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5L 2s/Temperaments: Difference between revisions

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Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[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 diatonic ([[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.  


== 5-limit meantone ==
== Meantone ==
{{main| Meantone }}
{{main| Meantone }}
Subgroup: 2.3.5
Subgroup: 2.3.5
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[[Tuning ranges]]:  
[[Tuning ranges]]:  


* [[diamond monotone]] range: [685.714, 720.000] (7 to 5)
* [[Diamond monotone]] range: [685.714, 720.000] (7 to 5)
* [[diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
* [[Diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
* diamond monotone and tradeoff range: [694.786, 701.955]
* Diamond monotone and tradeoff: [694.786, 701.955]


{{Vals|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
{{Vals|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
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[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/5
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/5
* [[9-odd-limit]]
* [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }}, {{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }}, {{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}]
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[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[diamond monotone]] range: [692.308, 694.737] (26 to 19)
* [[Diamond monotone]] range: [692.308, 694.737] (26 to 19)
* [[diamond tradeoff]] range: [692.353, 701.955]
* [[Diamond tradeoff]] range: [692.353, 701.955]
* diamond monotone and tradeoff range: [692.353, 694.737]
* Diamond monotone and tradeoff: [692.353, 694.737]


Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
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[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[diamond monotone]] range: [694.737, 700.000] (19 to 12)
* [[Diamond monotone]] range: [694.737, 700.000] (19 to 12)
* [[diamond tradeoff]] range: [694.786, 701.955]
* [[Diamond tradeoff]] range: [694.786, 701.955]
* diamond monotone and tradeoff range: [694.786, 700.000]
* Diamond monotone and tradeoff: [694.786, 700.000]


Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
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Tuning ranges:  
Tuning ranges:  
* [[diamond monotone]] range: [694.737, 696.774] (19 to 31)
* [[Diamond monotone]] range: [694.737, 696.774] (19 to 31)
* [[diamond tradeoff]] range: [691.202, 701.955]
* [[Diamond tradeoff]] range: [691.202, 701.955]
* diamond monotone and tradeoff range: [694.737, 696.774]
* Diamond monotone and tradeoff: [694.737, 696.774]


Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
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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, 99/98, 126/125


Mapping: [{{val| 1 0 -4 -13 -25 }}, {{val| 0 1 4 10 18 }}]
Mapping: [{{val| 1 0 -4 -13 -25 }}, {{val| 0 1 4 10 18 }}]
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Tuning ranges:  
Tuning ranges:  
* [[diamond monotone]] range: [696.774, 700.000] (31 to 12)
* [[Diamond monotone]] range: [696.774, 700.000] (31 to 12)
* [[diamond tradeoff]] range: [691.202, 701.955]
* [[Diamond tradeoff]] range: [691.202, 701.955]
* diamond monotone and tradeoff range: [696.774, 700.000]
* Diamond monotone and tradeoff: [696.774, 700.000]


[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
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== Schismic ==
== Schismic ==
Subgroup: 2.3.5.7.9.11.13.19
Subgroup: 2.3.5.7.11.13.19


Period: 1\1
Period: 1\1
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Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


{{Val list|legend=1| 41, 53, 94}}
{{Val list|legend=1| 41, 53, 94 }}


== Parapyth ==
== Parapyth ==
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[[Mapping|Period-generator mapping]]: [&lt;2: 1, 3: 0, 7: -21, 11: -14, 13: -9|, &lt;2: 0, 3: 1, 7: 15, 11: 11, 13: 8|]
[[Mapping|Period-generator mapping]]: [&lt;2: 1, 3: 0, 7: -21, 11: -14, 13: -9|, &lt;2: 0, 3: 1, 7: 15, 11: 11, 13: 8|]


Commas: 169/168, 352/351, 364/363
Comma list: 169/168, 352/351, 364/363


Gencom: [2 3/2; 169/169 352/351 364/363]
Gencom: [2 3/2; 169/169 352/351 364/363]
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Gencom mapping: [&lt;2: 1, 3: 1, 7: 4|, &lt;2: 0, 3: 1, 7: -2|]
Gencom mapping: [&lt;2: 1, 3: 1, 7: 4|, &lt;2: 0, 3: 1, 7: -2|]


EDOs: 5, 12, 17, 22, 27, 137bc
Vals: 5, 12, 17, 22, 27, 137bc


[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents
[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents
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Gencom mapping: [&lt;2: 1, 3: 1, 7: 4, 11: 7|, &lt;2: 0, 3: 1, 7: -2, 11: -6|]
Gencom mapping: [&lt;2: 1, 3: 1, 7: 4, 11: 7|, &lt;2: 0, 3: 1, 7: -2, 11: -6|]


EDOs: 5, 12, 17, 39c, 56c
Vals: 5, 12, 17, 39c, 56c


[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
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|}
|}
<references/></div></div>
<references/></div></div>
[[Mapping|Period-generator mapping]]: [&lt;1 0 -12 6|, &lt;0 1 9 -2|]
[[Mapping|Period-generator mapping]]: [{{val|1 0 -12 6}}, {{val|0 1 9 -2}}]


[[Comma]]s: 64/63, 245/243
[[Comma]]s: 64/63, 245/243
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[[Wedgie]]: {{wedgie|1 9 -2 12 -6 -30}}
[[Wedgie]]: {{wedgie|1 9 -2 12 -6 -30}}


EDOs: 5, 17, 22, 27, 49
Vals: 5, 17, 22, 27, 49


Badness: 0.0323
Badness: 0.0323
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