Wurschmidt: Difference between revisions

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'''Würschmidt''', '''Wurschmidt''', or '''Wuerschmidt''', 31 & 34, is a temperament that splits 6/1 into 8 slightly sharp 5/4's. It is notable for [[Canonical extension|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 resp. S50; however, adding prime 11 implies less extra damage than prime 17, so we'll treat Wurschmidt as canonically having prime 11.)
'''Wurschmidt''', '''Würschmidt''', or '''Wuerschmidt''', 2.3.5[{{e|31}} & {{e|34}}], is a temperament that splits 6/1 into 8 slightly sharp 5/4's. It is notable for [[Canonical extension|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.)
'''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.)
Line 6: Line 6:
* 1 gen = '''5/4'''
* 1 gen = '''5/4'''
* 2 gens = '''25/16''' ~ 36/23 (~ 11/7 ~ 14/9)
* 2 gens = '''25/16''' ~ 36/23 (~ 11/7 ~ 14/9)
* 3 gens = 2/1 complement of 46/45 ~ 47/46 ~ 48/47 ~ 49/48 ~ 50/49
* 3 gens = 44/45 ~ 45/46 ~ 46/47 ~ 47/48 ~ 48/49 ~ 49/50
* 4 gens = 11/9
* 4 gens = 11/9 ~ 27/22 ~ 49/40 ~ 60/49
* 5 gens = '''49/32'''
* 5 gens = 49/32 ~ 72/47
* 6 gens = 48/25 ~ 23/12 ~ 44/23
* 6 gens = 48/25 ~ 23/12 ~ 44/23
* 7 gens = 6/5
* 7 gens = 6/5
Line 22: Line 22:
* 17 gens = 45/32
* 17 gens = 45/32
* 18 gens = 225/128 ~ 81/46 (~ '''7/4''')
* 18 gens = 225/128 ~ 81/46 (~ '''7/4''')
* 19 gens = 11/10
* 19 gens = 11/10 ~ 54/49
* 20 gens = '''11/8'''
* 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!!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
|}
{{Navbox regtemp}}
{{Navbox regtemp}}
{{Cat|temperaments}}
{{Cat|temperaments}}

Latest revision as of 13:58, 14 April 2026

Wurschmidt, Würschmidt, or Wuerschmidt, 2.3.5[​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


View • Talk • EditRegular temperaments
Rank-2
Acot Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave)
Monocot Meantone • Schismic • Gentle-fifth temperaments • Archy
Complexity 2 Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot)
Complexity 3 Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot)
Complexity 4 Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot)
Complexity 5-6 Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot)
Higher complexity Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (epsilon-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot)
Straddle-3 A-Team (alter-tricot) • Machine (alter-monocot)
No-3 Mabilic (alpha-triseph) • Orgone (trimech) • Didacus (diseph)
No-octaves Sensamagic (monogem)
Exotemperaments Dicot • Mavila • Father
Higher-rank
Rank-3 Hemifamity • Marvel • Parapyth
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