71zpi: Difference between revisions

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== 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''' 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 [[#No-2s_zeta_peak|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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The nearest zeta peaks to 71zpi that surpass its strength are [[65zpi]] and [[75zpi]].

71zpi is distinguished by its extensive [[EDO-span|EDO-deviation]] and substantial zeta strength, qualifying it as a strong candidate for no-octave tuning systems. It is noteworthy that only [[19zpi]] exhibits both a greater octave error and stronger zeta height and integral than 71zpi, although 71zpi still has a more pronounced zeta gap. Other notable [[Zeta peak index|zeta peak ~~indexes~~]] in this category include [[61zpi]], [[84zpi]], [[110zpi]], [[137zpi]], [[151zpi]], [[222zpi]], and [[273zpi]], each demonstrating characteristics that make them suitable for similar applications.

71zpi is distinguished by its extensive [[EDO-span|EDO-deviation]] and substantial zeta strength, qualifying it as a strong candidate for no-octave tuning systems. It is noteworthy that only [[19zpi]] exhibits both a greater octave error and stronger zeta height and integral than 71zpi, although 71zpi still has a more pronounced zeta gap. Other notable [[Zeta peak index|zeta peak indices]] in this category include [[61zpi]], [[84zpi]], [[110zpi]], [[137zpi]], [[151zpi]], [[222zpi]], and [[273zpi]], each demonstrating characteristics that make them suitable for similar applications.

=== Harmonic series ===

{{Harmonics in cet|59.3329806724710|columns=15|title=Approximation of harmonics in 71zpi}}

{{Harmonics in cet|59.3329806724710|columns=17|start=16|title=Approximation of harmonics in 71zpi}}

==== No-2s zeta peak ====

{{Harmonics in cet|59.2711049327|intervals=odd|columns=17|title=Approximation of odd harmonics in local no-2s zeta peak}}

== Intervals ==

@@ Line 33: / Line 33: @@
 == 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''' 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 [[#No-2s_zeta_peak|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 39: / Line 39: @@
 The nearest zeta peaks to 71zpi that surpass its strength are [[65zpi]] and [[75zpi]].
-zpi is distinguished by its extensive [[EDO-span|EDO-deviation]] and substantial zeta strength, qualifying it as a strong candidate for no-octave tuning systems. It is noteworthy that only [[19zpi]] exhibits both a greater octave error and stronger zeta height and integral than 71zpi, although 71zpi still has a more pronounced zeta gap. Other notable [[Zeta peak index|zeta peak indexes]] in this category include [[61zpi]], [[84zpi]], [[110zpi]], [[137zpi]], [[151zpi]], [[222zpi]], and [[273zpi]], each demonstrating characteristics that make them suitable for similar applications.
+zpi is distinguished by its extensive [[EDO-span|EDO-deviation]] and substantial zeta strength, qualifying it as a strong candidate for no-octave tuning systems. It is noteworthy that only [[19zpi]] exhibits both a greater octave error and stronger zeta height and integral than 71zpi, although 71zpi still has a more pronounced zeta gap. Other notable [[Zeta peak index|zeta peak indices]] in this category include [[61zpi]], [[84zpi]], [[110zpi]], [[137zpi]], [[151zpi]], [[222zpi]], and [[273zpi]], each demonstrating characteristics that make them suitable for similar applications.
 === Harmonic series ===
 {{Harmonics in cet|59.3329806724710|columns=15|title=Approximation of harmonics in 71zpi}}
 {{Harmonics in cet|59.3329806724710|columns=17|start=16|title=Approximation of harmonics in 71zpi}}
+==== No-2s zeta peak ====
+{{Harmonics in cet|59.2711049327|intervals=odd|columns=17|title=Approximation of odd harmonics in local no-2s zeta peak}}
 == Intervals ==