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 | 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 |
