207zpi: Difference between revisions
A first introduction to 207zpi! However crude it is in this form, please give it a warm welcome to the website. |
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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. ]] | [[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 == | == 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 == | == Approximation of Harmonics == | ||
Revision as of 04:47, 6 April 2026
207 Zeta Peak Index (abbreviated 207zpi) is the equal-step tuning system derived from the 207th peak of the [Riemann Zeta Function].
Theory
207zpi is the strongest zeta peak corresponding to 45edo, and serves as a [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 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 ([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 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
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Absolute error (¢) | 4.289 | -1.855 | 8.577 | -3.069 | 2.433 | 3.182 | 12.866 | -3.710 | 1.220 | -3.213 | 6.722 | 1.959 | 7.471 | -4.924 | -9.607 |