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


=== Parameters ===
=== Parameters ===
All parameters are required.
All parameters are required.


; <code>{{{<nowiki />steps}}}</code>
; <code><nowiki>{{{steps}}}</nowiki></code>
: 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 {{w|Z function|Riemann–Siegel ''Z'' function}}, ''Z''({{frac|2π|ln(2)}}''x'').
: 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 {{w|Z function|Riemann–Siegel ''Z'' function}},  
 
$$ Z\left( \frac{2\pi}{\ln 2} \, x \right) $$
 
: For example, [[12edo]]'s local zeta-optimised step count is 12.0231830072926.
: For example, [[12edo]]'s local zeta-optimised step count is 12.0231830072926.


; <code>{{{<nowiki />step size}}}</code>
; <code><nowiki>{{{step size}}}</nowiki></code>
: The step size in [[cent]]s, which is 1200 divided by the aformentioned zeta-optimised step count.
: The step size in [[cent]]s, which is 1200 divided by the aformentioned zeta-optimised step count.


For the height of the zeta peak, you can use <code>tempered&nbsp;height</code> and <code>pure&nbsp;height</code> or just <code>height</code>.
For the height of the zeta peak, you can use <code>tempered height</code> and <code>pure height</code> or just <code>height</code>.


; <code>{{{<nowiki />tempered height}}}</code>
; <code><nowiki>{{{tempered height}}}</nowiki></code>
: The absolute value of ''Z'' evaluated at the zeta-optimised step count. For [[12edo]], this value is 5.193290 when evaluated at 12.0231830072926.
: 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 <code>pure&nbsp;height</code>, do not use with <code>height</code>.
: Must be used in conjunction with <code>pure height</code>, do not use with <code>height</code>.


; <code>{{{<nowiki />pure height}}}</code>
; <code><nowiki>{{{pure height}}}</nowiki></code>
: The absolute value of ''Z'' evaluated at the exact integer. For [[12edo]], this value is 5.084467 when evaluated at exactly 12.
: 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 <code>tempered&nbsp;height</code>, do not use with <code>height</code>.
: Must be used in conjunction with <code>tempered height</code>, do not use with <code>height</code>.


; <code>{{{<nowiki />height}}}</code>
; <code><nowiki>{{{height}}}</nowiki></code>
: The absolute value of ''Z'' evaluated at a ZPI that does not correspond strongly to an EDO. Do not use with <code>tempered&nbsp;height</code> or <code>pure&nbsp;height</code>.
: The absolute value of ''Z'' evaluated at a ZPI that does not correspond strongly to an edo. Do not use with <code>tempered height</code> or <code>pure height</code>.


; <code>{{{<nowiki />integral}}}</code>
; <code><nowiki>{{{integral}}}</nowiki></code>
: 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.
: 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.


; <code>{{{<nowiki />gap}}}</code>
; <code><nowiki>{{{gap}}}</nowiki></code>




; <code>{{{<nowiki />edo}}}</code> (optional)
; <code><nowiki>{{{edo}}}</nowiki></code> (optional)
: This parameter is optional. Pass in the EDO that the ZPI represents if the page's title does not contain it.
: This parameter is optional. Pass in the edo that the ZPI represents if the page's title does not contain it.


; <code>{{{<nowiki />octave}}}</code>
; <code><nowiki>{{{octave}}}</nowiki></code>
: The zeta-optimised octave size. For 12edo, this is 1197.68616940005.
: The zeta-optimised octave size. For 12edo, this is 1197.68616940005.


; <code>{{{<nowiki />consistent}}}</code>
; <code><nowiki>{{{consistent}}}</nowiki></code>
: The ZPI's [[consistency]] limit.
: The ZPI's [[consistency]] limit.


; <code>{{{<nowiki />distinct}}}</code>
; <code><nowiki>{{{distinct}}}</nowiki></code>
: The ZPI's ''distinct'' consistency limit.
: The ZPI's ''distinct'' consistency limit.


=== Example (12edo) ===
=== Example (12edo) ===
Here is the complete zeta peak index table for [[12edo]] [[12edo#Zeta_peak_index|as it appears on the page]]:
Here is the complete zeta peak index table for [[12edo]] [[12edo #Zeta peak index|as it appears on the page]]:


<pre>
<pre>

Latest revision as of 19:36, 20 May 2025

This template should not be substituted.

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,

$$ Z\left( \frac{2\pi}{\ln 2} \, x \right) $$

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}}} (optional)
This parameter is optional. Pass in the edo that the ZPI represents if the page's title does not contain it.
{{{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 Octave (cents) Integer limit
ZPI Steps
per 8ve 		Step size
(cents) 		Height Integral Gap Size Stretch Consistent Distinct
Tempered Pure
34zpi 12.023183 99.807181 5.19329 5.084467 1.269599 15.899282 1197.686169 −2.313831 10 6
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