Valentine
Valentine is a regular temperament generated by a small semitone that serves as both 21/20 and 25/24. It is a member of the starling temperaments, by tempering out 126/125, and the gamelismic clan, by tempering out 1029/1024. It extends naturally to the 11-limit by treating the generator as 22/21, tempering out 121/120, 176/175, 385/384, and 441/440.
Valentine can be viewed as a counterpart of miracle in several ways. Miracle splits the generator of slendric in two while valentine splits it in three. Miracle is generated by 15/14~16/15, the classical diatonic semitone tempered together with the septimal major semitone, while valentine is generated by 21/20~25/24, the classical chromatic semitone tempered together with the septimal minor semitone. Miracle is known for its efficiency; the same is true of valentine. The 11-odd-limit tonality diamond is covered by miracle with 22 generator steps, and by valentine with 21 generator steps.
Valentine is very closely related to Carlos Alpha, the rank-1 non-octave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in Beauty in the Beast suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank-1 temperament. Carlos tells us that "[t]he melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. Mos scales of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.
See Valentine extensions for a discussion on 13-limit extensions. See Gamelismic clan #Valentine for technical data.
Interval chain
In the following table, odd harmonics and subharmonics 1–21 are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 77.9 | 21/20, 22/21, 25/24 |
| 2 | 155.8 | 12/11, 11/10, 35/32 |
| 3 | 233.7 | 8/7 |
| 4 | 311.6 | 6/5 |
| 5 | 389.5 | 5/4 |
| 6 | 467.4 | 21/16 |
| 7 | 545.3 | 11/8, 15/11 |
| 8 | 623.2 | 10/7 |
| 9 | 701.1 | 3/2 |
| 10 | 779.0 | 11/7, 25/16 |
| 11 | 856.9 | 18/11 |
| 12 | 934.8 | 12/7 |
| 13 | 1012.7 | 9/5, 25/14 |
| 14 | 1090.6 | 15/8 |
| 15 | 1168.5 | 55/28, 63/32, 96/49, 108/55, 125/64 |
| 16 | 46.4 | 33/32, 36/35, 50/49 |
| 17 | 124.3 | 15/14, 27/25 |
| 18 | 202.2 | 9/8 |
| 19 | 280.1 | 33/28 |
| 20 | 358.0 | 27/22 |
| 21 | 435.9 | 9/7 |
| 22 | 513.8 | 27/20 |
| 23 | 591.7 | 45/32 |
| 24 | 669.6 | 72/49 |
| 25 | 747.5 | 54/35 |
| 26 | 825.4 | 45/28 |
| 27 | 903.3 | 27/16 |
| 28 | 981.2 | 99/56 |
| 29 | 1059.1 | 90/49 |
| 30 | 1137.0 | 27/14 |
| 31 | 14.9 | 81/80, 99/98 |
* In 11-limit CWE tuning
Chords
Tunings
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 1\16 | 75.000 | Lower bound of 7-odd-limit diamond monotone | |
| 11/6 | 75.319 | ||
| 15/11 | 76.707 | ||
| 7/4 | 77.058 | ||
| 7/5 | 77.186 | ||
| 5/4 | 77.263 | ||
| 2\31 | 77.419 | Lower bound of 9- and 11-odd-limit, 11-limit 15-, and 21-odd-limit diamond monotone | |
| 11/9 | 77.508 | ||
| 15/14 | 77.614 | ||
| 15/8 | 77.733 | ||
| 7/6 | 77.761 | 7-odd-limit minimax | |
| 9/7 | 77.861 | 9-odd-limit minimax | |
| 5\77 | 77.922 | ||
| 3/2 | 77.995 | 5-odd-limit minimax | |
| 11/7 | 78.249 | 11-odd-limit minimax | |
| 3\46 | 78.261 | ||
| 9/5 | 78.277 | ||
| 21/16 | 78.463 | ||
| 11/8 | 78.760 | ||
| 5/3 | 78.910 | ||
| 1\15 | 80.000 | Upper bound of 7-, 9- and 11-odd-limit, 11-limit 15-, and 21-odd-limti diamond monotone | |
| 21/11 | 80.537 | ||
| 11/10 | 82.502 | ||
| 21/20 | 84.867 |
* Besides the octave