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 [[~~The~~ Riemann zeta function ~~and tuning|Riemann Zeta Function~~]].

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 ==

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.

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.

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-Zeta Peak Index (abbreviated 173[[zpi]]) is the [[Equal-step tuning|equal-step]] [[tuning system]] derived from the 173rd peak of the [[The Riemann zeta function and tuning|Riemann Zeta Function]].
+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 ==
-zpi 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.
+zpi 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.