Meantone family: Difference between revisions
m →Trimean: Extend to lower, less accurate edos. |
Cmloegcmluin (talk | contribs) unchanged interval → unchanged-interval |
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[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} | * [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} | ||
: [[Eigenmonzo basis]]: 2.5 | : [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
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* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} | * [[7-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 }}] | : [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3 0 5/2 0 }}] | ||
: [[Eigenmonzo basis]]: 2.5 | : [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
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* 11-odd-limit: ~3/2 = {{Monzo| 9/16 -1/8 0 0 1/16 }} | * 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 }}] | : [{{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 basis: 2.11/9 | : Eigenmonzo basis (unchanged-interval basis): 2.11/9 | ||
Tuning ranges: | Tuning ranges: | ||
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Minimax tuning: | Minimax tuning: | ||
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }} | * [[13-odd-limit|13-]] and [[15-odd-limit]]: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }} | ||
: Eigenmonzo basis: 2.11/9 | : Eigenmonzo basis (unchanged-interval basis): 2.11/9 | ||
Optimal GPV sequence: {{Val list| 12f, 19e, 31 }} | Optimal GPV sequence: {{Val list| 12f, 19e, 31 }} | ||
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Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }} | ||
: Eigenmonzo basis: 2.13/7 | : Eigenmonzo basis (unchanged-interval basis): 2.13/7 | ||
Tuning ranges: | Tuning ranges: | ||
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Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }} | ||
: Eigenmonzo basis: 2.13/9 | : Eigenmonzo basis (unchanged-interval basis): 2.13/9 | ||
Optimal GPV sequence: {{Val list| 12f, 31f, 43 }} | Optimal GPV sequence: {{Val list| 12f, 31f, 43 }} | ||
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* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }} | * 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 }}] | : [{{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 basis: 2.5 | : Eigenmonzo basis (unchanged-interval basis): 2.5 | ||
Tuning ranges: | Tuning ranges: | ||
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Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }} | ||
: Eigenmonzo basis: 2.13/11 | : Eigenmonzo basis (unchanged-interval basis): 2.13/11 | ||
Tuning ranges: | Tuning ranges: | ||
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Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }} | ||
: Eigenmonzo basis: 2.11 | : Eigenmonzo basis (unchanged-interval basis): 2.11 | ||
Optimal GPV sequence: {{Val list| 12e, 19, 31f }} | Optimal GPV sequence: {{Val list| 12e, 19, 31f }} | ||
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* [[7-odd-limit]]: ~3/2 = {{Monzo| 8/13 0 1/13 -1/13 }} | * [[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 }}] | : [{{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]]: ~3/2 = {{Monzo| 6/11 2/11 0 -1/11 }} | * [[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 }}] | : [{{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]]: | ||
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* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{Monzo| 0 0 1/8 }} | * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{Monzo| 0 0 1/8 }} | ||
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}] | : [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}] | ||
: [[Eigenmonzo]]s (unchanged intervals): 2, 5 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 1,699: | Line 1,699: | ||
* [[11-odd-limit]]: ~11/9 = {{Monzo| 0 0 1/8 }} | * [[11-odd-limit]]: ~11/9 = {{Monzo| 0 0 1/8 }} | ||
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}] | : [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}] | ||
: Eigenmonzos (unchanged intervals): 2, 5 | : Eigenmonzos (unchanged-intervals): 2, 5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 1,836: | Line 1,836: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{Monzo| 0 0 1/12 }} | * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{Monzo| 0 0 1/12 }} | ||
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3 0 -1/12 0 }}] | : [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3 0 -1/12 0 }}] | ||
: [[Eigenmonzo]]s (unchanged intervals): 2, 5 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 5 | ||
{{Val list|legend=1| 5, 26, 31 }} | {{Val list|legend=1| 5, 26, 31 }} | ||
| Line 1,955: | Line 1,955: | ||
* 7- and 9-odd-limit: ~10/7 = {{Monzo| 1/3 0 1/12 }} | * 7- and 9-odd-limit: ~10/7 = {{Monzo| 1/3 0 1/12 }} | ||
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 2/3 0 11/12 0 }}] | : [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 2/3 0 11/12 0 }}] | ||
: [[Eigenmonzo]]s (unchanged intervals): 2, 5 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 5 | ||
[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges. | [[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges. | ||
| Line 2,080: | Line 2,080: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }} | * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }} | ||
: [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3/2 0 9/16 0 }}] | : [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3/2 0 9/16 0 }}] | ||
: [[Eigenmonzo]]s (unchanged intervals): 2, 5 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 5 | ||
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents. | [[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents. | ||