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. { class="wikitable sortable" !|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 |}