Wurschmidt: Difference between revisions
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''' | '''Wurschmidt''', '''Würschmidt''', 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 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, | '''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 == | == Interval chain == | ||
Interpretations in parentheses pertain to 2.3.7.11 Squares only. | |||
* 1 gen = '''5/4''' | * 1 gen = '''5/4''' | ||
* 2 gens = 25/16 ~ 36/23 | * 2 gens = '''25/16''' ~ 36/23 (~ 11/7 ~ 14/9) | ||
* 3 gens = | * 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 | ||
* 8 gens = '''3/2''' | * 8 gens = '''3/2''' | ||
* 9 gens = '''15/8''' | * 9 gens = '''15/8''' | ||
* 10 gens = 27/23 | * 10 gens = 27/23 (~ 7/6) | ||
* 11 gens = 47/32 | * 11 gens = 47/32 | ||
* 12 gens = 11/6 ~ 46/25 | * 12 gens = 11/6 ~ 46/25 | ||
* 13 gens = 23/20 | * 13 gens = 23/20 | ||
* 14 gens = 23/16 | * 14 gens = '''23/16''' | ||
* 15 gens = 9/5 | * 15 gens = 9/5 | ||
* 16 gens = '''9/8''' | * 16 gens = '''9/8''' | ||
* 17 gens = 45/32 | * 17 gens = 45/32 | ||
* 18 gens = 225/128 ~ 81/46 | * 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. | |||
{{Navbox regtemp}} | {{Navbox regtemp}} | ||
{{Cat|temperaments}} | {{Cat|temperaments}} | ||
Latest revision as of 01:12, 25 March 2026
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.
| 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 | Trismegistus (alpha-triseph) • Orgone (trimech) • Didacus (diseph) |
| No-octaves | Sensamagic (monogem) |
| Exotemperament | Dicot • Mavila • Father |
| Higher-rank | |
| Rank-3 | Hemifamity • Marvel • Parapyth |
