Fractional-octave temperaments - Revision history

Eliora: /* Disagreement between temperament catalog policy and fractional-octave practice */ showcasing that landscape temperaments are really the embodiment or the worst "offender" of such disagreeement

2026-08-05T01:24:27Z

Disagreement between temperament catalog policy and fractional-octave practice: showcasing that landscape temperaments are really the embodiment or the worst "offender" of such disagreeement

← Older revision		Revision as of 01:24, 5 August 2026
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	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]].		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.		Besides, on this wiki, temperament collection pages are used to collect temperaments that temper out a common comma. For example, the [[landscape microtemperaments]] are the most prominent example of this phenomenon, as the 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 ==

Eliora: /* Individual pages of temperaments by subtending equal division */ put legend right after the table it's supposed to explian

2026-07-13T17:06:22Z

Individual pages of temperaments by subtending equal division: put legend right after the table it's supposed to explian

← Older revision		Revision as of 17:06, 13 July 2026
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	\| [[100th-octave temperaments\|100]]		\| [[100th-octave temperaments\|100]]
	\|}		\|}

	~~=== 101 and up ===~~
	~~[[111th-octave temperaments\|111]], [[118th-octave temperaments\|118]], [[159th-octave temperaments\|159]], [[400th-octave temperaments\|400]], [[665th-octave temperaments\|665]]~~

	* 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 ==

FloraC: /* Temperaments discussed elsewhere */ cut the table (1/)

2026-07-03T16:08:06Z

Temperaments discussed elsewhere: cut the table (1/)

← Older revision		Revision as of 16:08, 3 July 2026
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	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:

	* [[2nd- to 4th-octave temperaments]]
	* 1\5 period temperaments		* 1\5 period temperaments
	** [[Quintile family\|Quintile temperaments]]		** [[Quintile family\|Quintile temperaments]]
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	** [[Cloudy comma #Temperaments\|Cloudy temperaments]]		** [[Cloudy comma #Temperaments\|Cloudy temperaments]]
	** [[Blackwood family]]		** [[Blackwood family]]
	* 1\6 period temperaments
	** [[Augmented family #Hexe\|Hexe]]
	** [[Landscape microtemperaments #Sextile\|Sextile]]
	** [[Stearnsmic clan #Stearnscape\|Stearnscape]]
	* [[Akjaysma\|Akjaysmic temperaments]] (1\7 period)
	** [[Ragismic microtemperaments #Brahmagupta\|Brahmagupta]]
	** [[Schismatic family #Septant\|Septant]]
	** [[Whitewood family #Whitewood\|Whitewood temperaments]]
	** [[Keemic temperaments #Sevond\|Sevond]]
	** [[Mistismic temperaments #Neutron\|Neutron]]
	* [[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)

FloraC: Page merge followup

2026-07-02T13:00:22Z

Page merge followup

← Older revision		Revision as of 13:00, 2 July 2026
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	\|+		\|+
	\|		\|
	\| [[2nd-~~octave temperaments\|2]]~~		\| colspan="3" \| [[2nd- to 4th-octave temperaments\|2–4]]
	~~\| [[3rd-octave temperaments\|3]]~~
	~~\| [[~~4th-octave temperaments\|4]]
	\| [[5th-octave temperaments\|5]]		\| [[5th-octave temperaments\|5]]
	\| [[6th-octave temperaments\|6]]		\| [[6th-octave temperaments\|6]]
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	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\2 period temperaments		* [[2nd- to 4th-octave temperaments]]
	** [[~~Diaschismic family\|Diaschismic temperaments]]~~
	** [[Vishnuzmic family\|Vishnuzmic temperaments]]
	** [[Jubilismic clan\|Jubilismic temperaments]]
	** [[Varunismic temperaments]]
	** [[Lokismic temperaments]]
	** [[Nimona\|Nimona temperament]]
	* 1\3 period temperaments
	** [[Augmented family\|Augmented temperaments]]
	** [[Misty family\|Misty temperaments]]
	** [[Landscape microtemperaments\|Landscape temperaments]]
	* 1\4 period temperaments
	** [[Diminished family\|Diminished temperaments]]
	** [[Undim family\|Undim temperaments]]
	* 1\5 period temperaments		* 1\5 period temperaments
	** [[Quintile family\|Quintile temperaments]]		** [[Quintile family\|Quintile temperaments]]

FloraC: /* Theory */ rework for neutrality

2026-06-28T12:08:38Z

Theory: rework for neutrality

← Older revision		Revision as of 12:08, 28 June 2026
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	== Theory ==		== Theory ==
	Fractional-octave temperaments are valuable with regards to [[~~Polysystemic\|polysystemicism~~]] and polychromatics. They are acoustically significant with regards to containing 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.		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)~~-ET~~ which repeats y times within that period, its "wireframe" is y~~-ET~~. 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 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 ~~strategy~~ and fractional-octave practice ===		=== Disagreement between temperament catalog policy and fractional-octave practice ===
	~~Traditional~~ regular temperament ~~perspective on~~ periods and generators ~~has a shortcoming when it comes to handling fractional-octave temperaments~~, as it treats divisions of periods (for example, what [[hemiennealimmal]] is to [[ennealimmal]]) as [[extension]]s of a temperament with a subset period. However, fractional-octave temperaments and scales are sought for being able to treat an 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]]~~. This facet is reflected by the distinction of strong and weak extensions~~.		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]].

	~~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.		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 ==

FloraC: Update

2026-06-28T11:42:53Z

Update

← Older revision		Revision as of 11:42, 28 June 2026
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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]] \|\| [[~~Limmic temperaments~~\|Blackwood]] \|\| 3 \|\| 5		\| [[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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	** [[Quintosec family\|Quintosec temperaments]]		** [[Quintosec family\|Quintosec temperaments]]
	** [[Trisedodge family\|Trisedodge temperaments]]		** [[Trisedodge family\|Trisedodge temperaments]]
	** [[Cloudy ~~clan~~\|Cloudy temperaments]]		** [[Cloudy comma #Temperaments\|Cloudy temperaments]]
	** [[~~Limmic temperaments~~]]		** [[Blackwood family]]
	* 1\6 period temperaments		* 1\6 period temperaments
	** [[Augmented family #Hexe\|Hexe]]		** [[Augmented family #Hexe\|Hexe]]

Xenllium at 07:09, 7 June 2026

2026-06-07T07:09:43Z

← Older revision		Revision as of 07:09, 7 June 2026
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	== Octave-splitting comma ==		== 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]].		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 ===		=== Examples ===

FloraC: - self-redirect

2026-05-20T09:34:06Z

- self-redirect

← Older revision		Revision as of 09:34, 20 May 2026
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	== See also ==		== See also ==
	* [[Map of rank-2 temperaments]]: Visual map of many of the temperaments listed here.		* [[Map of rank-2 temperaments]]: Visual map of many of the temperaments listed here.
	* [[Octave-splitting comma]]

	[[Category:Temperament collections]]		[[Category:Temperament collections]]

FloraC: Move Xenllium's "octave-splitting comma" concept here

2026-05-20T09:33:25Z

Move Xenllium's "octave-splitting comma" concept here

@@ Line 15: / Line 15: @@
 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.
 == Individual pages of temperaments by subtending equal division ==
 === 2 to 100 ===
 {| class="wikitable center-all"

Xenllium at 09:29, 20 May 2026

2026-05-20T09:29:13Z

← Older revision		Revision as of 09:29, 20 May 2026
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	== See also ==		== See also ==
	* [[Map of rank-2 temperaments]]: Visual map of many of the temperaments listed here.		* [[Map of rank-2 temperaments]]: Visual map of many of the temperaments listed here.
			* [[Octave-splitting comma]]

	[[Category:Temperament collections]]		[[Category:Temperament collections]]

← Older revision		Revision as of 16:08, 3 July 2026
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	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:

	* [[2nd- to 4th-octave temperaments]]
	* 1\5 period temperaments		* 1\5 period temperaments
	** [[Quintile family\|Quintile temperaments]]		** [[Quintile family\|Quintile temperaments]]
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	** [[Cloudy comma #Temperaments\|Cloudy temperaments]]		** [[Cloudy comma #Temperaments\|Cloudy temperaments]]
	** [[Blackwood family]]		** [[Blackwood family]]
	* 1\6 period temperaments
	** [[Augmented family #Hexe\|Hexe]]
	** [[Landscape microtemperaments #Sextile\|Sextile]]
	** [[Stearnsmic clan #Stearnscape\|Stearnscape]]
	* [[Akjaysma\|Akjaysmic temperaments]] (1\7 period)
	** [[Ragismic microtemperaments #Brahmagupta\|Brahmagupta]]
	** [[Schismatic family #Septant\|Septant]]
	** [[Whitewood family #Whitewood\|Whitewood temperaments]]
	** [[Keemic temperaments #Sevond\|Sevond]]
	** [[Mistismic temperaments #Neutron\|Neutron]]
	* [[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)

← Older revision		Revision as of 13:00, 2 July 2026
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	\|+		\|+
	\|		\|
	\| [[2nd-~~octave temperaments\|2]]~~		\| colspan="3" \| [[2nd- to 4th-octave temperaments\|2–4]]
	~~\| [[3rd-octave temperaments\|3]]~~
	~~\| [[~~4th-octave temperaments\|4]]
	\| [[5th-octave temperaments\|5]]		\| [[5th-octave temperaments\|5]]
	\| [[6th-octave temperaments\|6]]		\| [[6th-octave temperaments\|6]]
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	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\2 period temperaments		* [[2nd- to 4th-octave temperaments]]
	** [[~~Diaschismic family\|Diaschismic temperaments]]~~
	** [[Vishnuzmic family\|Vishnuzmic temperaments]]
	** [[Jubilismic clan\|Jubilismic temperaments]]
	** [[Varunismic temperaments]]
	** [[Lokismic temperaments]]
	** [[Nimona\|Nimona temperament]]
	* 1\3 period temperaments
	** [[Augmented family\|Augmented temperaments]]
	** [[Misty family\|Misty temperaments]]
	** [[Landscape microtemperaments\|Landscape temperaments]]
	* 1\4 period temperaments
	** [[Diminished family\|Diminished temperaments]]
	** [[Undim family\|Undim temperaments]]
	* 1\5 period temperaments		* 1\5 period temperaments
	** [[Quintile family\|Quintile temperaments]]		** [[Quintile family\|Quintile temperaments]]