191zpi
191 zeta peak index (abbreviated 191zpi), is the equal-step tuning system obtained from the 191st 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 |
| 191zpi | 42.187591425251 | 28.444382802136 | 3.591963 | 0.481940 | 11.612694 | 42edo | 1194.6640776897 | 3 | 3 |
Theory
191zpi can be used as an octave-compressed version of 42edo, with a much better 3/1 and 5/1 compared to 42edo, but at the expense of a worse 7/1.
It is a kind of opposite twin to the scale 189zpi, as they improve 42edo’s JI approximation by about the same amount, but in opposite directions (those harmonics which are slightly sharp in one are slightly flat in the other).
191zpi’s step size is very close to that of APS715jot and 42ed255/128.
See Table of stretched 42edo tunings for more.
Harmonic series
191zpi performs well in the no-7s 13-limit, or alternatively in the dual-7 13-limit.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.3 | +3.8 | +1.2 | -12.4 | +1.6 | -3.2 | -12.5 | -6.0 | +4.6 | +1.5 |
| Relative (%) | -18.8 | +13.4 | +4.3 | -43.6 | +5.5 | -11.3 | -44.0 | -21.0 | +16.2 | +5.3 | |
| Steps | 42 | 67 | 98 | 118 | 146 | 156 | 172 | 179 | 191 | 205 | |
Notation
191zpi can use most notation systems designed for 42edo. See 42edo#Notation.
Scala file
Tuning file for anything that supports Scala. Made with Scale Workshop.
! 191zpi.scl ! Created using Scale Workshop 3.0.1 ! ! https://scaleworkshop.plainsound.org/scale/H7m9KNfqS ! The 191st Zeta peak index 42 ! 28.444383 56.888766 85.333148 113.777531 142.221914 170.666297 199.110680 227.555062 255.999445 284.443828 312.888211 341.332594 369.776976 398.221359 426.665742 455.110125 483.554508 511.998890 540.443273 568.887656 597.332039 625.776422 654.220804 682.665187 711.109570 739.553953 767.998336 796.442718 824.887101 853.331484 881.775867 910.220250 938.664632 967.109015 995.553398 1023.997781 1052.442164 1080.886546 1109.330929 1137.775312 1166.219695 1194.664078
Scales
- Eugene/Tritikleismic[9]: 3 8 3 3 8 3 3 8 3
- Eugene/Tritikleismic[15]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3
- Lemba[16]: 3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3
- Qeema/Skateboard[15]: 2 5 2 2 2 5 2 2 2 5 2 2 2 5 2
- Qeema/Skateboard[19]: 2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2
- Seville/Sevond[14] 1st mode: 1 5 1 5 1 5 1 5 1 5 1 5 1 5
- Seville/Sevond[14] 2nd mode: 5 1 5 1 5 1 5 1 5 1 5 1 5 1
- Seville/Sevond[21]: 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4
- Subsets of MOS scales
(Names used are idiosyncratic.)
- Eugene/Tritikleismic[9]
- Groovy aeolian pentatonic: 11 6 8 3 14
- Otonal mixolydian pentatonic: 14 3 8 11 6
- Pseudo-equipentatonic: 11 6 8 6 11
- Septimal melodic minor pentatonic: 8 3 14 14 3
- Septimal Picardy pentatonic: 8 6 11 3 14
- Undecimal lydian-aeolian pentatonic: 8 14 3 11 6
- Yokai pentatonic: 3 14 8 3 14
Instruments
- Lumatone
- 191zpi can use most types of Lumatone mapping for 42edo