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| === Division of a thirteenth (e. g. 10/3) into n equal parts ===
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| 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.
| | 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. |
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| 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. |
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| 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}}. |
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| 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. |
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| 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]] |
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| 5&7: (Contra-alto) Chromatic Major
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| 6&6: Macro-Hexe
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| 10/3 being a major thirteenth, any way to treat it as an equivalence is a member of the Kiriage Mangan family:
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| (Tetrad and Pentatonic - Mangan Temperament
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| Hexa- and Heptatonic - Haneman Temperament
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| Enneatonic plus or minus one - Baiman Temperament
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| Hen- and dodecatonic - Sanbaiman Temperament)
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| Triskaidekatonic - Yakuman Temperament List
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| (1L 12s and 12L 1s - Kazoe Yakuman)
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| 7L 6s and 6L 7s - Daichīsei and Daisharin
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| 9L 4s and 4L 9s - Shōsūshī and Daisūshī
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| 10L 3s and 3L 10s - Shōsangen and Daisangen
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| 5L 8s and 8L 5s - Ryūīsō
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| 2L 11s and 11L 2s - Kokushimusō
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