15th-octave temperaments: Difference between revisions
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== Pentadecal == | == Pentadecal == | ||
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Badness: 0.055361 | Badness: 0.055361 | ||
=== Quindecic === | |||
Quindecic preserves the [[11-limit]] structure of [[15edo]], with an independent generator for [[13/1|harmonic 13]]. | |||
[[Subgroup]]: 2.3.5.7.11.13 | |||
[[Comma list]]: 28/27, 49/48, 55/54, 77/75 | |||
[[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }} | |||
: mapping generators: ~22/21, ~13 | |||
[[Optimal tuning]]s: | |||
* [[CTE]]: ~22/21 = 80.000, ~13/8 = 840.528 (~40/39 = 39.472) | |||
* [[POTE]]: ~22/21 = 80.000, ~13/8 = 852.924 (~40/39 = 27.076) | |||
{{Optimal ET sequence|legend=1| 15, 30 }} | |||
[[Badness]] (Smith): 0.028944 | |||
{{Fractional-octave footer}} | {{Fractional-octave footer}} | ||
Revision as of 15:50, 21 December 2024
Template:Fractional-octave navigation
Pentadecal
The pentadecal temperament tempers out the 15-5/3-comma (Quintriyo, pentadecatonic minor thirds comma), [-11 -15 15⟩. This temperament can be described as 15&60 temperament; in the 7-limit, it tempers out the cloudy comma, 16807/16384, and the keema, 875/864.
Subgroup: 2.3.5
Comma: 30517578125/29386561536
Mapping: [⟨15 0 11], ⟨0 1 1]]
POTE generator: ~3/2 = 702.652
Optimal ET sequence: 15, 45, 60, 75, 135, 210c, 345cc
Badness: 0.692042
7-limit
Subgroup: 2.3.5.7
Comma list: 875/864, 16807/16384
Mapping: [⟨15 0 11 42], ⟨0 1 1 0]]
- mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 700.223 (~126/125 = 19.777)
Optimal ET sequence: 15, 45, 60
Badness: 0.114833
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 385/384, 1372/1331
Mapping: [⟨15 0 11 42 52], ⟨0 1 1 0 0]]
- mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 702.733 (~56/55 = 17.267)
Optimal ET sequence: 15, 45, 60, 75de, 135de
Badness: 0.077420
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 144/143, 1372/1331
Mapping: [⟨15 0 11 42 52 8], ⟨0 1 1 0 0 2]]
- mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.715 (~91/90 = 18.285)
Optimal ET sequence: 15, 45f, 60, 135de, 195cddee
Badness: 0.051740
Quindecal
Subgroup: 2.3.5.7.11
Comma list: 121/120, 441/440, 875/864
Mapping: [⟨15 0 11 42 28], ⟨0 1 1 0 1]]
- mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 700.318 (~126/125 = 19.682)
Optimal ET sequence: 15, 45e, 60e
Badness: 0.044405
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 196/195, 352/351, 875/864
Mapping: [⟨15 0 11 42 28 103], ⟨0 1 1 0 1 -2]]
- mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.647 (~65/64 = 18.353)
Optimal ET sequence: 15f, 45e, 60e
Badness: 0.055361
Quindecic
Quindecic preserves the 11-limit structure of 15edo, with an independent generator for harmonic 13.
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 49/48, 55/54, 77/75
Mapping: [⟨15 24 35 42 52 0], ⟨0 0 0 0 0 1]]
- mapping generators: ~22/21, ~13
- CTE: ~22/21 = 80.000, ~13/8 = 840.528 (~40/39 = 39.472)
- POTE: ~22/21 = 80.000, ~13/8 = 852.924 (~40/39 = 27.076)
Badness (Smith): 0.028944