Valentine extensions
Valentine has various competing extensions to the 13-limit. The following temperaments are discussed in this article:
- Valentino (31 & 46) – tempering out 121/120, 126/125, 176/175, and 196/195;
- Lupercalia (31 & 46f) – tempering out 66/65, 105/104, 121/120, and 126/125;
- Dwynwen (31f & 46) – tempering out 91/90, 121/120, 126/125, and 176/175.
The most important of these is valentino, which tempers out 196/195 and may also be characterized by tempering out 351/350 or 352/351 instead. It is supported by 77. However, it does come at the cost of a way increased complexity level. A low-complexity low-accuracy mapping is given by lupercalia, where the ~13/8 is conflated with the ~18/11. Dwynwen is a more perculiar case: the mapping is not very simple to start with, and the approximation is not ideal if we just use the tunings optimized for valentine or valentino. However, a sharper tuning especially from those sharper than 46edo may do the trick.
Another path that relates a sense of compromise is to temper out 169/168, leading to semivalentine. This has the effect of slicing the period in two, and is supported by 30, 46, and 62. Yet another possibility is given by hemivalentine, with the effect of slicing the generator of 21/20~25/24 into two 40/39's. Finally, tempering out 676/675 leads to demivalentine, with the effect of slicing the generator of 21/10~25/12 (one octave above 21/20~25/24) into two 13/9's, and is supported by 62 and 77.
Interval chain
In the following table, odd harmonics and subharmonics 1–21 are in bold.
| # | Cents* | Approximate ratios | |||
|---|---|---|---|---|---|
| 11-limit | 13-limit extensions | ||||
| Valentino | Lupercalia | Dwynwen | |||
| 0 | 0.0 | 1/1 | |||
| 1 | 77.9 | 21/20, 22/21, 25/24 | 26/25 | 27/26 | |
| 2 | 155.8 | 12/11, 11/10, 35/32 | 13/12 | ||
| 3 | 233.7 | 8/7 | 15/13 | ||
| 4 | 311.6 | 6/5 | 13/11 | ||
| 5 | 389.5 | 5/4 | 26/21, 33/26 | ||
| 6 | 467.4 | 21/16 | 13/10 | ||
| 7 | 545.3 | 11/8, 15/11 | 18/13 | ||
| 8 | 623.2 | 10/7 | 56/39 | 13/9 | |
| 9 | 701.1 | 3/2 | 52/35 | ||
| 10 | 779.0 | 11/7, 25/16 | 52/33 | ||
| 11 | 856.9 | 18/11 | 64/39 | 13/8 | |
| 12 | 934.8 | 12/7 | 26/15 | ||
| 13 | 1012.7 | 9/5, 25/14 | |||
| 14 | 1090.6 | 15/8 | 13/7 | ||
| 15 | 1168.5 | 55/28, 63/32, 96/49, 108/55, 125/64 |
39/20 | ||
| 16 | 46.4 | 33/32, 36/35, 50/49 | 40/39 | 27/26 | 26/25 |
| 17 | 124.3 | 15/14, 27/25 | 14/13 | 13/12 | |
| 18 | 202.2 | 9/8 | |||
| 19 | 280.1 | 33/28 | 13/11 | ||
| 20 | 358.0 | 27/22 | 16/13 | 39/32 | 26/21 |
| 21 | 435.9 | 9/7 | 13/10 | ||
| 22 | 513.8 | 27/20 | 35/26 | ||
| 23 | 591.7 | 45/32 | 39/28 | ||
| 24 | 669.6 | 72/49 | 52/35 | ||
| 25 | 747.5 | 54/35 | 20/13 | ||
| 26 | 825.4 | 45/28 | 21/13 | 13/8 | |
| 27 | 903.3 | 27/16 | 22/13 | ||
| 28 | 981.2 | 99/56 | |||
| 29 | 1059.1 | 90/49 | 24/13 | 13/7 | |
| 30 | 1137.0 | 27/14 | 25/13 | 39/20 | |
| 31 | 14.9 | 81/80, 99/98 | |||
* In 11-limit CWE tuning