Breedsmic temperaments: Difference between revisions
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* ''[[Hemicountercomp]]'' (+52613349376/52301766015) → [[Countercomp family #Hemicountercomp|Countercomp family]] | * ''[[Hemicountercomp]]'' (+52613349376/52301766015) → [[Countercomp family #Hemicountercomp|Countercomp family]] | ||
Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, lockerbie, unthirds, neominor, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]]. | Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, lockerbie, unthirds, neominor, cotritone, fibo, quasimoha, mintone, gorgik, retrotonic, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]]. | ||
== Tertiaseptal == | == Tertiaseptal == | ||
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Badness (Sintel): 1.33 | Badness (Sintel): 1.33 | ||
== Retrotonic == | |||
Retrotonic may be described as the {{nowrap| 58 & 311 }} temperament, with a [[ploidacot]] of 14-sheared 18-cot, the same as [[#Gorgik|gorgik]], but is strongly a 58-form. It is the generated by the retroptolemaic whole tone, [[~]][[729/640]], whose [[octave complement]] is the retroptolemaic minor seventh, ~[[1280/729]], given in the data below. | |||
Retrotonic becomes much more interesting in the [[11-limit]], where it tempers out important commas such as the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), and the [[symbiotic comma]] (19712/19683). This leads to a 13-limit extension where [[1575/1573]], [[2080/2079]], and [[2200/2197]] are tempered out, enabling their corresponding [[essentially tempered chord]]s. Recommended retrotonic tunings include [[311edo]], [[369edo]] (in the 369f val) and [[680edo]] (in the 680ef val). | |||
Retrotonic was named by [[Flora Canou]] in 2026 after one of its possible generators, the retroptolemaic whole tone. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 2401/2400, {{monzo| 29 -28 3 3 }} | |||
{{Mapping|legend=1| 1 -13 -86 -45 | 0 18 109 59 }} | |||
: mapping generators: ~2, ~1280/729 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.9156{{c}}, ~1280/729 = 972.2848{{c}} | |||
: [[error map]]: {{val| -0.084 +0.269 -0.012 -0.225 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1280/729 = 972.3523{{c}} | |||
: error map: {{val| 0.000 +0.386 +0.088 -0.040 }} | |||
{{Optimal ET sequence|legend=1| 58, 195c, 253, 311, 369, 680 }} | |||
[[Badness]] (Sintel): 4.25 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 2401/2400, 4000/3993, 19712/19683 | |||
{{Mapping|legend=0| 1 -13 -86 -45 -80 | 0 18 109 59 103 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.9156{{c}}, ~135/77 = 972.1718{{c}} | |||
: error map: {{val| -0.099 +0.221 -0.196 -0.346 +0.577 }} | |||
* CWE: ~2 = 1200.0000{{c}}, ~135/77 = 972.3507{{c}} | |||
: error map: {{val| 0.000 +0.358 -0.083 -0.132 +0.808 }} | |||
{{Optimal ET sequence|legend=0| 58, 195ce, 253, 311, 680e, 991e }} | |||
Badness (Sintel): 1.69 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 1575/1573, 2080/2079, 2200/2197, 2401/2400 | |||
{{Mapping|legend=0| 1 -13 -86 -45 -80 -83 | 0 18 109 59 103 107 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.8883{{c}}, ~135/77 = 972.2604{{c}} | |||
: error map: {{val| -0.112 +0.184 -0.325 -0.436 +0.438 +0.605 }} | |||
* CWE: ~2 = 1200.0000{{c}}, ~135/77 = 972.3497{{c}} | |||
: error map: {{val| 0.000 +0.339 -0.201 -0.196 +0.697 +0.886 }} | |||
{{Optimal ET sequence|legend=0| 58, 195cef, 253, 311, 680ef, 991ef }} | |||
Badness (Sintel): 1.10 | |||
== Hemigoldis == | == Hemigoldis == | ||