5L 2s/Temperaments: Difference between revisions
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Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the 5L 2s range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music. | Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the 5L 2s range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music. | ||
== Meantone (12&19, 2.3.5) == | |||
{{main| Meantone }} | {{main| Meantone }} | ||
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<references/></div></div> | <references/></div></div> | ||
Comma list: 81/80 | Comma list: 81/80 | ||
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[[Badness]]: 0.00736 | [[Badness]]: 0.00736 | ||
=== Flattone (19&26, 2.3.5.7.13) === | |||
Period: 1\1 | Period: 1\1 | ||
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|} | |} | ||
<references/></div></div> | <references/></div></div> | ||
[[Comma]] list: 81/80, 525/512 | [[Comma]] list: 81/80, 525/512 | ||
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[[Badness]]: 0.0386 | [[Badness]]: 0.0386 | ||
=== Septimal meantone (19&12, 2.3.5.7) === | |||
Period: 1\1 | Period: 1\1 | ||
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<references/></div></div> | <references/></div></div> | ||
[[Comma]] list: 81/80, 126/125 | [[Comma]] list: 81/80, 126/125 | ||
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[[Badness]]: 0.0137 | [[Badness]]: 0.0137 | ||
==== Meanpop (31&50, 2.3.5.7.11) ==== | |||
Period: 1\1 | Period: 1\1 | ||
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Mapping: Same as septimal meantone, plus -13 gens = 11/8 | Mapping: Same as septimal meantone, plus -13 gens = 11/8 | ||
Comma list: 81/80, 126/125, 385/384 | Comma list: 81/80, 126/125, 385/384 | ||
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[[Badness]]: 0.0215 | [[Badness]]: 0.0215 | ||
==== Huygens (31&43, 2.3.5.7.11) ==== | |||
[[Period]]: 1\1 | [[Period]]: 1\1 | ||
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Mapping: Same as septimal meantone, plus 18 gens = 11/8 | Mapping: Same as septimal meantone, plus 18 gens = 11/8 | ||
Comma list: 81/80, 126/125, 99/98 | Comma list: 81/80, 126/125, 99/98 | ||
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Badness: 0.0170 | Badness: 0.0170 | ||
== Schismic (41&53, 2.3.5.7.11.13.19) == | |||
Period: 1\1 | Period: 1\1 | ||
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|} | |} | ||
<references/></div></div> | <references/></div></div> | ||
Comma list: 225/224, 275/273, 325/324, 385/384, 513/512 | Comma list: 225/224, 275/273, 325/324, 385/384, 513/512 | ||
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{{Val list|legend=1| 41, 53, 94}} | {{Val list|legend=1| 41, 53, 94}} | ||
== Parapyth (29&17, 2.3.7.11.13) == | |||
Period: 1\1 | Period: 1\1 | ||
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|} | |} | ||
<references/></div></div> | <references/></div></div> | ||
[[Mapping|Period-generator mapping]]: [<1 0 -21 -14 -9|, <0 1 15 11 8|] | [[Mapping|Period-generator mapping]]: [<1 0 -21 -14 -9|, <0 1 15 11 8|] | ||
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[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents | [[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents | ||
== Archy (17&5, 2.3.7) == | |||
Period: 1\1 | Period: 1\1 | ||
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|} | |} | ||
<references/></div></div> | <references/></div></div> | ||
[[Mapping|Period-generator mapping]]: [<1 2 2|, <0 -1 2|] | [[Mapping|Period-generator mapping]]: [<1 2 2|, <0 -1 2|] | ||
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[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents | [[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents | ||
=== Supra (17&22, 2.3.7.11) === | |||
Period: 1\1 | Period: 1\1 | ||
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|} | |} | ||
<references/></div></div> | <references/></div></div> | ||
[[Mapping|Period-generator mapping]]: [<1 0 6 13|, <0 1 -2 -6|] | [[Mapping|Period-generator mapping]]: [<1 0 6 13|, <0 1 -2 -6|] | ||
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[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents | [[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents | ||
=== Superpyth (22&27, 2.3.5.7) === | |||
Period: 1\1 | Period: 1\1 | ||
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|} | |} | ||
<references/></div></div> | <references/></div></div> | ||
[[Mapping|Period-generator mapping]]: [<1 0 -12 6|, <0 1 9 -2|] | [[Mapping|Period-generator mapping]]: [<1 0 -12 6|, <0 1 9 -2|] | ||
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Badness: 0.0323 | Badness: 0.0323 | ||
[[Category:Scales]] | [[Category:Scales]] | ||