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5L 2s/Temperaments: Difference between revisions

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[[Tuning ranges]]:  
[[Tuning ranges]]:  
 
* [[Diamond monotone]] range: [685.714, 720.000] (4\7 to 3\5)
* [[Diamond monotone]] range: [685.714, 720.000] (7 to 5)
* [[Diamond tradeoff]] range: [694.786, 701.955]
* [[Diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
* Diamond monotone and tradeoff: [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 }}


[[Badness]]: 0.00736
[[Badness]]: 0.007381


=== Flattone ===
=== Flattone ===
Subgroup: 2.3.5.7.13
Subgroup: 2.3.5.7 or 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]]: 65/64, 81/80, 105/104
 
[[Mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]


Mapping generators: ~2, ~3
[[Gencom]]: [2 3/2; 65/64 81/80 105/104]


[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}
[[Gencom]] [[mapping]]: [{{val|1 1 0 8 6}}, {{val|0 1 4 -9 -4}}]


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]: ~3/2 = {{Monzo| 8/13 0 1/13 -1/13 }}
: [{{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]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [{{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 }}]
: Eigenmonzos (unchanged intervals): 2, 9/7
: Eigenmonzos (unchanged intervals): 2, 9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: [692.308, 694.737] (26 to 19)
* [[Diamond monotone]] range: [692.308, 694.737] (15\26 to 11\19)
* [[Diamond tradeoff]] range: [692.353, 701.955]
* 7-odd-limit [[diamond tradeoff]]: [692.353, 701.955]
* Diamond monotone and tradeoff: [692.353, 694.737]
* 9-odd-limit diamond tradeoff: [691.202, 701.955]
* 7-odd-limit diamond monotone and tradeoff: [692.353, 694.737]
* 9-odd-limit diamond monotone and tradeoff: [692.308, 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.


{{Vals|legend=1| 7, 19, 26, 45 }}
{{Vals|legend=1| 7, 19, 26, 45f }}


[[Badness]]: 0.0386
[[Tp_tuning#T2 tuning|RMS error]]: 1.742 cents


=== Septimal meantone ===
=== Septimal meantone ===
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Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
[[Wedgie]]: {{wedgie| 1 4 10 4 13 12 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]
* 7- and [[9-odd-limit]]: ~3/2 = {{Monzo| 0 0 1/4 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: [694.737, 700.000] (19 to 12)
* [[Diamond monotone]] range: [694.737, 700.000] (11\19 to 7\12)
* [[Diamond tradeoff]] range: [694.786, 701.955]
* 7-odd-limit [[diamond tradeoff]]: [694.786, 701.955]
* Diamond monotone and tradeoff: [694.786, 700.000]
* 9-odd-limit diamond tradeoff: [691.202, 701.955]
* 7-odd-limit diamond monotone and tradeoff: [694.786, 700.000]
* 9-odd-limit diamond monotone and tradeoff: [694.737, 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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{{Vals|legend=1| 12, 19, 31, 81, 112b, 143b }}
{{Vals|legend=1| 12, 19, 31, 81, 112b, 143b }}


[[Badness]]: 0.0137
[[Badness]]: 0.013707


==== Meanpop ====
==== Meanpop ====
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Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]: 1/4 comma
* [[11-odd-limit]]: ~3/2 = {{Monzo| 0 0 1/4 }}
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| -3 0 5/2 0 0 }}, {{Monzo| 11 0 -13/4 0 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


Tuning ranges:  
Tuning ranges:  
* [[Diamond monotone]] range: [694.737, 696.774] (19 to 31)
* [[Diamond monotone]] range: [694.737, 696.774] (11\19 to 18\31)
* [[Diamond tradeoff]] range: [691.202, 701.955]
* [[Diamond tradeoff]] range: [691.202, 701.955]
* Diamond monotone and tradeoff: [694.737, 696.774]
* Diamond monotone and tradeoff: [694.737, 696.774]
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{{Vals|legend=1| 12e, 19, 31, 81 }}
{{Vals|legend=1| 12e, 19, 31, 81 }}


[[Badness]]: 0.0215
[[Badness]]: 0.021543


==== Huygens ====
==== Huygens ====
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Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]
* [[11-odd-limit]]: ~3/2 = {{Monzo| 9/16 -1/8 0 0 1/16 }}
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 25/16 -1/8 0 0 1/16 }}, {{Monzo| 9/4 -1/2 0 0 1/4 }}, {{Monzo| 21/8 -5/4 0 0 5/8 }}, {{Monzo| 25/8 -9/4 0 0 9/8 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 11/9
: [[Eigenmonzo]]s (unchanged intervals): 2, 11/9


Tuning ranges:  
Tuning ranges:  
* [[Diamond monotone]] range: [696.774, 700.000] (31 to 12)
* [[Diamond monotone]] range: [696.774, 700.000] (18\31 to 7\12)
* [[Diamond tradeoff]] range: [691.202, 701.955]
* [[Diamond tradeoff]] range: [691.202, 701.955]
* Diamond monotone and tradeoff: [696.774, 700.000]
* Diamond monotone and tradeoff: [696.774, 700.000]
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{{Vals|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}
{{Vals|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}


Badness: 0.0170
[[Badness]]: 0.017027


== Schismic ==
== Schismic ==
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|}
|}
<references/></div></div>
<references/></div></div>
Comma list: 225/224, 275/273, 325/324, 385/384, 513/512
[[Comma list]]: 190/189, 209/208, 225/224, 275/273, 325/324


Mapping: [{{val|2: 1, 3: 0, 5: 15, 7: 25, 11: -33, 13: -28, 19: 9 }}, {{val|2: 0, 3: 1, 5: -8, 7: -14, 11: 23, 13: 20, 19: -3 }}]
[[Gencom]]: [2 3/2; 190/189 209/208 225/224 275/273 325/324]


Mapping generators: ~2, ~3
[[Gencom]] [[mapping]]: [{{val|1 1 7 11 -10 -8 6 }}, {{val|0 1 -8 -14 23 20 -3 }}]


{{Val list|legend=1| 41, 53, 94 }}
{{Val list|legend=1| 41, 53, 94 }}
[[Tp_tuning#T2 tuning|RMS error]]: 0.6486 cents


== Parapyth ==
== Parapyth ==
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|}
<references/></div></div>
<references/></div></div>
[[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|]
[[Comma list]]: 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]


Gencom mapping: [&lt;2: 1, 3: 1, 7: -6, 11: -3, 13: -1|, &lt;2: 0, 3: 1, 7: 15, 11: 11, 13: 8|]
[[Gencom]] [[mapping]]: [{{val| 1 1 -6 -3 -1 }}, {{val| 0 1 15 11 8 }}]


EDOs: 17, 46, 63
{{Val list|legend=1| 17, 46, 63 }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents
[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents
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|}
<references/></div></div>
<references/></div></div>
[[Mapping|Period-generator mapping]]: [&lt;2: 1 3: 2, 7: 2|, &lt;2: 0, 3: -1, 7: 2|]
[[Comma]]: 64/63


Comma: 64/63
[[Gencom]]: [2 3/2; 64/63]


Gencom: [2 3/2; 64/63]
[[Gencom mapping]]: [{{val| 1 1 4 }}, {{val| 0 1 -2 }}]


Gencom mapping: [&lt;2: 1, 3: 1, 7: 4|, &lt;2: 0, 3: 1, 7: -2|]
{{Val list|legend=1| 5, 12, 17, 22, 27, 137bd }}


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
=== Supra ===
=== Supra ===
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11
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|}
<references/></div></div>
<references/></div></div>
[[Mapping|Period-generator mapping]]: [&lt;2: 1, 3: 0, 7: 6, 11: 13|, &lt;2: 0, 3: 1, 7: -2, 11: -6|]
Comma list: 64/63, 99/98
 
Commas: 64/63, 99/98


Gencom: [2 3/2; 64/63 99/98]
Gencom: [2 3/2; 64/63 99/98]


Gencom mapping: [&lt;2: 1, 3: 1, 7: 4, 11: 7|, &lt;2: 0, 3: 1, 7: -2, 11: -6|]
Gencom mapping: [{{val| 1 1 4 7 }}, {{val| 0 1 -2 -6 }}]


Vals: 5, 12, 17, 39c, 56c
Vals: {{Val list| 5, 12, 17, 39c, 56d }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
RMS error: 1.977 cents


=== Superpyth ===
=== Superpyth ===
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|}
|}
<references/></div></div>
<references/></div></div>
[[Mapping|Period-generator mapping]]: [{{val|1 0 -12 6}}, {{val|0 1 9 -2}}]
[[Comma list]]: 64/63, 245/243


[[Comma]]s: 64/63, 245/243
[[Mapping]]: [{{val|1 0 -12 6}}, {{val|0 1 9 -2}}]


[[Wedgie]]: {{wedgie|1 9 -2 12 -6 -30}}
[[Wedgie]]: {{wedgie|1 9 -2 12 -6 -30}}


Vals: 5, 17, 22, 27, 49
Vals: {{Val list| 5, 17, 22, 27, 49 }}


Badness: 0.0323
Badness: 0.032318


[[Category:Scales]]
[[Category:Scales]]
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