Didacus: Difference between revisions
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For technical data, see [[Hemimean clan #Didacus]]. | For technical data, see [[Hemimean clan #Didacus]]. | ||
== Interval chain == | == Theory == | ||
=== Interval chain === | |||
In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''. | In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''. | ||
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<nowiki/>* In [[CWE]] undecimal didacus | <nowiki/>* In [[CWE]] undecimal didacus | ||
=== The hexatonic framework === | |||
The 2.5.7 subgroup can be crudely approximated by [[6edo]], which is itself technically a didacus tuning as 5/4 spans 2 steps and 7/5 spans 3. Every other didacus tuning is essentially a dietic inflection of this basic hexatonic structure. Therefore, the intervals of didacus can be organized according to how many steps of [[6edo]], or equivalently the 6-note MOS, they correspond to. They can be labeled "wholetone", "ditone", "tritone", etc., and inflected so that "minor" intervals are those just below a step of 6edo, and "major" intervals are just above. Below are the intervals of the symmetric mode of Didacus[25] ([[6L 19s]]) in undecimal CEE tuning. | |||
{| class="wikitable" | |||
|- | |||
! Steps of 6edo | |||
| Unison | |||
| Wholetone | |||
| Ditone | |||
| Tritone | |||
| Tetratone | |||
| Pentatone | |||
| Hexatone | |||
|- | |||
! "Augmented" interval | |||
| 67.79 | |||
| 262.14 | |||
| 456.49 | |||
| 650.84 | |||
| 845.19 | |||
| 1039.54 | |||
| | |||
|- | |||
! JI intervals represented | |||
| 26/25 | |||
| 64/55, 65/56 | |||
| 13/10, 64/49 | |||
| 16/11 | |||
| 13/8 | |||
| 20/11, 64/35 | |||
| | |||
|- | |||
! "Major" interval | |||
| 33.89 | |||
| 228.24 | |||
| 422.60 | |||
| 616.95 | |||
| 811.30 | |||
| 1005.65 | |||
| ''1200.00'' | |||
|- | |||
! JI intervals represented | |||
| 50/49, 56/55, 65/64 | |||
| 8/7, 25/22 | |||
| 14/11, 32/25 | |||
| 10/7 | |||
| 8/5, 35/22 | |||
| 25/14 | |||
| ''2/1'' | |||
|- | |||
! "Minor" interval | |||
| ''0.00'' | |||
| 194.35 | |||
| 388.70 | |||
| 583.05 | |||
| 777.40 | |||
| 971.76 | |||
| 1166.11 | |||
|- | |||
! JI intervals represented | |||
| 1/1 | |||
| 28/25 | |||
| 5/4, 44/35 | |||
| 7/5 | |||
| 11/7, 25/16 | |||
| 7/4, 44/25 | |||
| 49/25, 55/28 | |||
|- | |||
! "Diminished" interval | |||
| | |||
| 160.46 | |||
| 354.81 | |||
| 549.16 | |||
| 743.51 | |||
| 937.86 | |||
| 1132.21 | |||
|- | |||
! JI intervals represented | |||
| | |||
| 11/10, 35/32 | |||
| 16/13, 49/40 | |||
| 11/8 | |||
| 20/13, 49/32 | |||
| 55/32 | |||
| 25/13 | |||
|} | |||
== Tunings == | == Tunings == | ||