View source for 207zpi

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. ]]

== 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 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).  

== Approximation of Harmonics == 

{| class="wikitable"
|-
! 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
|}

== Scala file ==

<pre>
! 207zpi.scl
! 
207 zpi
45
!
26.7619697233
53.5239394466
80.2859091699
107.0478788932
133.8098486165
160.5718183398
187.3337880631
214.0957577864
240.8577275097
267.6196972330
294.3816669563
321.1436366796
347.9056064029
374.6675761262
401.4295458495
428.1915155728
454.9534852961
481.7154550194
508.4774247427
535.2393944660
562.0013641893
588.7633339126
615.5253036359
642.2872733592
669.0492430825
695.8112128058
722.5731825291
749.3351522524
776.0971219757
802.8590916990
829.6210614223
856.3830311456
883.1450008689
909.9069705922
936.6689403155
963.4309100388
990.1928797621
1016.9548494854
1043.7168192087
1070.4787889320
1097.2407586553
1124.0027283786
1150.7646981019
1177.5266678252
1204.2886375485 
</pre>

[[Category:Zeta peak indexes]]