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 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.