45zpi: Difference between revisions
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45zpi is characterized by a very broad octave error, yet it maintains a quite decent zeta strength. This combination makes it an ideal candidate for no-octave tuning applications. | 45zpi is characterized by a very broad octave error, yet it maintains a quite decent zeta strength. This combination makes it an ideal candidate for no-octave tuning applications. | ||
No other [[Zeta peak index|zeta peak indexes]] exhibit both a larger octave error and greater zeta | No other [[Zeta peak index|zeta peak indexes]] exhibit both a larger octave error and greater zeta height than 45zpi. | ||
45zpi supports a complex chord structure with ratios of 1:3:4:5:7:9:13:15:18:19:20:21:22:23:24:25, which further exemplifies its capabilities. | 45zpi supports a complex chord structure with ratios of 1:3:4:5:7:9:13:15:18:19:20:21:22:23:24:25, which further exemplifies its capabilities. | ||
Revision as of 09:09, 23 April 2024
45 zeta peak index (abbreviated 45zpi), is the equal-step tuning system obtained from the 45th peak of the Riemann zeta function.
| Tuning | Strength | Closest EDO | Integer limit | ||||||
|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per octave | Step size (cents) | Height | Integral | Gap | EDO | Octave (cents) | Consistent | Distinct |
| 45zpi | 14.5944346577250 | 82.2231232756126 | 2.097730 | 0.344839 | 10.594800 | 15edo | 1233.34684913419 | 2 | 2 |
Theory
45zpi is characterized by a very broad octave error, yet it maintains a quite decent zeta strength. This combination makes it an ideal candidate for no-octave tuning applications.
No other zeta peak indexes exhibit both a larger octave error and greater zeta height than 45zpi.
45zpi supports a complex chord structure with ratios of 1:3:4:5:7:9:13:15:18:19:20:21:22:23:24:25, which further exemplifies its capabilities.
The closest zeta peak indexes to 45zpi that exceed its strength are 42zpi and 47zpi, though 43zpi is nearly as strong as 45zpi.
Harmonic series
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +33.3 | -10.8 | -15.5 | +9.3 | +22.5 | +2.3 | +17.8 | -21.6 | -39.6 | -40.2 | -26.4 | -0.5 | +35.7 | -1.6 | -31.1 |
| Relative (%) | +40.6 | -13.2 | -18.9 | +11.3 | +27.4 | +2.8 | +21.7 | -26.3 | -48.2 | -48.8 | -32.1 | -0.6 | +43.4 | -1.9 | -37.8 | |
| Step | 15 | 23 | 29 | 34 | 38 | 41 | 44 | 46 | 48 | 50 | 52 | 54 | 56 | 57 | 58 | |
| Harmonic | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +28.4 | +11.7 | +0.3 | -6.3 | -8.5 | -6.8 | -1.5 | +7.0 | +18.5 | +32.9 | -32.5 | -13.2 | +8.3 | +31.8 | -25.0 | +2.3 |
| Relative (%) | +34.6 | +14.2 | +0.4 | -7.6 | -10.3 | -8.3 | -1.9 | +8.5 | +22.6 | +40.0 | -39.5 | -16.1 | +10.1 | +38.7 | -30.4 | +2.8 | |
| Step | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 69 | 70 | 71 | 72 | 72 | 73 | |
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