173zpi: Difference between revisions
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== Theory == | == Theory == | ||
173zpi is the strongest [[Zeta peak index|zeta peak]] within ±0.5 [[EDO|divisions]] of [[39edo]], and serves as a [[Stretched and compressed tuning|compressed-octave]] version thereof ([[2/1]] ≈ 1196.204¢), narrowing the octave by ~3.796¢. 39edo, being a [[zeta valley edo|zeta ''valley'' EDO]], poorly approximates [[Just intonation]] relative to its size. Its harmonics routinely exceed 30% [[relative error]], so using 173zpi instead—the strongest zeta ''peak'' within 39edo's vicinity—is a logical way of correcting for this, if desired. | 173zpi is the strongest [[Zeta peak index|zeta peak]] within ±0.5 [[EDO|divisions]] of [[39edo]], and serves as a [[Stretched and compressed tuning|compressed-octave]] version thereof ([[2/1]] ≈ 1196.204¢), narrowing the octave by ~3.796¢. 39edo, being a [[zeta valley edo|zeta ''valley'' EDO]], poorly approximates [[Just intonation]] relative to its size. Its approximations of harmonics routinely exceed 30% [[relative error]], so using 173zpi instead—the strongest zeta ''peak'' within 39edo's vicinity—is a logical way of correcting for this, if desired. | ||
Revision as of 06:54, 13 April 2026
173 Zeta Peak Index (abbreviated 173zpi) is the equal-step tuning system derived from the 173rd peak of the Riemann Zeta Function.
Theory
173zpi is the strongest zeta peak within ±0.5 divisions of 39edo, and serves as a compressed-octave version thereof (2/1 ≈ 1196.204¢), narrowing the octave by ~3.796¢. 39edo, being a zeta valley EDO, poorly approximates Just intonation relative to its size. Its approximations of harmonics routinely exceed 30% relative error, so using 173zpi instead—the strongest zeta peak within 39edo's vicinity—is a logical way of correcting for this, if desired.