Meantone family: Difference between revisions
m Text replacement - "Eigenmonzo (unchanged-interval)" to "Unchanged-interval (eigenmonzo)" |
m Text replacement - "eigenmonzo (unchanged-interval) " to "unchanged-interval (eigenmonzo) " |
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[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma) | * [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma) | ||
: [[Eigenmonzo basis| | : [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 101: | Line 101: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma) | * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma) | ||
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }} | : [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }} | ||
: [[eigenmonzo basis| | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 133: | Line 133: | ||
* 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 }} | ||
: projection map: [{{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 }}] | : projection map: [{{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 }}] | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11/9 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 161: | Line 161: | ||
Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11/9 | ||
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }} | {{Optimal ET sequence|legend=0| 12f, 19e, 31 }} | ||
| Line 333: | Line 333: | ||
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 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.13/9 | ||
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }} | {{Optimal ET sequence|legend=0| 12f, 31f, 43 }} | ||
| Line 543: | Line 543: | ||
* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }} | * 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }} | ||
: projection map: [{{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 }}] | : projection map: [{{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 }}] | ||
: | : unchanged-interval (eigenmonzo) basis: 2.5 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 572: | Line 572: | ||
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 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.13/11 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 655: | Line 655: | ||
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 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11 | ||
{{Optimal ET sequence|legend=0| 12e, 19, 31f }} | {{Optimal ET sequence|legend=0| 12e, 19, 31f }} | ||
| Line 1,100: | Line 1,100: | ||
* [[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 }} | ||
: [[projection map]]: [{{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 }}] | : [[projection map]]: [{{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 basis| | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 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 }} | ||
: [[projection map]]: [{{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 }}] | : [[projection map]]: [{{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 }}] | ||
: [[eigenmonzo basis| | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 1,574: | Line 1,574: | ||
* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }} | * 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }} | ||
: projection map: [{{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 }}] | : projection map: [{{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 }}] | ||
: | : unchanged-interval (eigenmonzo) basis: 2.5 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 1,714: | Line 1,714: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }} | * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }} | ||
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 2/3 0 11/12 0 }} | : [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 2/3 0 11/12 0 }} | ||
: [[eigenmonzo basis| | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 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 1,904: | Line 1,904: | ||
* [[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 }} | ||
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3/2 0 9/16 0 }} | : [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3/2 0 9/16 0 }} | ||
: [[eigenmonzo basis| | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 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. | ||