Würschmidt

Würschmidt is a rank-2 temperament and parent of the würschmidt family, characterized by tempering out the würschmidt comma, (393216/390625). It can be treated as analogous to schismatic with the roles of the primes 3 and 5 reversed, since würschmidt is generated by a classical major third (5/4), very slightly sharpened so that eight of them make the sixth harmonic (6/1), giving 3/2 the same complexity 5/4 does in schismatic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval 625/512, which is equated to 768/625 and functions as a neutral third.

For technical data, see Würschmidt family #Würschmidt.

Extensions

Another useful interpretation of the würschmidt comma is that it makes the interval of 25/24 equal to two-thirds the size of 16/15. This can be exploited, as 16/15 factorizes into near-2:1 parts as (24/23)×(46/45), and therefore it is illogical not to set 25/24 equal to 24/23 (and 128/125 equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering S24 = 576/575 and S46² × S47 = 12167/12150 in the 2.3.5.23 subgroup. 14 generators turn out to stack to 23/1, and notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma 279936/279841.

Strong extensions to the 7-limit include septimal würschmidt, worschmidt, and whirrschmidt, but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension hemiwürschmidt, which splits the ~5/4 generator into two ~28/25's. Therefore, it may be advisable to consider würschmidt a no-sevens system, specifically in the 2.3.5.11 subgroup, where an extension that equates 128/125 with 45/44 and therefore 625/512 with 11/9 (by tempering out 243/242 and 5632/5625), finding the 11th harmonic at 20 generators up, is highly natural, in addition to the aforementioned extension to prime 23.

Interval chains

In the below, octave-reduced harmonics below 125 are indicated in bold.

Würschmidt

#	Cents*	Approximate Ratios
		2.3.5.23 subgroup	Add-11 extension
-8	497.59	4/3	162/121
-7	885.39	5/3, 192/115	92/55
-6	73.19	24/23, 25/24	23/22, 288/275
-5	460.99	30/23, 125/96	72/55, 176/135
-4	848.79	75/46, 368/225, 625/384	18/11, 44/27
-3	36.60	46/45, 128/125	45/44, 55/54
-2	424.40	23/18, 32/25	88/69, 225/176
-1	812.20	8/5, 115/72	110/69
0	0.0	1/1
1	387.80	5/4, 144/115	69/55
2	775.60	25/16, 36/23	69/44, 352/225
3	1163.40	45/23, 125/64, 736/375	88/45, 108/55
4	351.21	92/75, 225/184, 625/512	11/9, 27/22
5	739.01	23/15, 192/125	55/36, 135/88
6	1126.81	23/12, 48/25	44/23, 275/144
7	314.61	6/5, 115/96	55/46
8	702.41	3/2	121/81
9	1090.21	15/8, 216/115	207/110, 253/135
10	278.01	27/23, 75/64	88/75, 207/176
11	665.82	184/125, 135/92, 375/256	22/15, 81/55
12	1053.62	46/25, 675/368	11/6, 81/44
13	241.42	23/20, 144/125	55/48, 132/115
14	629.22	23/16, 36/25	33/23, 275/192
15	1017.02	9/5, 115/64	165/92, 242/135
16	204.82	9/8	121/108
17	592.62	45/32, 162/115	253/180
18	980.43	81/46, 225/128	44/25
19	168.23	138/125, 405/368	11/10, 243/220
20	556.03	69/50, 864/625	11/8, 243/176

Tunings

Optimized tunings

Prime-Optimized Tunings

Weight-skew\Order	Euclidean
Weil	(2.3.5) CWE: ~5/4 = 387.776¢
Tenney	(2.3.5) POTE: ~5/4 = 387.7993¢
Weil	(2.3.5.23) CWE: ~6/5 = 387.781¢
Tenney	(2.3.5.23) POTE: ~6/5 = 387.8051¢

Tuning spectrum

The below assumes the 2.3.5.11.23 subgroup extension. Note that "e" and "i" are the warts for primes 11 and 23, respectively.

Edo Generator	Eigenmonzo (Unchanged-interval)*	Generator (¢)	Comments
9\28		385.7143	28ei val
	11/9	386.3137	1/4 vishdel comma
	5/4	386.8520	Untempered tuning, lower bound of 5-odd-limit diamond tradeoff
10\31		387.0968	Lower bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone
	23/22	387.1739
	375/256	387.3542	1/11-comma
41\127		387.4016	127e val
	11/6	387.4469
	75/64	387.4582	1/10-comma
31\96		387.5000
	11/8	387.5659
52\161		387.5776
	15/8	387.5854	1/9-comma
73\226		387.6106
	11/10	387.6318
	45/32	387.6602	2/17-comma
21\65		387.6923
	23/12	387.7199
	23/16	387.7338
116\359		387.7437	359ee val
	3/2	387.7444	1/8-comma
74\229		387.7729	229e val
53\164		387.8049	164e val
	23/18	387.8178	1/2 S24
85\263		387.8327	263ee val
	9/5	387.8393	2/15-comma
	23/20	387.8431
32\99		387.8788	99e val
75\232		387.9310	232eei val
	6/5	387.9490	1/7-comma, upper bound of 5-odd-limit diamond tradeoff
43\133		387.9699	133e val
	46/25	387.9706
	23/15	388.0011
54\167		388.0240	167eei val
	48/25	388.2213	1/6-comma, upper bound of 2.3.5.23-subgroup 25-odd-limit diamond tradeoff
11\34		388.2353
	192/125	388.6028	1/5-comma
23\71		388.7324	71eei val
	768/625	389.1750	1/4-comma
12\37		389.1892	37eei val
1\3		400.0000	Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone

* besides the octave

Other tunings

• DKW (2.3.5): ~2 = 1\1, ~5/4 = 387.8015