Breedsmic temperaments: Difference between revisions
→Tertiaseptal: move no-11 13- and 17-limit extensions here |
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Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the {{nowrap| 31 & 171 }} temperament, and [[256/245]], [[1029/1024]] less than [[21/20]], serves as its generator. Three of these fall short of [[8/7]] by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. The [[ploidacot]] for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure). | Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the {{nowrap| 31 & 171 }} temperament, and [[256/245]], [[1029/1024]] less than [[21/20]], serves as its generator. Three of these fall short of [[8/7]] by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. The [[ploidacot]] for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure). | ||
[[171edo]] makes for an excellent tuning, although [[140edo]] ({{nowrap| {{=}} 171 - 31 }}) also makes sense, and | Tertiaseptal does not extend well to the the [[11-limit]], with three possible mappings for the [[11/1|11th harmonic]]: tertia (109 & 140), tertiaseptia (140 & 171), and septia (171 & 202). Their useful tuning ranges are indicated by the equal-temperament joins. However, since the entire range can use the same mapping for [[13/1|13]] and [[17/1|17]], we can consider tertiaseptal a no-11 13- and 17-limit temperament, in which [[625/624]], [[833/832]], [[1225/1224]], and [[4096/4095]] are tempered out. | ||
[[171edo]] makes for an excellent tuning, although [[140edo]] ({{nowrap| {{=}} 171 - 31 }}) also makes sense, and for the add-13 add-17 extension [[311edo]] ({{nowrap| {{=}} 140 + 171 }}) is especially notable. The 15- or 16-note [[mos]] can be used to explore no-3's harmony, and the 31-note mos gives plenty of room for those as well. | |||
=== 7-limit === | === 7-limit === | ||
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* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8230{{c}} (~68/65 = 77.1770{{c}}) | * CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8230{{c}} (~68/65 = 77.1770{{c}}) | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 31, 109g, 140, 171, 311, 482 }} | ||
Badness (Sintel): 0.574 | Badness (Sintel): 0.574 | ||
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=== Tertiaseptia === | === Tertiaseptia === | ||
This extension was considered by [[Gene Ward Smith]] as a 41-limit temperament<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9274.html Yahoo! Tuning Group | ''A 41-limit temperament'']</ref>. It can be extended as such by tempering out 875/874, 714/713, 703/702 and 697/696, and mapping 19, 31, 37 and 41 to 94, 105, -81 and +10 steps, respectively. | This extension is enormously complex, but preserves much of the accuracy. It was considered by [[Gene Ward Smith]] as a 41-limit temperament<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9274.html Yahoo! Tuning Group | ''A 41-limit temperament'']</ref>. It can be extended as such by tempering out 875/874, 714/713, 703/702 and 697/696, and mapping 19, 31, 37 and 41 to 94, 105, -81 and +10 steps, respectively. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||