Cathartic temperaments: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
This is a collection of rank-2 temperaments that temper out the [[ | This is a collection of rank-2 temperaments that temper out the [[cathartic comma]] ({{monzo|legend=1| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969). For the rank-3 temperament, see [[Cathartic family]]. | ||
Temperaments discussed elsewhere are: | Temperaments discussed elsewhere are: | ||
* [[Godzilla]] (+49/48 or 81/80) → [[ | * [[Godzilla]] (+49/48 or 81/80) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]] | ||
* ''[[Satin]]'' (+2100875/2097152) → [[Garischismic clan #Satin|Garischismic clan]] | |||
* ''[[Kleischismic]]'' (+32805/32768) → [[Schismatic family #Kleischismic|Schismatic family]] | |||
* ''[[Pentaorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentaorwell|Orwellismic temperaments]] | |||
* ''[[Septiquarter]]'' (+5120/5103) → [[Aberschismic temperaments #Septiquarter|Aberschismic temperaments]] | |||
* ''[[Hemiquindromeda]]'' (+67108864/66976875) → [[Quindromeda family #Hemiquindromeda|Quindromeda family]] | |||
* ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]] | * ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]] | ||
* | * [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]] | ||
* ''[[ | * ''[[Turkey (temperament)|Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]] | ||
* [[ | * ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]] | ||
* ''[[ | * ''[[Amicable]]'' (+2401/2400) → [[Amity family #Amicable|Amity family]] | ||
* ''[[Marthirds]]'' (+15625/15552) → [[Kleismic family #Marthirds|Kleismic family]] | * ''[[Marthirds]]'' (+15625/15552) → [[Kleismic family #Marthirds|Kleismic family]] | ||
* ''[[ | * ''[[Semiluna]]'' (+95703125/95551488) → [[Luna family #Semiluna|Luna family]] | ||
* ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]] | * ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]] | ||
Considered below is superlimmal. | |||
== Superlimmal == | == Superlimmal == | ||
Superlimmal is essentially an 80-form, and may be described as the {{nowrap| 80 & 311 }} temperament. It uses an ever slightly sharpened [[27/25|large limma]] as the generator, nine exceed the octave by [[126/125]]. Note that in the data that follow, the generator is its [[octave complement]], [[~]][[50/27]], so that 57 of them [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]. | Superlimmal is essentially an 80-form, and may be described as the {{nowrap| 80 & 311 }} temperament. It uses an ever slightly sharpened [[27/25|large limma]] as the generator, nine exceed the octave by [[126/125]]. Note that in the data that follow, the generator is its [[octave complement]], [[~]][[50/27]], so that 57 of them [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]. | ||
Superlimmal gets all the primes up to [[29/1|29]] reasonably covered, but is acceptable just as a 13-limit microtemperament, given a relatively simple [[comma basis]]. It can also be extended to include prime [[37/1|37]] by tempering out ([[27/25]])/([[40/37]]) = [[1000/999]] | Superlimmal gets all the primes up to [[29/1|29]] reasonably covered, but is acceptable just as a 13-limit microtemperament, given a relatively simple [[comma basis]]. It can also be extended to include prime [[37/1|37]] by mapping it to 87 generator steps, tempering out ([[27/25]])/([[40/37]]) = [[1000/999]]. Since 40/37 is the mediant of [[27/25]] and [[13/12]], this extension further consolidates the sharpened limma. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Comma list: 3025/3024, 4000/3993, 1479016/1476225 | Comma list: 3025/3024, 4000/3993, 1479016/1476225 | ||
{{Mapping|legend=0| 1 -49 -74 -117 -56 | 0 57 86 135 67 }} | |||
Optimal tuning: | Optimal tuning: | ||
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Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455 | Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455 | ||
{{Mapping|legend=0| 1 -49 -74 -117 -56 25 | 0 57 86 135 67 -24 }} | |||
Optimal tuning: | Optimal tuning: | ||
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Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224 | Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224 | ||
{{Mapping|legend=0| 1 -49 -74 -117 -56 25 -11 | 0 57 86 135 67 -24 17 }} | |||
Optimal tuning: | Optimal tuning: | ||
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Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499 | Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499 | ||
{{Mapping|legend=0| 1 -49 -74 -117 -56 25 -11 -49 | 0 57 86 135 67 -24 17 60 }} | |||
Optimal tuning: | Optimal tuning: | ||
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Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495 | Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495 | ||
{{Mapping|legend=0| 1 -49 -74 -117 -56 25 -11 -49 -15 | 0 57 86 135 67 -24 17 60 22 }} | |||
Optimal tuning: | Optimal tuning: | ||
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Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495 | Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495 | ||
{{Mapping|legend=0| 1 -49 -74 -117 -56 25 -11 -49 -15 -83 | 0 57 86 135 67 -24 17 60 22 99 }} | |||
Optimal tuning: | Optimal tuning: | ||
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Badness (Sintel): 1.09 | Badness (Sintel): 1.09 | ||
[[Category:Cathartic temperaments| ]] <!-- main article --> | |||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||