191zpi: Difference between revisions

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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]].
-{| 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 ==
 zpi 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 opposite twin to the scale [[1ed28.7c|APS720jot]]. 191zpi makes 42edo into a better 2.3.5.11.13 tuning, while APS720jot 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).
-zpi is almost identical to [[ed255/128#42ed255/128|42ed255/128]]. 191zpi does slightly better at approximating most harmonics but 42ed255/128 does slightly better at [[3/1]].
+zpi’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 ===
 zpi 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 ==
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 .664078
 </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 ==
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 [[Category:42-tone scales]]
 [[Category:Tempered scales]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Zeta peak indexes]]
 [[Category:42edo]]
 [[Category:Pages with Scala files]]