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 (tempering out 225/224, finding 7 at +18 generator steps), worschmidt (tempering out 126/125, finding 7 at -13 generator steps), and whirrschmidt (tempering out 4375/4374, finding 7 at +52 generator steps), 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 by tempering out 3136/3125 alongside 6144/6125 (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma S161 = 25921/25920).
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 chain
In the below, octave-reduced harmonics 1–125 are indicated in bold. All intervals are in the 625-odd limit.
| # | Cents* | Approximate ratios | |
|---|---|---|---|
| 2.3.5.23 subgroup | Add-11 extension | ||
| 0 | 0.00 | 1/1 | |
| 1 | 387.78 | 5/4, 144/115 | 69/55 |
| 2 | 775.55 | 25/16, 36/23 | 69/44, 352/225 |
| 3 | 1163.33 | 45/23, 125/64, 736/375 | 88/45, 108/55 |
| 4 | 351.10 | 92/75, 225/184, 625/512 | 11/9, 27/22 |
| 5 | 738.88 | 23/15, 192/125 | 55/36, 135/88 |
| 6 | 1126.66 | 23/12, 48/25 | 44/23 |
| 7 | 314.43 | 6/5, 115/96 | 55/46 |
| 8 | 702.20 | 3/2 | 121/81 |
| 9 | 1089.98 | 15/8, 216/115 | 207/110, 253/135 |
| 10 | 277.76 | 27/23, 75/64 | 88/75, 207/176 |
| 11 | 665.54 | 184/125, 135/92, 375/256 | 22/15, 81/55 |
| 12 | 1053.31 | 46/25, 675/368 | 11/6, 81/44 |
| 13 | 241.09 | 23/20, 144/125 | 55/48, 132/115 |
| 14 | 628.86 | 23/16, 36/25 | 33/23 |
| 15 | 1016.64 | 9/5, 115/64 | 165/92, 242/135 |
| 16 | 204.42 | 9/8 | 121/108 |
| 17 | 592.19 | 45/32, 162/115 | 253/180 |
| 18 | 979.97 | 81/46, 225/128 | 44/25 |
| 19 | 167.74 | 138/125, 405/368 | 11/10, 243/220 |
| 20 | 555.52 | 69/50, 864/625 | 11/8, 243/176 |
| 21 | 555.52 | 69/40, 216/125 | 55/32 |
| 22 | 555.52 | 27/25, 69/64 | 99/92 |
| 23 | 555.52 | 27/20, 345/256 | |
| 24 | 555.52 | 27/16 | 253/150 |
| 25 | 555.52 | 135/128, 243/230 | 132/125, 253/240 |
| 26 | 555.52 | 243/184, 828/625 | 33/25, 253/192 |
| 27 | 555.52 | 207/125 | 33/20 |
| 28 | 555.52 | 207/200, 648/625 | 33/32 |
| 29 | 555.52 | 162/125, 207/160 | 165/128 |
| 30 | 555.52 | 81/50, 207/128 | 121/75 |
| 31 | 555.52 | 81/80 | 121/120 |
* In 5-limit CWE tuning
Tunings
Optimized tunings
| Weight-skew\Order | Euclidean |
|---|---|
| Tenney | (2.3.5) CTE: ~5/4 = 387.734¢ |
| Weil | (2.3.5) CWE: ~5/4 = 387.776¢ |
| Tenney | (2.3.5.23) CTE: ~6/5 = 387.734¢ |
| Weil | (2.3.5.23) CWE: ~6/5 = 387.781¢ |
| Weight-skew\Order | Euclidean |
|---|---|
| Tenney | (2.3.5) POTE: ~5/4 = 387.7993¢ |
| Tenney | (2.3.5.23) POTE: ~6/5 = 387.8051¢ |
| Optimized chord | Generator value | Polynomial | Further notes |
|---|---|---|---|
| 3:4:5 (+1 +1) | ~5/4 = 387.4975 | g8 + 8g - 16 = 0 | 1-3-5 equal-beating tuning, close to 3/29-comma |
| 4:5:6 (+1 +1) | ~5/4 = 388.1207 | g8 - 8g + 8 = 0 | 1-3-5 equal-beating tuning, close to 3/19-comma |
| 10:12:15 (+2 +3) | ~5/4 = 388.2216 | g8 - 2g7 + 4 = 0 | Close to 1/6-comma |
| 15:18:23 (+3 +5) | ~5/4 = 387.9215 | 4g7 - 3g5 - 10 = 0 |
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 | ||
| 3/2 | 387.7444 | 1/8-comma | |
| 95\294 | 387.7551 | 294e val | |
| 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 | |
| 5/3 | 387.9490 | 1/7-comma, upper bound of 5-odd-limit diamond tradeoff | |
| 43\133 | 387.9699 | 133e val | |
| 25/23 | 387.9706 | ||
| 23/15 | 388.0011 | ||
| 54\167 | 388.0240 | 167eei val | |
| 25/24 | 388.2213 | 1/6-comma, upper bound of 2.3.5.23-subgroup 25-odd-limit diamond tradeoff | |
| 11\34 | 388.2353 | ||
| 125/96 | 388.6028 | 1/5-comma | |
| 23\71 | 388.7324 | 71eei val | |
| 625/384 | 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
- 5-odd-limit minimax: ~2 = 1\1, ~5/4 = 387.7444 (eigenmonzo 3/2 aka 1/8-comma, generator = 61/8)
Music
- Extrospection (2013) – play | SoundCloud – Würschmidt[16] in 31edo tuning.