User:BudjarnLambeth/116zpi
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116zpi, the 116th zeta peak index, is a stretched-octaves version of 29edo. It can be thought of as 80ed1202.49c or as 41.465cet.
It has a step size of 41.465 cents, and the octave (2/1) is tuned slightly impurely to 80ed1202.49 cents.
29edo tunes almost all simple harmonics slightly flat by roughly the same amount, so 116zpi is one possible way of correcting for this.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | |
|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.5 | +5.4 | -8.2 | -10.2 | -4.8 | -3.8 | -12.1 |
| Relative (%) | +6.0 | +13.1 | -19.7 | -24.5 | -11.6 | -9.1 | -29.1 | |
| Step | 29 | 46 | 67 | 81 | 100 | 107 | 118 | |
| Harmonic | 19 | 23 | 29 | 31 | 37 | 41 | 43 | |
|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.7 | +3.6 | +17.0 | -15.5 | +9.9 | -2.0 | -1.5 |
| Relative (%) | +6.5 | +8.8 | +41.0 | -37.5 | +23.8 | -4.8 | -3.6 | |
| Step | 123 | 131 | 141 | 143 | 151 | 155 | 157 | |
For primes up to 29:
435zpi approximates these with less than 7 cents error (<17% relative error):
- 2, 3, 11, 13, 19, 23
...these with 7-14 cents error (17-34% relative error):
- 5, 7, 17
...and these with more than 14 cents error (>34% relative error):
- 29
This makes it usable as a full 29-limit tuning, or as a more accurate 2.3.5.11.13 subgroup tuning.
Scales
Any scales from 29edo should also be useable here.
Instruments
All instruments listed under 29edo#Instruments also work for 116zpi.