207zpi: Difference between revisions

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207 Zeta Peak Index (abbreviated 207zpi) is the [[Equal-step tuning|equal-step]] [[tuning system]] derived from the 207th peak of the [https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning| Riemann Zeta Function].

207 Zeta Peak Index (abbreviated 207zpi) is the [[Equal-step tuning|equal-step]] [[tuning system]] derived from the 207th peak of the [[https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning| Riemann Zeta Function]].

[[File:Riemann Zeta Function around 45edo, Desmos.png|thumb|The Riemann Zeta Function around 45edo. The highest peak, right to the left of 45, corresponds to 207zpi, demonstrating its relative strength as a tuning. ]]

== Theory ==

207zpi is the strongest zeta peak corresponding to [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning| stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It substantially improves on 45edo's [[harmonic]] accuracy, with no non-powers of 2/1 below [[16/1]] exceeding 7.5¢ [[error]].

207zpi is the strongest zeta peak corresponding to [[45edo]], and serves as a [[https://en.xen.wiki/w/Stretched_and_compressed_tuning| stretched-octave]] version thereof ([[2/1]] ≈ 1204.289¢). It substantially improves on 45edo's [[harmonic]] accuracy, with no non-powers of 2/1 below [[16/1]] exceeding 7.5¢ [[error]].

Crucially, due to the octave stretch, the 207zpi [[Patent val|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([https://en.xen.wiki/w/Interval_class|''k'']=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[5L 2s|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).

Crucially, due to the octave stretch, the 207zpi [[Patent val|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([[https://en.xen.wiki/w/Interval_class|''k'']]=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[5L 2s|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).

== Approximation of Harmonics ==

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-Zeta Peak Index (abbreviated 207zpi) is the [[Equal-step tuning|equal-step]] [[tuning system]] derived from the 207th peak of the [https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning| Riemann Zeta Function].
+Zeta Peak Index (abbreviated 207zpi) is the [[Equal-step tuning|equal-step]] [[tuning system]] derived from the 207th peak of the [[https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning| Riemann Zeta Function]].
 [[File:Riemann Zeta Function around 45edo, Desmos.png|thumb|The Riemann Zeta Function around 45edo. The highest peak, right to the left of 45, corresponds to 207zpi, demonstrating its relative strength as a tuning. ]]
 == Theory ==
-zpi is the strongest zeta peak corresponding to [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning| stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It substantially improves on 45edo's [[harmonic]] accuracy, with no non-powers of 2/1 below [[16/1]] exceeding 7.5¢ [[error]].
+zpi is the strongest zeta peak corresponding to [[45edo]], and serves as a [[https://en.xen.wiki/w/Stretched_and_compressed_tuning| stretched-octave]] version thereof ([[2/1]] ≈ 1204.289¢). It substantially improves on 45edo's [[harmonic]] accuracy, with no non-powers of 2/1 below [[16/1]] exceeding 7.5¢ [[error]].
-Crucially, due to the octave stretch, the 207zpi [[Patent val|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([https://en.xen.wiki/w/Interval_class|''k'']=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[5L 2s|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).
+Crucially, due to the octave stretch, the 207zpi [[Patent val|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([[https://en.xen.wiki/w/Interval_class|''k'']]=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[5L 2s|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).
 == Approximation of Harmonics ==