173zpi - Revision history

Pizzabagel shef: Switched wording around to make clearer emphases, no new information yet.

2026-04-28T06:06:59Z

Switched wording around to make clearer emphases, no new information yet.

← Older revision		Revision as of 06:06, 28 April 2026
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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~~, 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.

Pizzabagel shef: Removed comment about 39edo'a relative error for brevity

2026-04-13T07:14:20Z

Removed comment about 39edo'a relative error for brevity

← Older revision		Revision as of 07:14, 13 April 2026
Line 2:		Line 2:

	== 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 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, being a [[zeta valley edo\|zeta ''valley'' EDO]], poorly approximates [[Just intonation]] relative to its size, so using 173zpi instead—the strongest zeta ''peak'' within 39edo's vicinity—is a logical way of correcting for this, if desired.

Pizzabagel shef: /* Theory */

2026-04-13T06:54:56Z

Theory

← Older revision		Revision as of 06:54, 13 April 2026
Line 2:		Line 2:

	== 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.

Pizzabagel shef: /* Theory */

2026-04-13T06:51:12Z

Theory

← Older revision		Revision as of 06:51, 13 April 2026
Line 2:		Line 2:

	== 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 ~~step~~ 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 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.

Pizzabagel shef: The creation of the page for 173zpi! It's sparse at the moment due to personal time constraints, but more will be added soon.

2026-04-13T06:48:30Z

The creation of the page for 173zpi! It's sparse at the moment due to personal time constraints, but more will be added soon.

New page

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]].

== 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 step 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.