Fractional-octave temperaments: Difference between revisions
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A particularly strong offender of this is the [[landscape microtemperaments]] list, which features temperaments which are all supersets of 3edo, but from a composer's perspective it contains wildly different temperaments due to the fact that edo multiples of 3 themselves are different. For example, magnesium (12), and zinc (30), are both landscape systems due to being multiples of 3, but 30edo is drastically different from 12edo in terms of composition, and therefore such temperaments are not alike at all. | A particularly strong offender of this is the [[landscape microtemperaments]] list, which features temperaments which are all supersets of 3edo, but from a composer's perspective it contains wildly different temperaments due to the fact that edo multiples of 3 themselves are different. For example, magnesium (12), and zinc (30), are both landscape systems due to being multiples of 3, but 30edo is drastically different from 12edo in terms of composition, and therefore such temperaments are not alike at all. | ||
== Octave-splitting comma == | |||
An '''octave-splitting comma''' is a rational [[interval]] which induces a fractional-octave temperament. If tempering out the comma leads to splitting the [[octave]] into ''N'' equal parts (''N'' > 1), the comma is an octave-splitting comma which induces an ''N''th-octave temperament. This terminology was developed by [[Xenllium]]. | |||
=== Property === | |||
While a comma is given in the ''p''-limit [[monzo]] form {{monzo| ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>5</sub> … ''a''<sub>''p''</sub> }}, as a rational interval, it is an octave-splitting comma if and only if GCD(''a''<sub>2</sub>, ''a''<sub>3</sub>, ''a''<sub>5</sub>, …, ''a''<sub>''p''</sub>) = 1 and ''N'' = GCD(''a''<sub>3</sub>, ''a''<sub>5</sub>, …, ''a''<sub>''p''</sub>) > 1, and leads to splitting the octave into ''N'' equal parts. | |||
=== Examples === | |||
Below is a list of octave-splitting commas for common temperaments: | |||
{| class="wikitable" | |||
|- | |||
! Comma !! Associated <br>temperament !! Harmonic <br>limit !! Splitting <br>order | |||
|- | |||
| [[256/243]] || [[Limmic temperaments|Blackwood]] || 3 || 5 | |||
|- | |||
| [[2187/2048]] || [[Apotome family|Whitewood]] || 3 || 7 | |||
|- | |||
| [[531441/524288]] || [[Compton family|Compton]] || 3 || 12 | |||
|- | |||
| [[2048/2025]] || [[Diaschismic family|Diaschismic]] || 5 || 2 | |||
|- | |||
| [[128/125]] || [[Augmented family|Augmented]] || 5 || 3 | |||
|- | |||
| [[648/625]] || [[Diminished family|Diminished]] || 5 || 4 | |||
|- | |||
| [[50/49]] || [[Jubilismic clan|Jubilismic]] || 7 || 2 | |||
|- | |||
| [[250047/250000]] || [[Landscape microtemperaments|Landscape]] || 7 || 3 | |||
|- | |||
| [[9801/9800]] || [[Kalismic temperaments|Kalismic]] || 11 || 2 | |||
|- | |||
| [[289/288]] || [[Semitonismic]] || 17 || 2 | |||
|} | |||
== Individual pages of temperaments by subtending equal division == | == Individual pages of temperaments by subtending equal division == | ||
=== 2 to 100 === | === 2 to 100 === | ||
{| class="wikitable center-all" | |||
{| class="wikitable" | |||
|+ | |+ | ||
| | | | ||
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|- | |- | ||
| [[11th-octave temperaments|11]] | | [[11th-octave temperaments|11]] | ||
| [[12th-octave temperaments|12]] | | [[12th-octave temperaments|12]]/[[Compton family|C]] | ||
| [[13th-octave temperaments|13]] | | [[13th-octave temperaments|13]] | ||
| [[14th-octave temperaments|14]] | | [[14th-octave temperaments|14]] | ||
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| [[40th-octave temperaments|40]] | | [[40th-octave temperaments|40]] | ||
|- | |- | ||
| [[41st-octave temperaments|41]] / [[Countercomp family| | | [[41st-octave temperaments|41]]/[[Countercomp family|CC]] | ||
| [[42nd-octave temperaments|42]] | | [[42nd-octave temperaments|42]] | ||
| [[43rd-octave temperaments|43]] | | [[43rd-octave temperaments|43]] | ||
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| [[51st-octave temperaments|51]] | | [[51st-octave temperaments|51]] | ||
| [[52nd-octave temperaments|52]] | | [[52nd-octave temperaments|52]] | ||
| [[53rd-octave temperaments|53]] / [[Mercator family|M]] | | [[53rd-octave temperaments|53]]/[[Mercator family|M]] | ||
| [[54th-octave temperaments|54]] | | [[54th-octave temperaments|54]] | ||
| [[55th-octave temperaments|55]] | | [[55th-octave temperaments|55]] | ||
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[[111th-octave temperaments|111]], [[118th-octave temperaments|118]], [[159th-octave temperaments|159]], [[400th-octave temperaments|400]], [[665th-octave temperaments|665]] | [[111th-octave temperaments|111]], [[118th-octave temperaments|118]], [[159th-octave temperaments|159]], [[400th-octave temperaments|400]], [[665th-octave temperaments|665]] | ||
* C = compton family | |||
C = | * CC = countercomp family | ||
* M = mercator family equated with 53rd-octave temperaments until otherwise documented, also contains 106th-octave temperaments | |||
M = | |||
== Temperaments discussed elsewhere == | == Temperaments discussed elsewhere == | ||
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[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||