Meantone family: Difference between revisions
Reflect difference of 7-odd-limit diamond tradeoff and 9-odd-limit diamond tradeoff on some temperaments |
Cmloegcmluin (talk | contribs) adding parenthetical "(unchanged intervals)" to eigenmonzo lists |
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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]]s: 2, 5 | : [[Eigenmonzo]]s (unchanged intervals): 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]]s: 2, 5 | : [[Eigenmonzo]]s (unchanged intervals): 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 }}] | ||
: Eigenmonzos: 2, 11/9 | : Eigenmonzos (unchanged intervals): 2, 11/9 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 128: | Line 128: | ||
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 }} | ||
: Eigenmonzos: 2, 11/9 | : Eigenmonzos (unchanged intervals): 2, 11/9 | ||
Vals: {{Val list| 12f, 19e, 31 }} | Vals: {{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 }} | ||
: Eigenmonzos: 2, 14/13 | : Eigenmonzos (unchanged intervals): 2, 14/13 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 167: | Line 167: | ||
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 }} | ||
: Eigenmonzos: 2, 18/13 | : Eigenmonzos (unchanged intervals): 2, 18/13 | ||
Vals: {{Val list| 12f, 31f, 43 }} | Vals: {{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 }}] | ||
: Eigenmonzos: 2, 5 | : Eigenmonzos (unchanged intervals): 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 }} | ||
: Eigenmonzos: 2, 13/11 | : Eigenmonzos (unchanged intervals): 2, 13/11 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 256: | Line 256: | ||
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 }} | ||
: Eigenmonzos: 2, 11 | : Eigenmonzos (unchanged intervals): 2, 11 | ||
Vals: {{Val list| 12e, 19, 31f, 50ff, 81fff }} | Vals: {{Val list| 12e, 19, 31f, 50ff, 81fff }} | ||
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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: 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: 2, 9/7 | : Eigenmonzos (unchanged intervals): 2, 9/7 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 1,293: | Line 1,293: | ||
* [[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: 2, 5 | : [[Eigenmonzo]]s (unchanged intervals): 2, 5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 1,324: | Line 1,324: | ||
* [[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: 2, 5 | : Eigenmonzos (unchanged intervals): 2, 5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 1,508: | Line 1,508: | ||
* [[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: 2, 5 | : [[Eigenmonzo]]s (unchanged intervals): 2, 5 | ||
{{Val list|legend=1| 5, 26, 31 }} | {{Val list|legend=1| 5, 26, 31 }} | ||
| Line 1,625: | Line 1,625: | ||
* 7- and 9-odd-limit: ~9/7 = {{Monzo| 1/2 0 -1/16 }} | * 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: 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. | ||
| Line 1,760: | Line 1,760: | ||
* 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: 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. | ||