207zpi: Difference between revisions

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. ]]

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).