Decic
Decic is a temperament for the 7, 11, 13, and 17 prime limits. It is a member of marvel temperaments, cloudy clan, and linus temperaments. It has a period of 1/10 octave and tempers out 225/224 and 16807/16384. The fifth of decic in size is a meantone fifth, but four of them are not used to reach the 5th harmonic. Instead, 14/13, 15/14 and 16/15 are equated to 1/10 of an octave, and from this it derives its name. Not only the meantone fifth (flat 3/2) or fourth (sharp 4/3), but also the magic major third (flat 5/4) can be used as a generator.
There are three mappings for 11, 13 and 17-limit that are comparable in complexity and error: decic (10&50), splendic (10e&50) and prodigic (10&50e).
Temperament data
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
Mapping: [⟨10 16 23 28 35 37 41], ⟨0 -1 1 0 -2 0 -1]]
- 7-limit: ~15/14 = 120.00000, ~3/2 = 698.69596
- 11-limit: ~15/14 = 120.00000, ~3/2 = 696.79119
- 13-limit: ~14/13 = 120.00000, ~3/2 = 696.99342
- 17-limit: ~14/13 = 120.00000, ~3/2 = 697.08527
- 7-limit: ~15/14 = 120.18411, ~3/2 = 699.76795
- 11-limit: ~15/14 = 120.14165, ~3/2 = 697.61366
- 13-limit: ~14/13 = 120.11775, ~3/2 = 697.67733
- 17-limit: ~14/13 = 120.12744, ~3/2 = 697.82559
Diamond monotone ranges:
- 7-odd-limit: ~3/2 = [680.00000, 720.00000] (17\30 to 6\10)
- 9-odd-limit: ~3/2 = [696.00000, 720.00000] (29\50 to 6\10)
- 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)
Diamond tradeoff ranges:
- 7 and 9-odd-limit: ~3/2 = [693.12909, 702.51219]
- 11, 13, and 15-odd-limit: ~3/2 = [689.36294, 702.51219]
- 17-odd-limit: ~3/2 = [689.36294, 704.95541]
Diamond monotone and tradeoff ranges:
- 7-odd-limit: ~3/2 = [693.12909, 702.51219]
- 9-odd-limit: ~3/2 = [696.00000, 702.51219]
- 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]
- 7-limit: 10, 30b, 40, 50, 60, 110d, 170cdd
- 11 and 13-limit: 10, 30be, 40, 50
- 17-limit: 10, 30beg, 40, 50
- 7-limit: 0.089135
- 11-limit: 0.063900
- 13-limit: 0.036880
- 17-limit: 0.025064
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
Mapping: [⟨10 16 23 28 34 37 41], ⟨0 -1 1 0 3 0 -1]]
- 11-limit: ~15/14 = 120.00000, ~3/2 = 698.51792
- 13-limit: ~14/13 = 120.00000, ~3/2 = 698.36508
- 17-limit: ~14/13 = 120.00000, ~3/2 = 698.37473
- 11-limit: ~15/14 = 120.18780, ~3/2 = 699.61110
- 13-limit: ~14/13 = 120.15710, ~3/2 = 699.27937
- 17-limit: ~14/13 = 120.15778, ~3/2 = 699.29299
Diamond monotone ranges:
- 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)
Diamond tradeoff ranges:
- 11, 13, and 15-odd-limit: ~3/2 = [693.12909, 702.51219]
- 17-odd-limit: ~3/2 = [693.12909, 704.95541]
Diamond monotone and tradeoff ranges:
- 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]
- 11-limit: 0.059802
- 13-limit: 0.037989
- 17-limit: 0.026050
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288
Mapping: [⟨10 16 23 28 35 37 41], ⟨0 -1 1 0 -3 0 -1]]
- 11-limit: ~15/14 = 120.00000, ~3/2 = 700.20964
- 13-limit: ~14/13 = 120.00000, ~3/2 = 700.50272
- 17-limit: ~14/13 = 120.00000, ~3/2 = 700.47431
- 11-limit: ~15/14 = 120.20310, ~3/2 = 701.39473
- 13-limit: ~14/13 = 120.16631, ~3/2 = 701.47358
- 17-limit: ~14/13 = 120.15866, ~3/2 = 701.40045
Diamond monotone ranges:
- 11, 13, 15, and 17-odd-limit: ~3/2 = [700.00000, 702.85714] (35\60 to 41\70)
Diamond tradeoff ranges:
- 11, 13, 15, and 17-odd-limit: ~3/2 = [693.12909, 707.40794]
Diamond monotone and tradeoff ranges:
- 11, 13, 15, and 17-odd-limit: ~3/2 = [700.00000, 702.85714]
- 11-limit: 0.066600
- 13-limit: 0.041788
- 17-limit: 0.027590
Interval chains
| Generator | -3 | -2 | -1 | 0 | 1 | 2 | 3 | |
|---|---|---|---|---|---|---|---|---|
| Period 0 | Cents* | 1131.256 | 1154.171 | 1177.085 | 0.000 | 22.915 | 45.829 | 68.744 |
| Ratios | 1/1 | 40/39 | 28/27, 25/24 | |||||
| Period 1 | Cents* | 51.256 | 74.171 | 97.085 | 120.000 | 142.915 | 165.829 | 188.744 |
| Ratios | 36/35, 33/32 | 21/20 | 18/17, 17/16 | 16/15, 15/14, 14/13 | 13/12, 12/11 | 10/9 | ||
| Period 2 | Cents* | 171.256 | 194.171 | 217.085 | 240.000 | 262.915 | 285.829 | 308.744 |
| Ratios | 11/10 | 9/8 | 17/15 | 8/7, 15/13 | 7/6 | 20/17, 13/11 | ||
| Period 3 | Cents* | 291.256 | 314.171 | 337.085 | 360.000 | 382.915 | 405.829 | 428.744 |
| Ratios | 6/5 | 17/14 | 11/9, 16/13, 21/17 | 26/21, 5/4 | 14/11 | |||
| Period 4 | Cents* | 411.256 | 434.171 | 457.085 | 480.000 | 502.915 | 525.829 | 548.744 |
| Ratios | 9/7 | 22/17, 13/10, 17/13, 21/16 | 4/3 | 15/11 | ||||
| Period 5 | Cents* | 531.256 | 554.171 | 577.085 | 600.000 | 622.915 | 645.829 | 668.744 |
| Ratios | 11/8, 18/13 | 7/5 | 24/17, 17/12 | 10/7 | 13/9, 16/11 | |||
* in 17-limit POTE tuning