Würschmidt
Würschmidt, is a rank-2 temperament and parent of the würschmidt family, characterized by tempering out the würschmidt comma, (393216/390625). It is generated by a classical major third (5/4), eight of which make the sixth harmonic (6/1). Four generators, therefore, reach the interval 625/512, which is equated to 768/625 and functions as a neutral third.
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.
For technical data, see Würschmidt family #Würschmidt.
Interval chains
In the below, octave-reduced harmonics below 125 are indicated in bold.
| # | Cents* | Approximate Ratios | |
|---|---|---|---|
| 2.3.5.23 subgroup | Add-11 extension | ||
| -8 | 497.59 | 4/3 | 162/121 |
| -7 | 885.39 | 5/3 | 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 | 110/69 |
| 0 | 0.0 | 1/1 | |
| 1 | 387.80 | 5/4 | 69/55 |
| 2 | 775.60 | 25/16, 36/23 | 69/44, 352/225 |
| 3 | 1163.40 | 45/23, 125/64 | 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 | 55/46 |
| 8 | 702.41 | 3/2 | 121/81 |
| 9 | 1090.21 | 15/8 | 207/110, 253/135 |
| 10 | 278.01 | 27/23, 75/64 | 88/75, 207/176 |
| 11 | 665.82 | 135/92, 184/125, 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 | 165/92, 242/135 |
| 16 | 204.82 | 9/8 | 121/108 |
| 17 | 592.62 | 45/32 | 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 |
Other tunings
- DKW (2.3.5): ~2 = 1\1, ~5/4 = 387.8015
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