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, | 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 approximates [[Just intonation]] rather poorly for its size, enough to designate it as a [[zeta valley edo|zeta ''valley'' EDO]], 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 06:06, 28 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 approximates Just intonation rather poorly for its size, enough to designate it as a zeta valley EDO, so using 173zpi instead—the strongest zeta peak within 39edo's vicinity—is a logical way of correcting for this, if desired.