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== 11-limit optimizations of 31-EDO == | == 11-limit optimizations of 31-EDO == | ||
=== Euler-product === | |||
{{Harmonics in cet|38.743026608201324738|columns=11|intervals=prime|title=Approximation of prime harmonics in 11-limit Euler-product (σ = 1.0) optimization of 31-EDO}} | {{Harmonics in cet|38.743026608201324738|columns=11|intervals=prime|title=Approximation of prime harmonics in 11-limit Euler-product (σ = 1.0) optimization of 31-EDO}} | ||
{{Harmonics in cet|38.748558591979841975|columns=11|intervals=prime|title=Approximation of prime harmonics in 11-limit Tenney-Euclidian optimization of 31-EDO}}121-ED15 offers a good trade-off between these two optimization methods for 11-limit in 31-EDO.{{Harmonics in equal|121|15|1|columns=11|intervals=prime|title=Approximation of prime harmonics in 121-ED15}} | === Tenney-Euclidian regular temperament === | ||
{{Harmonics in cet|38.748558591979841975|columns=11|intervals=prime|title=Approximation of prime harmonics in 11-limit Tenney-Euclidian optimization of 31-EDO}} | |||
=== 121-ED15 is a good trade-off === | |||
121-ED15 offers a good trade-off between these two optimization methods for 11-limit in 31-EDO. | |||
{{Harmonics in equal|121|15|1|columns=11|intervals=prime|title=Approximation of prime harmonics in 121-ED15}} | |||
Revision as of 21:06, 28 September 2025
11-limit optimizations of 31-EDO
Euler-product
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.0 | -3.5 | +3.2 | +1.8 | -5.8 | +14.9 | +15.4 | +16.6 | -4.3 | -18.1 | -17.4 |
| Relative (%) | +2.7 | -9.2 | +8.2 | +4.7 | -15.0 | +38.5 | +39.8 | +42.8 | -11.0 | -46.8 | -44.8 | |
| Step | 31 | 49 | 72 | 87 | 107 | 115 | 127 | 132 | 140 | 150 | 153 | |
Tenney-Euclidian regular temperament
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.2 | -3.3 | +3.6 | +2.3 | -5.2 | +15.6 | +16.1 | +17.3 | -3.5 | -17.3 | -16.5 |
| Relative (%) | +3.1 | -8.5 | +9.2 | +5.9 | -13.5 | +40.1 | +41.6 | +44.6 | -9.0 | -44.6 | -42.6 | |
| Step | 31 | 49 | 72 | 87 | 107 | 115 | 127 | 132 | 140 | 150 | 153 | |
121-ED15 is a good trade-off
121-ED15 offers a good trade-off between these two optimization methods for 11-limit in 31-EDO.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.1 | -3.4 | +3.4 | +2.1 | -5.5 | +15.3 | +15.8 | +17.0 | -3.8 | -17.7 | -16.9 |
| Relative (%) | +2.9 | -8.8 | +8.8 | +5.4 | -14.2 | +39.4 | +40.8 | +43.8 | -9.9 | -45.6 | -43.6 | |
| Steps (reduced) |
31 (31) |
49 (49) |
72 (72) |
87 (87) |
107 (107) |
115 (115) |
127 (6) |
132 (11) |
140 (19) |
150 (29) |
153 (32) | |