Template:ZPI/doc

This template generates a table for a particular EDO's zeta peak index.

Parameters

All parameters are required.

{{{steps}}}

The zeta-optimised number of steps per octave, corresponding to the location of a particular EDO's nearest local maximum or minimum according to the Riemann–Siegel Z function, [math]\displaystyle{ Z\left(\frac{2\pi}{\ln 2}x\right) }[/math]. For example, 12edo's local zeta-optimised step count is 12.0231830072926.

{{{step size}}}

The step size in cents, which is 1200 divided by the aformentioned zeta-optimised step count.

For the height of the zeta peak, you can use tempered height and pure height or just height.

{{{tempered height}}}

The absolute value of Z evaluated at the zeta-optimised step count. For 12edo, this value is 5.193290 when evaluated at 12.0231830072926.

Must be used in conjunction with pure height, do not use with height.

{{{pure height}}}

The absolute value of Z evaluated at the exact integer. For 12edo, this value is 5.084467 when evaluated at exactly 12.

Must be used in conjunction with tempered height, do not use with height.

{{{height}}}

The absolute value of Z evaluated at a ZPI that does not correspond strongly to an EDO. Do not use with tempered height or pure height.

{{{integral}}}

The absolute value of the integral between the two roots (zeroes) of Z closest to the given EDO. For 12edo, this value is 1.269599.

{{{gap}}}

{{{edo}}}

{{{octave}}}

The zeta-optimised octave size. For 12edo, this is 1197.68616940005.

{{{consistent}}}

The ZPI's consistency limit.

{{{distinct}}}

The ZPI's distinct consistency limit.

Example (12edo)

Here is the complete zeta peak index table for 12edo as it appears on the page:

{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

Tuning			Strength				Closest edo			Integer limit
ZPI	Steps per 8ve	Step size (cents)	Height		Integral	Gap	Edo	Octave (cents)		Consistent	Distinct
			Size	Stretch				Tempered	Pure
34zpi	12.023183	99.807181	5.19329	5.084467	1.269599	15.899282	12edo	1197.686169	−2.313831	10	6