207zpi

207 Zeta Peak Index (abbreviated 207zpi) is the equal-step tuning system derived from the 207th peak of the Riemann Zeta Function.

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 in the vicinity of 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

Scala file text

! 207zpi.scl 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