173zpi: Difference between revisions
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173 Zeta Peak Index (abbreviated 173[[zpi]]) is the [[Equal-step tuning|equal-step]] [[tuning system]] derived from the 173rd peak of the [[ | 173 Zeta Peak Index (abbreviated 173[[zpi]]) is the [[Equal-step tuning|equal-step]] [[tuning system]] derived from the 173rd peak of the [[Riemann zeta function]]. | ||
== 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. | ||
[[Category:Zeta peak indexes]][[Category:39edo]] | |||
Latest revision as of 09:07, 3 October 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.