User:BudjarnLambeth/127zpi
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127zpi, the 127th zeta peak index, is a stretched-octaves version of 31edo. It can be thought of as 31ed1200.84c or as 38.737cet.
It has a step size of 38.737 cents, and the octave (2/1) is tuned slightly impurely to 1200.84 cents.
31edo tunes most simple harmonics slightly flat by roughly the same amount, so 127zpi is one possible way of correcting for this.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.8 | -3.8 | +2.8 | +1.3 | -6.5 | +14.2 | +14.6 | +15.8 | -5.1 | -19.0 |
| Relative (%) | +2.2 | -9.9 | +7.1 | +3.3 | -16.7 | +36.7 | +37.8 | +40.7 | -13.2 | -49.1 | |
| Step | 31 | 49 | 72 | 87 | 107 | 115 | 127 | 132 | 140 | 150 | |
For primes up to 29:
127zpi approximates these with less than 4 cents error (<10% relative error):
- 2, 3, 5, 7
...these with 4-8 cents error (10-21% relative error):
- 11, 23
...these with 8-16 cents error (21-41% relative error):
- 13, 17, 19
...and these with more than 16 cents error (>41% relative error):
- 29
This makes it usable as a full 23-limit tuning, or as a more accurate full 11-limit tuning.
Scales
Any scales from 31edo should also be useable here.
Instruments
All instruments listed under 31edo#Instruments also work for 127zpi.