Valentine

Valentine

Subgroups	2.3.5.7, 2.3.5.7.11
Comma basis	126/125, 1029/1024 (7-limit); 121/120, 126/125, 176/175 (11-limit)
Reduced mapping	⟨1; 9 5 -3 7]
Edo join	15 & 16
Generator (CTE)	~21/20 = 77.9 ¢
MOS scales	..., 1L 14s, 15L 1s, 15L 16s
Ploidacot	enneacot
Pergen	(P8, P5/9)
Minimax error	(7-odd limit) 4.60 ¢; ((11-limit) 21-odd limit) 9.01 ¢
Target scale size	(7-odd limit) 31 notes; ((11-limit) 21-odd limit) 46 notes

Valentine is a regular temperament that divides a tempered 3/2 into 9 equal generators which are small semitones close to 78 cents; a stack of 3 generators is interpreted as 8/7 and a stack of 5 generators is interpreted as 5/4. The generator 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 valentine might be a better description of Alpha than the rank-1 temperament that we know as Alpha today. 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. See Valentine scales for scales in this temperament.

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

Main article: Chords of valentine

Tunings

Prime-optimized tunings

7-limit Prime-Optimized Tunings

	Euclidean
	Unskewed	Skewed
Equilateral	CEE: ~15/14 = 77.7625¢	CSEE: ~15/14 = 77.7211¢
Tenney	CTE: ~15/14 = 77.8776¢	CWE: ~15/14 = 77.8673¢
Benedetti, Wilson	CBE: ~15/14 = 77.9062¢	CSBE: ~15/14 = 77.9075¢

11-limit Prime-Optimized Tunings

	Euclidean
	Unskewed	Skewed
Equilateral	CEE: ~15/14 = 78.0604¢	CSEE: ~15/14 = 77.9698¢
Tenney	CTE: ~15/14 = 77.9633¢	CWE: ~15/14 = 77.9007¢
Benedetti, Wilson	CBE: ~15/14 = 77.9388¢	CSBE: ~15/14 = 77.9028¢

Tuning spectrum

Edo generator	Eigenmonzo (unchanged-interval)*	Generator (¢)	Comments
1\16		75.000	Lower bound of 7-odd-limit diamond monotone
	12/11	75.319
3\47		76.596	47e val
	15/11	76.707
	8/7	77.058
	10/7	77.186
	5/4	77.263	Lower bound of 5-odd-limit diamond tradeoff
2\31		77.419	Lower bound of 9- and 11-odd-limit, 11-limit 15- and 21-odd-limit diamond monotone
	18/11	77.508
	15/14	77.614
9\139		77.698	139e val
	[26 25 3 -23⟩	77.707	7-odd-limit least squares
	15/8	77.733
	12/7	77.761	7-odd-limit minimax
7\108		77.778	108e val
	9/7	77.861	9-odd-limit minimax
5\77		77.922
	[23 -13 -1⟩	77.965	5-odd-limit least squares, Dave Benson's optimized tuning for Alpha
	3/2	77.995	5-odd-limit minimax
8\123		78.049	123e val
	11/7	78.249	11-odd-limit minimax
3\46		78.261
	9/5	78.277
	21/16	78.463
4\61		78.689
	11/8	78.760
	6/5	78.910	Upper bound of 5-odd-limit diamond tradeoff
1\15		80.000	Upper bound of 7-, 9- and 11-odd-limit, 11-limit 15- and 21-odd-limit diamond monotone
	22/21	80.537
	11/10	82.502
	21/20	84.867

* Besides the octave