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	23 extension	Generator	3/2 tuning	11/8 tuning	23/16 tuning
28			385.714	685.714	557.143	642.857
31	31 & 34	31 & 34	387.097	696.774	541.935	619.355
158			387.342	698.734	554.430	630.380
127	31 & 34	31 & 34	387.402	699.213	548.031	623.622
223	31 & 34		387.444	699.552	548.879	629.596
96	31 & 34	31 & 34	387.500	700.000	550.000	625.000
353	31 & 34		387.535	700.283	550.708	628.895
257	31 & 34		387.549	700.389	550.973	630.350
161	31 & 34	31 & 34	387.578	700.621	551.553	626.087
322	31 & 34		387.578	700.621	551.553	629.814
387	31 & 34		387.597	700.775	551.938	629.457
226	31 & 34	31 & 34	387.611	700.885	552.212	626.549
452	31 & 34		387.611	700.885	552.212	629.204
291	31 & 34	31 & 34	387.629	701.031	552.577	626.804
356	31 & 34	31 & 34	387.640	701.124	552.809	626.966
421		31 & 34	387.648	701.188	550.119	627.078
65	31 & 34	31 & 34	387.692	701.538	553.846	627.692
359		31 & 34	387.744	701.950	551.532	628.412
294		31 & 34	387.755	702.041	551.020	628.571
229		31 & 34	387.773	702.183	550.218	628.821
393		31 & 34	387.786	702.290	552.672	629.008
164		31 & 34	387.805	702.439	548.780	629.268
263		31 & 34	387.833	702.662	552.091	629.658
362		31 & 34	387.845	702.762	550.276	629.834
99		31 & 34	387.879	703.030	545.455	630.303
297			387.879	703.030	549.495	626.263
331			387.915	703.323	551.057	627.190
232			387.931	703.448	553.448	625.862
365			387.945	703.562	552.329	627.945
133		31 & 34	387.970	703.759	550.376	631.579
266			387.970	703.759	550.376	627.068
167			388.024	704.192	553.293	625.150
201			388.060	704.478	549.254	626.866
34	31 & 34	31 & 34	388.235	705.882	564.706	635.294
71			388.732	709.859	557.746	625.352
37			389.189	713.514	551.351	616.216
3		31 & 34	400.000	800.000	400.000	800.000