Wurschmidt

Wurschmidt, Würschmidt, or Wuerschmidt, 31 & 34, is a temperament that splits 6/1 into 8 slightly sharp 5/4's. It is notable for extending structurally into higher-limit subgroups by tempering together a sequence of diesis-sized superparticulars: 46/45 ~ 47/46 ~ 48/47 ~ 49/48 ~ 50/49 (all equated with 128/125). (Primes 11 and 17 can be added by tempering out S45 (equivalently, tempering out 243/242) and S50 respectively; however, adding prime 11 implies less extra damage than prime 17, so we'll treat Wurschmidt as canonically having prime 11 but not 17.)

Squares, 17 & 31, is an index-2 subtemperament, which can be considered a 2.3.7.11.23.25 temperament, with a generator of 32/25~23/18~14/11~9/7. However, it is most accurately interpreted as a no-7s temperament. (You can't add 7 to a strong extension of Wurschmidt without high damage.)

Interval chain

Interpretations in parentheses pertain to 2.3.7.11 Squares only.

• 1 gen = 5/4
• 2 gens = 25/16 ~ 36/23 (~ 11/7 ~ 14/9)
• 3 gens = 44/45 ~ 45/46 ~ 46/47 ~ 47/48 ~ 48/49 ~ 49/50
• 4 gens = 11/9 ~ 27/22 ~ 49/40 ~ 60/49
• 5 gens = 49/32 ~ 72/47
• 6 gens = 48/25 ~ 23/12 ~ 44/23
• 7 gens = 6/5
• 8 gens = 3/2
• 9 gens = 15/8
• 10 gens = 27/23 (~ 7/6)
• 11 gens = 47/32
• 12 gens = 11/6 ~ 46/25
• 13 gens = 23/20
• 14 gens = 23/16
• 15 gens = 9/5
• 16 gens = 9/8
• 17 gens = 45/32
• 18 gens = 225/128 ~ 81/46 (~ 7/4)
• 19 gens = 11/10 ~ 54/49
• 20 gens = 11/8
• 21 gens = 55/32
• 22 gens = 99/92
• 23 gens = 27/20
• 24 gens = 27/16
• 25 gens = 135/128
• 26 gens = 33/25
• 27 gens = 33/20
• 28 gens = 33/32
• 29 gens = 165/128
• 30 gens = 81/50
• 31 gens = 81/80

Extensions

Hemiwurschmidt is a weak extension that divides the 5/4 into two 28/25's. As such, it provides a high-accuracy extension of Didacus.

Patent vals

The following patent vals support Wurschmidt. Vals contorted in 2.3.5 are not included.

Edo	11 extension	Generator	3/2 tuning	11/8 tuning
28		385.714	685.714	557.143
31	31 & 34	387.097	696.774	541.935
158		387.342	698.734	554.430
127	31 & 34	387.402	699.213	548.031
223	31 & 34	387.444	699.552	548.879
96	31 & 34	387.500	700.000	550.000
353	31 & 34	387.535	700.283	550.708
257	31 & 34	387.549	700.389	550.973
161	31 & 34	387.578	700.621	551.553
387	31 & 34	387.597	700.775	551.938
226	31 & 34	387.611	700.885	552.212
291	31 & 34	387.629	701.031	552.577
356	31 & 34	387.640	701.124	552.809
421		387.648	701.188	550.119
65	31 & 34	387.692	701.538	553.846
359		387.744	701.950	551.532
294		387.755	702.041	551.020
229		387.773	702.183	550.218
393		387.786	702.290	552.672
164		387.805	702.439	548.780
263		387.833	702.662	552.091
362		387.845	702.762	550.276
99		387.879	703.030	545.455
331		387.915	703.323	551.057
232		387.931	703.448	553.448
365		387.945	703.562	552.329
133		387.970	703.759	550.376
167		388.024	704.192	553.293
201		388.060	704.478	549.254
34	31 & 34	388.235	705.882	564.706
71		388.732	709.859	557.746
37		389.189	713.514	551.351
3		400.000	800.000	400.000