Fractional-octave temperaments: Difference between revisions
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== Theory == | == Theory == | ||
Fractional-octave temperaments are valuable with regards to [[ | Fractional-octave temperaments are valuable with regards to [[polysystemic]]ism and polychromatics. They are acoustically significant with regards to containing {{w|modes of limited transposition}}, as well as their ability to expand on the harmony of the equal division they are a superset of. Such temperaments are also a way of introducing less common and harmonically less performing equal divisions into music that prefers consonance and is based on regular temperament theory. | ||
=== Terminology === | === Terminology === | ||
The terminology was developed by [[Eliora]]. The equal division containing the mos scale of such a temperament, starting from the tonic, is referred to as a ''wireframe'', and individual notes of that equal division are called ''hinges''. Thus in this context, the wireframe is the tuning consisting of only stacks of the period and no stacks of the generator. Temperament-agnostically, this can be used to refer to any structure embedded in an (x,y) | The terminology was developed by [[Eliora]]. The equal division containing the mos scale of such a temperament, starting from the tonic, is referred to as a ''wireframe'', and individual notes of that equal division are called ''hinges''. Thus in this context, the wireframe is the tuning consisting of only stacks of the period and no stacks of the generator. Temperament-agnostically, this can be used to refer to any structure embedded in an (''x'', ''y'') equal temperament which repeats ''y'' times within that period, its "wireframe" is ''y'' equal temperament. If an equal division is a subset of a temperament, it is said to ''subtend'' the temperament, just how hinges on a ferris wheel subtend the structure to make it rotate and function. | ||
The most common way to produce a fractional-octave temperament is through an excellent approximation of an interval relative to the size of the wireframe edo. For example, [[compton family]] tempers out the Pythagorean comma and maps 7 steps of 12edo to [[3/2]]. Likewise, a lot of 10th-octave temperaments have a [[13/8]] as 7\10, and 26th-octave temperaments often have a [[7/4]] for 21\26. | The most common way to produce a fractional-octave temperament is through an excellent approximation of an interval relative to the size of the wireframe edo. For example, [[compton family]] tempers out the Pythagorean comma and maps 7 steps of 12edo to [[3/2]]. Likewise, a lot of 10th-octave temperaments have a [[13/8]] as 7\10, and 26th-octave temperaments often have a [[7/4]] for 21\26. | ||
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However, an equal division does not have to be harmonically decent to be a wireframe for a fractional-octave temperament. If an equal division has multiples which are high in consistency or are zeta equal divisions or otherwise harmonically strong, it can produce a lot of such temperaments—notable examples being [[20edo]] or [[32edo]]. Likewise, proximity of a step of equal division to a comma is often a source of these temperaments—for example [[56edo]]'s step being directly close to [[81/80]], and 44edo's step being extremely close to [[64/63]]. | However, an equal division does not have to be harmonically decent to be a wireframe for a fractional-octave temperament. If an equal division has multiples which are high in consistency or are zeta equal divisions or otherwise harmonically strong, it can produce a lot of such temperaments—notable examples being [[20edo]] or [[32edo]]. Likewise, proximity of a step of equal division to a comma is often a source of these temperaments—for example [[56edo]]'s step being directly close to [[81/80]], and 44edo's step being extremely close to [[64/63]]. | ||
=== Disagreement between temperament catalog | === Disagreement between temperament catalog policy and fractional-octave practice === | ||
Vanilla regular temperament theory does not distinguish periods and generators, so it treats divisions of periods (for example, what [[hemiennealimmal]] is to [[ennealimmal]]) as [[extension]]s of a temperament with a subset period, just like how it treats divisions of generators. However, fractional-octave temperaments and scales are sometimes sought for being able to treat each equal division as an entity in its own right, so a composer might find hemiennealimmal to be a drastically different system to ennealimmal in line with [[18edo]] being very different from [[9edo]]. | |||
Besides, on this wiki, temperament collection pages are used to collect temperaments that temper out a common comma. For example, the [[landscape microtemperaments]] list features temperaments which all temper out the [[landscape comma]], as they are all related by that. A fractional-octave temperament user might consider that 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 == | == Octave-splitting comma == | ||
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! Comma !! Associated <br>temperament !! Harmonic <br>limit !! Splitting <br>order | ! Comma !! Associated <br>temperament !! Harmonic <br>limit !! Splitting <br>order | ||
|- | |- | ||
| [[256/243]] || [[ | | [[256/243]] || [[Blackwood family|Blackwood]] || 3 || 5 | ||
|- | |- | ||
| [[2187/2048]] || [[Apotome family|Whitewood]] || 3 || 7 | | [[2187/2048]] || [[Apotome family|Whitewood]] || 3 || 7 | ||
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|+ | |+ | ||
| | | | ||
| [[2nd- | | colspan="3" | [[2nd- to 4th-octave temperaments|2–4]] | ||
| [[5th-octave temperaments|5]] | | [[5th-octave temperaments|5]] | ||
| [[6th-octave temperaments|6]] | | [[6th-octave temperaments|6]] | ||
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| [[100th-octave temperaments|100]] | | [[100th-octave temperaments|100]] | ||
|} | |} | ||
* C = compton family | * C = compton family | ||
* CC = countercomp family | * CC = countercomp family | ||
* M = mercator family equated with 53rd-octave temperaments until otherwise documented, also contains 106th-octave temperaments | * M = mercator family equated with 53rd-octave temperaments until otherwise documented, also contains 106th-octave temperaments | ||
=== 101 and up === | |||
[[111th-octave temperaments|111]], [[118th-octave temperaments|118]], [[159th-octave temperaments|159]], [[400th-octave temperaments|400]], [[665th-octave temperaments|665]] | |||
== Temperaments discussed elsewhere == | == Temperaments discussed elsewhere == | ||
Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include: | Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include: | ||
* 1\5 period temperaments | * 1\5 period temperaments | ||
** [[Quintile family|Quintile temperaments]] | ** [[Quintile family|Quintile temperaments]] | ||
** [[Quintosec family|Quintosec temperaments]] | ** [[Quintosec family|Quintosec temperaments]] | ||
** [[Trisedodge family|Trisedodge temperaments]] | ** [[Trisedodge family|Trisedodge temperaments]] | ||
** [[Cloudy | ** [[Cloudy comma #Temperaments|Cloudy temperaments]] | ||
** [[ | ** [[Blackwood family]] | ||
* [[Ragismic microtemperaments #Octoid|Octoid]], [[Schismatic family #Octant|octant]] (1\8 period) | * [[Ragismic microtemperaments #Octoid|Octoid]], [[Schismatic family #Octant|octant]] (1\8 period) | ||
* [[Septiennealimmal clan|Septiennealimmal temperaments]] (1\9 period) | * [[Septiennealimmal clan|Septiennealimmal temperaments]] (1\9 period) | ||