Canousmic temperaments: Difference between revisions
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Note that 4802000/4782969 = 2×([[10/9]])<sup>3</sup>/([[9/7]])<sup>4</sup>, these intervals tend to have lower complexity. | Note that 4802000/4782969 = 2×([[10/9]])<sup>3</sup>/([[9/7]])<sup>4</sup>, these intervals tend to have lower complexity. | ||
Temperaments not dicussed here include [[Meantone family #Godzilla| | Temperaments not dicussed here include: | ||
* [[Sycamore family #Betic|Betic]] | * [[Meantone family #Godzilla|Godzilla]] ({49/48, 81/80}) | ||
* [[Amicable]] | * [[Sycamore family #Betic|Betic]] ({225/224, 1071875/1062882}) | ||
* [[Ragismic microtemperaments #Parakleismic|Parakleismic]] | * [[Orwellismic temperaments #Pentorwell|Pentorwell]] ({1728/1715, 179200/177147}) | ||
* [[Hemifamity temperaments #Septiquarter|Septiquarter]] | * [[Breedsmic temperaments #Amicable|Amicable]] ({2401/2400, 1600000/1594323}) | ||
* [[Schismatic family #Kleischismic|Kleischismic]] | * [[Ragismic microtemperaments #Parakleismic|Parakleismic]] ({3136/3125, 4375/4374}) | ||
* [[Hemifamity temperaments #Septiquarter|Septiquarter]] ({5120/5103, 420175/419904}) | |||
* [[Schismatic family #Kleischismic|Kleischismic]] ({32805/32768, 1562500/1492992}) | |||
Discussed below are satin and semiluna. | Discussed below are satin and semiluna. | ||
Revision as of 05:50, 25 April 2021
These are rank-2 temperaments that temper out the canousma, 4802000/4782969 = [4 -14 3 4⟩. For the rank-3 temperament, see Canou family.
Note that 4802000/4782969 = 2×(10/9)3/(9/7)4, these intervals tend to have lower complexity.
Temperaments not dicussed here include:
- Godzilla ({49/48, 81/80})
- Betic ({225/224, 1071875/1062882})
- Pentorwell ({1728/1715, 179200/177147})
- Amicable ({2401/2400, 1600000/1594323})
- Parakleismic ({3136/3125, 4375/4374})
- Septiquarter ({5120/5103, 420175/419904})
- Kleischismic ({32805/32768, 1562500/1492992})
Discussed below are satin and semiluna.
Satin
5-limit
Subgroup: 2.3.5
Comma list: [104 -70 3⟩
Mapping: [⟨1 2 12], ⟨0 -3 -70]]
POTE generator: ~[-34 23 -1⟩ = 165.907
Badness: 2.8530
7-limit
Subgroup: 2.3.5.7
Comma list: 2100875/2097152, 4802000/4782969
Mapping: [⟨1 2 12 -3], ⟨0 -3 -70 42]]
POTE generator: ~8575/7776 = 165.913
Badness: 0.1972
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4000/3993, 19712/19683, 41503/41472
Mapping: [⟨1 2 12 -3 13], ⟨0 -3 -70 42 -69]]
POTE generator: ~11/10 = 165.915
Badness: 0.0580
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689
Mapping: [⟨1 2 12 -3 13 -1], ⟨0 -3 -70 42 -69 34]]
POTE generator: ~11/10 = 165.914
Badness: 0.0303
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
Mapping: [⟨1 2 12 -3 13 -1 11], ⟨0 -3 -70 42 -69 34 -50]]
POTE generator: ~11/10 = 165.913
Badness: 0.0200
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573
Mapping: [⟨1 2 12 -3 13 -1 11 16], ⟨0 -3 -70 42 -69 34 -50 -85]]
POTE generator: ~11/10 = 165.913
Badness: 0.0145
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155
Mapping: [⟨1 2 12 -3 13 -1 11 16 16], ⟨0 -3 -70 42 -69 34 -50 -85 -83]]
POTE generator: ~11/10 = 165.914
Badness: 0.0122
Semiluna
Subgroup: 2.3.5.7
Comma list: 4802000/4782969, 95703125/95551488
Mapping: [⟨2 8 4 23], ⟨0 -15 2 -54]]
POTE generator: ~2187/1960 = 193.1725
Badness: 0.1922
11-limit
Subgroup: 2.3.5.7.11
Comma list: 5632/5625, 9801/9800, 14641/14580
Mapping: [⟨2 8 4 23 14], ⟨0 -15 2 -54 -22]]
POTE generator: ~121/108 = 193.1732
Badness: 0.0678