71zpi: Difference between revisions

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'''71 zeta peak index''' (abbreviated '''71zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 71st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].

{~~| class="wikitable"~~

{{ZPI

~~! colspan="3"~~ | ~~Tuning~~

| zpi = 71

~~! colspan~~=~~"3" | Strength~~

| steps = 20.2248393119540

~~! colspan="2"~~ | ~~Closest EDO~~

| step size = 59.3329806724710

~~! colspan~~=~~"2" | Integer limit~~

| height = 3.531097

|-

| integral = 0.613581

~~! ZPI~~

| gap = 12.986080

~~! Steps per octave~~

| octave = 1186.65961344942

~~! Step size (cents)~~

| consistent = 6

~~! Height~~

| distinct = 6

~~! Integral~~

}}

~~! Gap~~

~~! EDO~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[71zpi]]~~

| 20.2248393119540

| 59.3329806724710

| 3.531097

| 0.613581

| 12.986080

| ~~[[20edo]]~~

| 1186.65961344942

| 6

|}

[[File:71zpi.png|thumb|right|The Riemann zeta function around 71zpi]]

== Theory ==

'''71zpi''' marks the most prominent [[zeta peak index]] in the [[vicinity]] of [[20edo]]. While [[70zpi]] is the nearest peak to [[20edo]] and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength. 71zpi may also be viewed as a tritave compression of [[32edt]], a [[The_Riemann_zeta_function_and_tuning#Removing_primes|no-2s zeta peak EDT]] (consistent in the ~~no-2s~~ [[Odd_limit#Nonoctave_equaves|21-~~throdd~~-limit]]), but with less extreme stretch than [[71zpi#Record on the Riemann zeta function with prime 2 removed|the no-2s peak]] at 59.271105 cents.

'''71zpi''' marks the most prominent [[zeta peak index]] in the [[vicinity]] of [[20edo]]. While [[70zpi]] is the nearest peak to [[20edo]] and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength. 71zpi may also be viewed as a tritave compression of [[32edt]], a [[The_Riemann_zeta_function_and_tuning#Removing_primes|no-2s zeta peak EDT]] (consistent in the [[Odd_limit#Nonoctave_equaves|no-2s 19-integer-limit]]), but with less extreme stretch than [[71zpi#Record on the Riemann zeta function with prime 2 removed|the no-2s peak]] at 59.271105 cents.

71zpi features a good 3:5:9:11:14:15:16:19:25:26:33 chord, which differs a lot from the harmonic characteristics of [[20edo]].

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! Cents

! Ratios

! Ups and ~~Downs Notation~~

! Ups and downs notation

! Step

|-

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! Cents

! Ratios

! Ups and ~~Downs Notation~~

! Ups and downs notation

! Step

|-

@@ Line 1: / Line 1: @@
 '''71 zeta peak index''' (abbreviated '''71zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 71st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].
-{| class="wikitable"
+{{ZPI
-! colspan="3" | Tuning
+| zpi = 71
-! colspan="3" | Strength
+| steps = 20.2248393119540
-! colspan="2" | Closest EDO
+| step size = 59.3329806724710
-! colspan="2" | Integer limit
+| height = 3.531097
-|-
+| integral = 0.613581
-! ZPI
+| gap = 12.986080
-! Steps per octave
+| octave = 1186.65961344942
-! Step size (cents)
+| consistent = 6
-! Height
+| distinct = 6
-! Integral
+}}
-! Gap
-! EDO
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[71zpi]]
-| 20.2248393119540
-| 59.3329806724710
-| 3.531097
-| 0.613581
-| 12.986080
-| [[20edo]]
-| 1186.65961344942
-| 6
-| 6
-|}
 [[File:71zpi.png|thumb|right|The Riemann zeta function around 71zpi]]
 == Theory ==
-'''71zpi''' marks the most prominent [[zeta peak index]] in the [[vicinity]] of [[20edo]]. While [[70zpi]] is the nearest peak to [[20edo]] and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength. 71zpi may also be viewed as a tritave compression of [[32edt]], a [[The_Riemann_zeta_function_and_tuning#Removing_primes|no-2s zeta peak EDT]] (consistent in the no-2s [[Odd_limit#Nonoctave_equaves|21-throdd-limit]]), but with less extreme stretch than [[71zpi#Record on the Riemann zeta function with prime 2 removed|the no-2s peak]] at 59.271105 cents.
+'''71zpi''' marks the most prominent [[zeta peak index]] in the [[vicinity]] of [[20edo]]. While [[70zpi]] is the nearest peak to [[20edo]] and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength. 71zpi may also be viewed as a tritave compression of [[32edt]], a [[The_Riemann_zeta_function_and_tuning#Removing_primes|no-2s zeta peak EDT]] (consistent in the [[Odd_limit#Nonoctave_equaves|no-2s 19-integer-limit]]), but with less extreme stretch than [[71zpi#Record on the Riemann zeta function with prime 2 removed|the no-2s peak]] at 59.271105 cents.
 zpi features a good 3:5:9:11:14:15:16:19:25:26:33 chord, which differs a lot from the harmonic characteristics of [[20edo]].
@@ Line 64: / Line 47: @@
 ! Cents
 ! Ratios
-! Ups and Downs Notation
+! Ups and downs notation
 ! Step
 |-
@@ Line 3,536: / Line 3,519: @@
 ! Cents
 ! Ratios
-! Ups and Downs Notation
+! Ups and downs notation
 ! Step
 |-