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.

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