth (e.g. [[10/3]]) into n equal parts ===
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.  
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 and the complete ambitus of three, later five, of the church modes (Dorian G below low D to E above high D, Phrygian A below low E to F above high E and Mixolydian C below low G to A above above high G and later Aeolian D below low A to B(b) above high A and Ionian F below low C to D above high C; it is unknown whether a scale on Bb was within the question before the Baroque period). 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.  


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 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 the top of the upper structure of jazz voicings, as well as a fairly trivial point to split the difference between the [[EDT|tritave]] and the [[Ed4|double octave]]. 10/3 is also the complete ambitus of three, later five, of the church modes. Many, though not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.


The branches of the Bijou family are named thus:
Incidentally, one way to treat 10/3 as an equivalence 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 10/3.3.5 subgroup). This [[regular temperament]] yields monolarge MOS with 1-12 notes, followed by a 13-note [[12L 1s]] MOS.


2&10: Macro-Injera and Macro-Shrutar and Macro-[[Diaschismic family|srutal/pajara]] (Quadrifold Symmetric and Hexachordal Major)
3&9: Macro-augmented (Trifold Symmetric and Pentachordal Major)
4&8: Macro-diminished (Bifold Symmetric and Tetrachordal Major)
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
[[Category:Edonoi]]
[[Category:Edonoi]]
[[Category:Todo:cleanup]]
[[Category:Todo:improve layout]]
[[Category:Todo:improve readability]]
[[Category:Todo:improve synopsis]]

Revision as of 03:00, 6 July 2023

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 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 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. Many, though not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.

Incidentally, one way to treat 10/3 as an equivalence 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 10/3.3.5 subgroup). This regular temperament yields monolarge MOS with 1-12 notes, followed by a 13-note 12L 1s MOS.

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