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]].

'''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 [[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]]~~

| 42.187591425251

| 28.444382802136

| 3.591963

| 0.481940

| 11.612694

| [[42edo]]

| 1194.6640776897

| 3

|}

== Theory ==

~~''See also:~~ [[~~Table~~ of ~~stretched~~ 42edo ~~tunings~~]]''

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 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).

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~~.

191zpi’s step size is very close to that of [[1ed28.5c|APS715jot]] and [[ed255/128#42ed255/128|42ed255/128]].

~~191zpi 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~~]].

See [[Table of stretched 42edo tunings]] for more.

=== Harmonic series ===

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 ==

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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]]

@@ Line 1: / Line 1: @@
-'''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 [[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]]
-| 42.187591425251
-| 28.444382802136
-| 3.591963
-| 0.481940
-| 11.612694
-| [[42edo]]
-| 1194.6640776897
-| 3
-| 3
-|}
 == Theory ==
-''See also: [[Table of stretched 42edo tunings]]''
+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]].
-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 [[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).
-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.
+zpi’s step size is very close to that of [[1ed28.5c|APS715jot]] and [[ed255/128#42ed255/128|42ed255/128]].
-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]].
+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 ==
@@ Line 131: / Line 115: @@
 [[Category:42-tone scales]]
 [[Category:Tempered scales]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Zeta peak indexes]]
 [[Category:42edo]]
 [[Category:Pages with Scala files]]