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'''191 zeta peak index''' (abbreviated '''191zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 191st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].
'''191 zeta peak index''' (abbreviated '''191zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 191st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].


{| class="wikitable"
{{ZPI
! colspan="3" | Tuning
| zpi = 191
! colspan="3" | Strength
| steps = 42.187591425251
! colspan="2" | Closest EDO
| step size = 28.444382802136
! colspan="2" | Integer limit
| height = 3.591963
|-
| integral = 0.481940
! ZPI
| gap = 11.612694
! Steps per octave
| edo = 42edo
! Step size (cents)
| octave = 1194.6640776897
! Height
| consistent = 3
! Integral
| distinct = 3
! Gap
}}
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[191zpi]]
| 42.1875914252516
| 28.4443828021368
| 3.591963
| 0.481940
| 11.612694
| [[42edo]]
| 1194.66407768974
| 3
| 3
|}


== Theory ==
== Theory ==
191zpi can be used as an [[Octave shrinking|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]].
191zpi can be used as an [[Octave shrinking|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 twin to the scale [[Ed257/128#42ed257/128|42ed257/128]]. 191zpi makes 42edo into a better 2.3.5.11.13 tuning, while 42ed257/128 makes 42edo into a better 2.3.5.7.17 tuning. They both improve 42edo but in opposite directions.
It is a kind of opposite twin to the scale [[zpi|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 [[1ed28.5c|APS715jot]] and [[ed255/128#42ed255/128|42ed255/128]].
 
See [[Table of stretched 42edo tunings]] for more.


=== Harmonic series ===
=== Harmonic series ===
[[191zpi]] performs well in the no-7s [[13-limit]], or alternatively in the [[dual-n|dual-7]] 13-limit.
191zpi performs well in the no-7s [[13-limit]], or alternatively in the [[dual-n|dual-7]] 13-limit.
{{Harmonics in cet|28.4443828021368|intervals=prime|title=Approximation of harmonics in 186zpi}}
{{Harmonics in cet|28.4443828021368|intervals=prime|title=Approximation of harmonics in 191zpi}}


== Notation ==
== Notation ==
Line 96: Line 84:
  1194.664078
  1194.664078
</pre>
</pre>
==Scales==
;[[MOS scale]]s
* 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 [[Template:Idiosyncratic|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 ==
== Instruments ==
* [[Lumatone]]
* [[Lumatone]]
** 191 can use most types of [[Lumatone mapping for 42edo]]
** 191zpi can use most types of [[Lumatone mapping for 42edo]]


[[Category:42-tone scales]]
[[Category:42-tone scales]]
[[Category:Tempered scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Zeta peak indexes]]
[[Category:Zeta peak indexes]]
[[Category:42edo]]
[[Category:42edo]]
[[Category:Pages with Scala files]]
[[Category:Pages with Scala files]]
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