207zpi: Difference between revisions
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207 Zeta Peak Index (abbreviated | 207 Zeta Peak Index (abbreviated 207[[zpi]]) 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, 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, to the left of 45, corresponds to 207zpi, demonstrating its relative strength as a tuning. ]] | ||
== Theory == | == Theory == | ||
207zpi is the strongest [[Zeta peak index|zeta peak]] in the vicinity of [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It substantially | 207zpi is the strongest [[Zeta peak index|zeta peak]] in the vicinity of [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It improves substantially on 45edo's [[harmonic]] accuracy, with no harmonics (excluding powers of 2/1) below [[16/1]] exceeding 7.5¢ absolute [[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]] [[Diatonic scale|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]] [[Diatonic scale|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). | ||
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== Scala file | == Scala file == | ||
<pre> | |||
! 207zpi.scl | ! 207zpi.scl | ||
! | |||
207 zpi | |||
45 | 45 | ||
! | |||
26.7619697233 | 26.7619697233 | ||
53.5239394466 | 53.5239394466 | ||
| Line 95: | Line 99: | ||
1150.7646981019 | 1150.7646981019 | ||
1177.5266678252 | 1177.5266678252 | ||
1204.2886375485 | 1204.2886375485 | ||
</pre> | |||
[[Category:Zeta peak indexes]] | |||