Xenharmonic Wiki - Recent changes [en]

13-limit

Eufalesio — Fri, 21 Aug 2026 12:55:01 GMT

← Older revision		Revision as of 12:55, 21 August 2026
Line 14:		Line 14:
	As prime limits increase, ratios containing different primes over [[3/1\|3]] in the numerator and denominator become more and more numerous. Such ratios include [[15/14]], [[14/13]], [[11/10]], [[15/13]], [[13/11]], [[14/11]], [[13/10]], [[15/11]], [[7/5]], and their [[octave complement]]s. An example of a way to use these intervals is to build {{w\|tertian harmony\|tertian}} triads such as [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. Other examples include the [[neogothic major and minor]] triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit via vanishing of [[364/363]], but can also be used as they are.		As prime limits increase, ratios containing different primes over [[3/1\|3]] in the numerator and denominator become more and more numerous. Such ratios include [[15/14]], [[14/13]], [[11/10]], [[15/13]], [[13/11]], [[14/11]], [[13/10]], [[15/11]], [[7/5]], and their [[octave complement]]s. An example of a way to use these intervals is to build {{w\|tertian harmony\|tertian}} triads such as [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. Other examples include the [[neogothic major and minor]] triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit via vanishing of [[364/363]], but can also be used as they are.

	The subgroup can be conveniently rank-reduced into ~~the~~ 7-limit without much loss in accuracy by tempering out [[2080/2079]] and [[4096/4095]], resulting in the [[olympic]] temperament, which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32. Other notable rank-reductions include [[orthoschismic]] and [[~~cassaschismic~~]], which rank-~~reduces~~ olympic by equating 81/80 or 64/63 respectively with the [[Pythagorean comma]].		The subgroup can be conveniently rank-reduced into a 7-limit basis without much loss in accuracy by tempering out [[2080/2079]] and [[4096/4095]], resulting in the [[olympic]] temperament, which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32, among other properties. Other notable rank-reductions include [[orthoschismic]] and [[septischismic]], which rank-reduce olympic by equating 81/80 or 64/63 respectively with the [[Pythagorean comma]] and thus rank-reducing the 13-limit into a 2.3.7 basis or a 5-limit basis.

	== Edo approximation ==		== Edo approximation ==

User:Lucius Chiaraviglio/Keyboard Layout Lab/Various other Lumatone mappings

Lucius Chiaraviglio — Fri, 21 Aug 2026 06:43:36 GMT

Neutral Thirds (compressed) Lumatone mappings: Fog-Related Neutral Seconds Lumatone mappings: Sevond-related Lumatone mappings: Kleismic + Würschmidt Lumatone mappings: Move these down to their proper places

Show changes

User:Lucius Chiaraviglio/Keyboard Layout Lab

Lucius Chiaraviglio — Fri, 21 Aug 2026 06:28:06 GMT

Amavil Lumatone mappings: Insert Porcupine (Porky) Lumatone mappings above this (temporarily moved up for visibility), starting with Bryan Deister's flipped Porcupine Lumatone mapping for 51edo

← Older revision		Revision as of 06:28, 21 August 2026
Line 2:		Line 2:

	Moved here from [[User:Lucius Chiaraviglio/Musical Mad Science\|Musical Mad Science]]: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 06:54, 28 March 2025 (UTC)		Moved here from [[User:Lucius Chiaraviglio/Musical Mad Science\|Musical Mad Science]]: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 06:54, 28 March 2025 (UTC)

			== Porcupine (Porky) Lumatone mappings ==

			=== 51edo flipped (demonstrated to work but awaiting approval) ===

			The flipped [[porcupine]] ([[porky]]) mapping demonstrated in [[Bryan Deister]]'s [https://www.youtube.com/shorts/-ZFMDsE8jy4 ''51edo groove''] (2026) increases the octave slope and decreases the range (to just over four octaves) relative to the normal mapping for this temperament, but might be needed for easy reach to certain chords.

			{{Lumatone EDO mapping\|n=51\|start=22\|xstep=7\|ystep=-9}}

			Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 06:27, 21 August 2026 (UTC)

	== Amavil Lumatone mappings ==		== Amavil Lumatone mappings ==

Lumatone mapping for 72edo

Lucius Chiaraviglio — Fri, 21 Aug 2026 06:04:29 GMT

Catakleismic-related rank-3 (Harmonic table) Lumatone mappings: Fix section heading

← Older revision		Revision as of 06:04, 21 August 2026
(One intermediate revision by the same user not shown)
Line 29:		Line 29:
	This mapping uses [[ennealimmal]] temperament; it maps one axis to 1\9 and one axis to ~21/20, an ennealimmal generator.		This mapping uses [[ennealimmal]] temperament; it maps one axis to 1\9 and one axis to ~21/20, an ennealimmal generator.
	{{Lumatone EDO mapping\|n=72\|start=0\|xstep=8\|ystep=-5}}		{{Lumatone EDO mapping\|n=72\|start=0\|xstep=8\|ystep=-5}}

			== Catakleismic-related rank-3 (Harmonic table) ==
			[[Bryan Deister]] has demonstrated a [[1L 1s]] (42:23 step ratio) mapping for [[72edo]] in [https://www.youtube.com/shorts/a07Q_zQHe9M ''72edo groove''] (2026). The rightward generator 42\72 is the near-just perfect fifth ~[[3/2]]; the upward generator 19\72 is the near-just classic minor third ~[[6/5]] (as in [[catakleismic]] temperament); this makes the down-right generator 23/72 the near-just classic major third ~[[5/4]] (being derived from the other generators, this does not expand the temperament beyond rank-3). That makes this mapping sort of a harmonic table mapping, although different from that for [[12edo]] since neither the classic major third nor the classic minor third divide 72edo evenly, thus causing comma pumping if either of these is stacked with itself. With such a compressed layout, most notes are not present in any one octave, but all notes are present in some octave, even within the limited part of the Lumatone that was actually mapped in the demonstration video; the synthesized sound used in the demonstration video also ensures that all of the notes are audible, even pushing the brain to do automatic octave-reduction.
			{{Lumatone EDO mapping\|n=72\|start=64\|xstep=42\|ystep=-19}}

	{{Navbox Lumatone}}		{{Navbox Lumatone}}

User:Lucius Chiaraviglio/Keyboard Layout Lab/Various rank-3 temperament Lumatone mappings

Lucius Chiaraviglio — Fri, 21 Aug 2026 06:03:06 GMT

Cantonismic-Werckismic rank-3 temperament Lumatone mappings: Insert Catakleismic-related rank-3 Lumatone mappings after this, starting with Bryan Deister's Lumatone mapping for 72edo

@@ Line 42: / Line 42: @@
 Added:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 22:02, 12 June 2025 (UTC)<br>
 Last Modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:01, 15 August 2026 (UTC)
 == Char-based Diatonic + Erato + Injera + Diatonicized Chromaticism (divided 11/8) + Passion/Freivald Lumatone mappings ==

Just intonation

Overthink — Thu, 20 Aug 2026 22:39:24 GMT

Approximating JI with temperaments: add this

← Older revision		Revision as of 22:39, 20 August 2026
(One intermediate revision by the same user not shown)
Line 46:		Line 46:
	; Other approaches		; Other approaches
	Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.		Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.

			== Approximating JI with temperaments ==
			{{Main\|Temperament}}
			There are a lot of JI intervals, and it's difficult to keep track of all of them. As such, people often use simpler systems to approximate JI intervals, known as [[temperament]]s. A popular choice is [[equal temperament]]s; for example the predominant [[12edo\|12et]], which is widely used to approximate [[5-limit]] JI. Other equal temperaments exist, for example [[19edo\|19et]], [[22edo\|22et]], and [[31edo\|31et]]. Besides equal temperaments, other temperaments exist, such as [[regular temperament]]s and [[well temperament]]s.

			Temperaments also create new structures not found in JI; for example, [[meantone]] temperament (which 12et [[support]]s) tempers out the syntonic comma [[81/80]], making [[5/4]] the same as the major third obtained by stacking four fifths, [[81/64]]; this structural feature is often assumed without thinking in western music. There are many temperaments in various limits, including the 5-limit, the [[7-limit]], and beyond, as well as many subgroups.

			[[Microtemperaments]] are very accurate and temper out [[unnoticeable comma]]s, and add minimal damage to the intervals of JI, while also simplifying its structure. For example, [[schismic]] flattens the fifth by around 0.2 cents to equate the [[Pythagorean comma]] with the syntonic comma. Microtemperaments are also very useful in higher prime limits, where JI itself becomes too complex.

	== Instruments ==		== Instruments ==

51edo

Lucius Chiaraviglio — Thu, 20 Aug 2026 19:16:17 GMT

Music: Add Bryan Deister's ''51edo groove'' (2026)

← Older revision		Revision as of 19:16, 20 August 2026
Line 700:		Line 700:
	* [https://www.youtube.com/shorts/SJW-JTHyeIA ''51edo prelude''] (2026)		* [https://www.youtube.com/shorts/SJW-JTHyeIA ''51edo prelude''] (2026)
	* [https://www.youtube.com/watch?v=k3NOBYbiqpo ''51edo improv''] (2026-04-22)		* [https://www.youtube.com/watch?v=k3NOBYbiqpo ''51edo improv''] (2026-04-22)
			* [https://www.youtube.com/shorts/-ZFMDsE8jy4 ''51edo groove''] (2026)

	; [[Frédéric Gagné]]		; [[Frédéric Gagné]]

User:Lucius Chiaraviglio/Keyboard Layout Lab/Various other Lumatone mappings

Lucius Chiaraviglio — Thu, 20 Aug 2026 19:01:24 GMT

Hemikleismic + Semaja Lumatone mappings: Add Bryan Deister's forward oversize chroma version of this; rework to accommodate it

← Older revision		Revision as of 19:01, 20 August 2026
Line 274:		Line 274:
	== Hemikleismic + Semaja Lumatone mappings ==		== Hemikleismic + Semaja Lumatone mappings ==

	=== 53edo (demonstrated to work) ===		=== 53edo (2 mappings demonstrated to work) ===

	[[Bryan Deister]] has demonstrated ~~a mapping~~ for [[53edo]] ~~in [https://www.youtube.com/shorts/WtSaDQCyfVc ''Waltz in 53edo''] (2026)~~ that functions for [[Kleismic_family#Hemikleismic\|hemikleismic]] ([[7L 1s]] scale with a 7:4 step ratio ~~right and right-up~~ with the small step stretched) and [[~~Porwell_temperaments~~#Semaja\|semaja]] (rotated [[5L 1s]] with a 10:3 step ratio down-right and down); a [[5L 8s (4/1-equivalent)]] scale with a 10:7 step ratio is also available moving right and down-right. The range is a bit over 4⅓ octaves, with no missed notes and just one repeated note in each octave, but the upper left and lower right corners have many additional non-contiguous notes, and the octaves are not in consistent places; on the other hand, classic thirds both minor (two steps right) and major (one step right plus one step down-right) are in very easy reach as long as they do not pass through a vertical wraparound (it may be necessary to adjust note 0 position to avoid this for a particular performance, such as in the linked video which has note 0 where note 3 is here).		[[Bryan Deister]] has demonstrated two mappings for [[53edo]] that functions for [[Kleismic_family#Hemikleismic\|hemikleismic]] ([[7L 1s]] (pine) scale with a 7:4 step ratio with the small step stretched) and [[Gariboh_clan#Semaja\|semaja]] (rotated [[5L 1s]] scale with a 10:3 step ratio); both versions favor hemikleismic over semaja. For hemikleismic, the generator 7\53 is half of a near-just classic minor third ~[[6/5]], which functions as a flat undecimal submajor second ~[[11/10]], a sharp undecimal neutral second ~[[12/11]], a slightly sharp septimal neutral second ~[[35/32]], and a near-just large vicesimotertial neutral second ~[[23/21]]. Four of these make a slightly flat tridecimal diminished fifth ~[[13/9]]; five of them make a somewhat flat Boethius' minor sixth ~[[12/9]]; after octave-reduction, ten of them make a flat flat septimal major second ~[[8/7]], which at 10\53 is the generator for semaja; several undevicesimal ratios are also available. Two of the semaja generator make a slightly flat Barbados third ~[[13/10]]; after octave-reduction, eight of them make a near-just classic major third ~[[5/4]]; several tridecimal ratios are also available.

			==== Reverse chroma ====

			In the reverse-chroma version demonstrated in [https://www.youtube.com/shorts/WtSaDQCyfVc ''Waltz in 53edo''] (2026), the hemikleismic 7L 1s scale is played by moving right and right-up with the small step stretched, and the semaja rotated [[5L 1s]] scale is played by moving down-right and down; a [[5L 8s (4/1-equivalent)]] scale with a 10:7 step ratio is also available moving right and down-right. The range is a bit over 4⅓ octaves, with no missed notes and just one repeated note in each octave, but the upper left and lower right corners have many additional non-contiguous notes, and the octaves are not in consistent places; on the other hand, classic thirds both minor (two steps right) and major (one step right plus one step down-right) are in very easy reach as long as they do not pass through a vertical wraparound (it may be necessary to adjust note 0 position to avoid this for a particular performance, such as in the linked video which has note 0 where note 3 is here).

	{{Lumatone EDO mapping\|n=53\|start=7\|xstep=7\|ystep=3}}		{{Lumatone EDO mapping\|n=53\|start=7\|xstep=7\|ystep=3}}

			==== Forward oversize chroma ====

			In the forward oversize chroma version (very similar to the [[Lumatone_mapping_for_porcupine#Expanded\|expanded Lumatone mapping for porcupine]]), demonstrated in [https://www.youtube.com/shorts/7YfZZ7aMvH8 ''53edo groove''] (2026), the hemikleismic 7L 1s scale is played by moving right and right-down-right with the small step stretched, and the semaja rotated [[5L 1s]] scale is played by moving up and up-left; the 5L 8s (4/1-equivalent) scale is not practical in this version. The range is less than that of the reverse chroma version, at somewhat under three complete octaves, but in exchange, more repeated notes are available to mitigate vertical wraparounds, and the upwards slope of the octaves is less.

			{{Lumatone EDO mapping\|n=53\|start=32\|xstep=7\|ystep=-10}}

	Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 09:41, 27 February 2026 (UTC)<br>		Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 09:41, 27 February 2026 (UTC)<br>
	Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 12:12, ~~1 March~~ 2026 (UTC)		Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 19:01, 20 August 2026 (UTC)

	== Infraorwell Lumatone mappings ==		== Infraorwell Lumatone mappings ==

Lumatone mapping for 53edo

Lucius Chiaraviglio — Thu, 20 Aug 2026 19:00:46 GMT

Hemikleismic + Semaja: Add Bryan Deister's forward oversize chroma version of this; rework to accommodate it

← Older revision		Revision as of 19:00, 20 August 2026
Line 18:		Line 18:

	== Hemikleismic + Semaja ==		== Hemikleismic + Semaja ==
	[[Bryan Deister]] has demonstrated ~~a mapping~~ for [[53edo]] ~~in [https://www.youtube.com/shorts/WtSaDQCyfVc ''Waltz in 53edo''] (2026)~~ that functions for [[Kleismic_family#Hemikleismic\|hemikleismic]] ([[7L 1s]] scale with a 7:4 step ratio ~~right and right-up~~ with the small step stretched) and [[~~Porwell_temperaments~~#Semaja\|semaja]] (rotated [[5L 1s]] with a 10:3 step ratio down-right and down); a [[5L 8s (4/1-equivalent)]] scale with a 10:7 step ratio is also available moving right and down-right. The range is a bit over 4⅓ octaves, with no missed notes and just one repeated note in each octave, but the upper left and lower right corners have many additional non-contiguous notes, and the octaves are not in consistent places; on the other hand, classic thirds both minor (two steps right) and major (one step right plus one step down-right) are in very easy reach as long as they do not pass through a vertical wraparound (it may be necessary to adjust note 0 position to avoid this for a particular performance, such as in the linked video which has note 0 where note 3 is here).		[[Bryan Deister]] has demonstrated two mappings for [[53edo]] that functions for [[Kleismic_family#Hemikleismic\|hemikleismic]] ([[7L 1s]] (pine) scale with a 7:4 step ratio with the small step stretched) and [[Gariboh_clan#Semaja\|semaja]] (rotated [[5L 1s]] scale with a 10:3 step ratio); both versions favor hemikleismic over semaja. For hemikleismic, the generator 7\53 is half of a near-just classic minor third ~[[6/5]], which functions as a flat undecimal submajor second ~[[11/10]], a sharp undecimal neutral second ~[[12/11]], a slightly sharp septimal neutral second ~[[35/32]], and a near-just large vicesimotertial neutral second ~[[23/21]]. Four of these make a slightly flat tridecimal diminished fifth ~[[13/9]]; five of them make a somewhat flat Boethius' minor sixth ~[[12/9]]; after octave-reduction, ten of them make a flat flat septimal major second ~[[8/7]], which at 10\53 is the generator for semaja; several undevicesimal ratios are also available. Two of the semaja generator make a slightly flat Barbados third ~[[13/10]]; after octave-reduction, eight of them make a near-just classic major third ~[[5/4]]; several tridecimal ratios are also available.

			=== Reverse chroma ===
			In the reverse-chroma version demonstrated in [https://www.youtube.com/shorts/WtSaDQCyfVc ''Waltz in 53edo''] (2026), the hemikleismic 7L 1s scale is played by moving right and right-up with the small step stretched, and the semaja rotated [[5L 1s]] scale is played by moving down-right and down; a [[5L 8s (4/1-equivalent)]] scale with a 10:7 step ratio is also available moving right and down-right. The range is a bit over 4⅓ octaves, with no missed notes and just one repeated note in each octave, but the upper left and lower right corners have many additional non-contiguous notes, and the octaves are not in consistent places; on the other hand, classic thirds both minor (two steps right) and major (one step right plus one step down-right) are in very easy reach as long as they do not pass through a vertical wraparound (it may be necessary to adjust note 0 position to avoid this for a particular performance, such as in the linked video which has note 0 where note 3 is here).
	{{Lumatone EDO mapping\|n=53\|start=7\|xstep=7\|ystep=3}}		{{Lumatone EDO mapping\|n=53\|start=7\|xstep=7\|ystep=3}}

			=== Forward oversize chroma ===
			In the forward oversize chroma version (very similar to the [[Lumatone_mapping_for_porcupine#Expanded\|expanded Lumatone mapping for porcupine]]), demonstrated in [https://www.youtube.com/shorts/7YfZZ7aMvH8 ''53edo groove''] (2026), the hemikleismic 7L 1s scale is played by moving right and right-down-right with the small step stretched, and the semaja rotated [[5L 1s]] scale is played by moving up and up-left; the 5L 8s (4/1-equivalent) scale is not practical in this version. The range is less than that of the reverse chroma version, at somewhat under three complete octaves, but in exchange, more repeated notes are available to mitigate vertical wraparounds, and the upwards slope of the octaves is less.
			{{Lumatone EDO mapping\|n=53\|start=32\|xstep=7\|ystep=-10}}

	== Buzzard ==		== Buzzard ==

53edo

Lucius Chiaraviglio — Thu, 20 Aug 2026 18:00:40 GMT

Music: Fix section heading

← Older revision		Revision as of 18:00, 20 August 2026
Line 1,185:		Line 1,185:
	* [https://www.youtube.com/watch?v=DCENVnxH6bI ''Bande-announce''] – rendered by MortisTheneRd (2024)		* [https://www.youtube.com/watch?v=DCENVnxH6bI ''Bande-announce''] – rendered by MortisTheneRd (2024)

	==== 21st century ====		=== 21st century ===
	; [[ALLY195]]		; [[ALLY195]]
	* [https://www.bilibili.com/video/BV1f54y1r7XG/ ''My Soul adaptation''] (2020)		* [https://www.bilibili.com/video/BV1f54y1r7XG/ ''My Soul adaptation''] (2020)

SandBox

Lucius Chiaraviglio — Thu, 20 Aug 2026 17:53:44 GMT

← Older revision		Revision as of 17:53, 20 August 2026
Line 24:		Line 24:
	{{MOS genchain\|Scale Signature=29L 12s}}		{{MOS genchain\|Scale Signature=29L 12s}}

	==== Modes ===		==== Modes ====
	{{MOS mode degrees\|Scale Signature=29L 12s}}		{{MOS mode degrees\|Scale Signature=29L 12s}}

	=== Scale tree ===		=== Scale tree ===
	{{MOS tuning spectrum\|Scale Signature=29L 12s}}		{{MOS tuning spectrum\|Scale Signature=29L 12s}}

7th-octave temperaments

Eliora — Thu, 20 Aug 2026 17:16:19 GMT

← Older revision		Revision as of 17:16, 20 August 2026
Line 4:		Line 4:

	A comma that frequently appears in 7th-octave temps is [[akjaysma]], which sets [[105/64]] to be equal to 5\7.		A comma that frequently appears in 7th-octave temps is [[akjaysma]], which sets [[105/64]] to be equal to 5\7.

			[[Jackpot]] is another highly notable temperament, tempering out the [[7-29-comma]], associating [[29/16]] to 6\7.

	Temperaments discussed elsewhere include:		Temperaments discussed elsewhere include:

2072edo

Eliora — Thu, 20 Aug 2026 17:01:16 GMT

← Older revision		Revision as of 17:01, 20 August 2026
Line 2:		Line 2:
	{{ED intro}}		{{ED intro}}

	2072edo is [[consistent]] in the [[17-odd-limit]], as well as a strong 5-limit tuning, tempering out [[kwazy]], {{monzo\|-53 10 16}}, {{monzo\|-33 97 -52}}, and [[barium comma]], {{monzo\|-225 224 -56}}, equating [[81/80]] to 1/56th of the octave. It provides the [[optimal patent val]] for the [[barium]] temperament in the 13-limit. It tempers out the [[euzenius]] comma in the 7-limit.		2072edo is [[consistent]] in the [[17-odd-limit]], containing the 2.7.11 mapping of [[296edo]], as well as a strong 5-limit tuning, tempering out [[kwazy]], {{monzo\|-53 10 16}}, {{monzo\|-33 97 -52}}, and [[barium comma]], {{monzo\|-225 224 -56}}, equating [[81/80]] to 1/56th of the octave. It provides the [[optimal patent val]] for the [[barium]] temperament in the 13-limit. It tempers out the [[euzenius]] comma in the 7-limit.

	~~2072edo contains the 2.7.11 mapping of [[296edo]]~~.

	=== Prime harmonics ===		=== Prime harmonics ===

Talk:Sensipent family

Godtone — Thu, 20 Aug 2026 16:28:26 GMT

clarification

← Older revision		Revision as of 16:28, 20 August 2026
(One intermediate revision by the same user not shown)
Line 12:		Line 12:

	: —[[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 07:55, 20 August 2026 (UTC)		: —[[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 07:55, 20 August 2026 (UTC)

			:: If you agree a subgroup extension is canonical s.t there's no ambiguity (meaning it's unique among alternatives) then I believe it should be presented as such (as an extension proper, but with a subheader indicating it's a subgroup extension, as it was before). Furthermore, your reason seems backwards to me; shouldn't a subgroup extension that isn't in the increasing order of primes be prioritised at the very top, as it represents unique information? --[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]])

User:Eufalesio/Good scales

Eufalesio — Thu, 20 Aug 2026 15:52:06 GMT

The Tridecimal Skirmish [SV3, CS]

← Older revision		Revision as of 15:52, 20 August 2026
Line 73:		Line 73:

	=== The Tridecimal Skirmish [SV3, CS] ===		=== The Tridecimal Skirmish [SV3, CS] ===
	It's called a Skirmish because I like to arpeggiate it. Either the RH prime or the mode on 4/3 is good. I shy away from the utonal versions. It's a subset of the scale I call Ghyef~~. There is also a pyth, pental and undecimal version, but they aren't as important to me~~.		It's called a Skirmish because I like to arpeggiate it. Either the RH prime or the mode on 4/3 is good. I shy away from the utonal versions. It's a subset of the scale I call Ghyef.
	{\| class="wikitable"		{\| class="wikitable"
	! colspan="5" \|LH		! colspan="5" \|LH
Line 82:		Line 82:
	\|-		\|-
	!13/12		!13/12
	!39/32		!4/3
	!3/2		!3/2
	!13/8		!13/8
	! rowspan="5" \|2		! rowspan="5" \|2
	!13/12		!13/12
	!4/3		!39/32
	!3/2		!3/2
	!13/8		!13/8
	! rowspan="5" \|2		! rowspan="5" \|2
	\|-		\|-
	\|<small>9/8</small>		\|<small>16/13</small>
	\|<small>18/13</small>		\|<small>18/13</small>
	\|<small>3/2</small>		\|<small>3/2</small>
	\|<small>24/13</small>		\|<small>24/13</small>
	\|<small>16/13</small>		\|<small>9/8</small>
	\|<small>18/13</small>		\|<small>18/13</small>
	\|<small>3/2</small>		\|<small>3/2</small>
	\|<small>24/13</small>		\|<small>24/13</small>
	\|-		\|-
			\|<small>9/8</small>
			\|<small>39/32</small>
			\|<small>3/2</small>
			\|<small>13/8</small>
	\|<small>16/13</small>		\|<small>16/13</small>
	\|<small>4/3</small>		\|<small>4/3</small>
	\|<small>64/39</small>		\|<small>64/39</small>
	\|<small>16/9</small>		\|<small>16/9</small>
	~~\|<small>9/8</small>~~
	~~\|<small>39/32</small>~~
	~~\|<small>3/2</small>~~
	~~\|<small>13/8</small>~~
	\|-		\|-
	\|<small>13/12</small>		\|<small>13/12</small>
	\|<small>4/3</small>		\|<small>4/3</small>
	\|<small>13/9</small>		\|<small>13/9</small>
	\|<small>13/8</small>		\|<small>16/9</small>
	\|<small>13/12</small>		\|<small>13/12</small>
	\|<small>4/3</small>		\|<small>4/3</small>
	\|<small>13/9</small>		\|<small>13/9</small>
	\|<small>16/9</small>		\|<small>13/8</small>
	\|-		\|-
	\|<small>16/13</small>		\|<small>16/13</small>
	\|<small>4/3</small>		\|<small>4/3</small>
	\|<small>3/2</small>		\|<small>64/39</small>
	\|<small>24/13</small>		\|<small>24/13</small>
	\|<small>16/13</small>		\|<small>16/13</small>
	\|<small>4/3</small>		\|<small>4/3</small>
	\|<small>64/39</small>		\|<small>3/2</small>
	\|<small>24/13</small>		\|<small>24/13</small>
			\|}

			=== A general Skirmish ===
			Any scale of this form for any cent value 0 < C < 4/3. I declare to be a Skirmish. RH Skirmish have sqrt(4/3) < C < 4/3, and LH 1 < C < sqrt(4/3). This also means that the above scales, Nicepent, Pyth pentic modes, and semiquartal scales are Skirmishes, though... the term is better for C ≈ 13/12, you know, "minor second-ish" intervals.
			{\| class="wikitable"
			\|-
			! colspan="5" \|Scale degrees
			\|-
			!C
			!4/3
			!3/2
			!3/2C
			! rowspan="5" \|2
			\|-
			\|4/(3C)
			\|3/(2C)
			\|<small>3/2</small>
			\|2/C
			\|-
			\|<small>9/8</small>
			\|9/8C
			\|<small>3/2</small>
			\|3/2C
			\|-
			\|C
			\|<small>4/3</small>
			\|4/3C
			\|<small>16/9</small>
			\|-
			\|4/(3C)
			\|<small>4/3</small>
			\|16/(9C)
			\|2/C
	\|}		\|}

Line 234:		Line 267:
	\|}		\|}

	=== ~~Aura's Zarlinoid~~ [MV4, CS, SP] ===		=== Didymus [MV4, CS, SP] ===
	A scale that Aura really likes to use. It's good in some places, but Zarlino is better generally. The LH version is good basically only for its minor scale, because the major is kind of harsh with 10/9 and 40/27.		A scale that Aura really likes to use. It's good in some places, but Zarlino is better generally. The LH version is good basically only for its minor scale, because the major is kind of harsh with 10/9 and 40/27.
	{\| class="wikitable"		{\| class="wikitable"

Buzzardsma

Eufalesio — Thu, 20 Aug 2026 15:33:48 GMT

More definitions!

← Older revision		Revision as of 15:33, 20 August 2026
Line 4:		Line 4:
	\| Comma = yes		\| Comma = yes
	\|Color name=s4rM, saquaruma}}		\|Color name=s4rM, saquaruma}}
	The '''buzzardsma''' or '''buzzard comma''' ({{monzo\|legend=1\| 16 -3 0 -4 }}, [[ratio]]: 65536/64827) is a [[small comma\|small]] [[7-limit]] (specifically [[2.3.7 subgroup\|2.3.7-subgroup]]) [[comma]] measuring about 18.8 [[cent]]s. It is the amount by which a perfect twelfth ([[3/1]]) exceeds four grave fourths of [[21/16]].		The '''buzzardsma''' or '''buzzard comma''' ({{monzo\|legend=1\| 16 -3 0 -4 }}, [[ratio]]: 65536/64827) is a [[small comma\|small]] [[7-limit]] (specifically [[2.3.7 subgroup\|2.3.7-subgroup]]) [[comma]] measuring about 18.8 [[cent]]s. It is the amount by which a perfect twelfth ([[3/1]]) exceeds four grave fourths of [[21/16]], the difference between a stack of four [[8/7]]'s and 27/16, and the difference between a stack of 4 [[64/63]] and 256/243. In terms of commas it can be expressed as (64/63)/([[1029/1024]]).

	== Temperaments ==		== Temperaments ==

User:Francium/5113edo

Francium — Thu, 20 Aug 2026 14:02:32 GMT

Regular temperament properties: +music

← Older revision		Revision as of 14:02, 20 August 2026
Line 64:		Line 64:
	\| 4.26		\| 4.26
	\|}		\|}

			== Music ==
			; [[Francium]]
			* "ameliorific" from ''with our kindness.'' (2026) – [https://francium223.bandcamp.com/track/ameliorific Bandcamp] \| [https://www.youtube.com/watch?v=p779wKPTKQY YouTube]

Jofur

FloraC — Thu, 20 Aug 2026 09:48:29 GMT

Recategorize (jove extension)

← Older revision		Revision as of 09:48, 20 August 2026
Line 1:		Line 1:
	#redirect [[Breed family#Jofur]]		#redirect [[Breed family #Jofur]]

	[[Category:~~Temperaments~~]]		[[Category:Jove]]

Technologismic

FloraC — Thu, 20 Aug 2026 09:47:37 GMT

Recategorize

← Older revision		Revision as of 09:47, 20 August 2026
Line 1:		Line 1:
	#REDIRECT [[Very high accuracy temperaments#Technologismic ~~or Trisa-quinbiru-agu (441 & 1848 & 3125)~~]]		#REDIRECT [[Very high accuracy temperaments #Technologismic]]

	[[Category:~~Temperaments~~]]		[[Category:Rank-3 temperaments]]
			[[Category:Microtemperaments]]

Lisa

FloraC — Thu, 20 Aug 2026 09:45:53 GMT

Recategorize (liese extension)

← Older revision		Revision as of 09:45, 20 August 2026
Line 1:		Line 1:
	#redirect [[Meantone family#Lisa]]		#redirect [[Meantone family #Lisa]]

	[[Category:~~Temperaments]]~~		[[Category:Liese]]
	~~[[Category:Meantone family~~]]

Liesel

FloraC — Thu, 20 Aug 2026 09:45:32 GMT

Recategorize (liese extension)

← Older revision		Revision as of 09:45, 20 August 2026
Line 1:		Line 1:
	#redirect [[Meantone family#Liesel]]		#redirect [[Meantone family #Liesel]]

	[[Category:~~Temperaments]]~~		[[Category:Liese]]
	~~[[Category:Meantone family~~]]

Lahoh

FloraC — Thu, 20 Aug 2026 09:41:41 GMT

Recategorize (injera extension)

← Older revision		Revision as of 09:41, 20 August 2026
Line 1:		Line 1:
	#redirect [[Meantone family #Lahoh]]		#redirect [[Meantone family #Lahoh]]

	[[Category:~~Temperaments]]~~		[[Category:Injera]]
	~~[[Category:Meantone family~~]]

Laconic

FloraC — Thu, 20 Aug 2026 09:40:53 GMT

Recategorize

← Older revision		Revision as of 09:40, 20 August 2026
Line 1:		Line 1:
	#redirect [[Syntonic–diatonic equivalence continuum #Laconic]]		#redirect [[Syntonic–diatonic equivalence continuum #Laconic]]

	[[Category:~~Temperaments~~]]		[[Category:Rank-2 temperaments]]

Ragismic microtemperaments

FloraC — Thu, 20 Aug 2026 09:34:45 GMT

Chlorine

← Older revision		Revision as of 09:34, 20 August 2026
Line 501:		Line 501:
	== Chlorine ==		== Chlorine ==
	: ''For the 5-limit version, see [[17th-octave temperaments #Chlorine]].''		: ''For the 5-limit version, see [[17th-octave temperaments #Chlorine]].''

			Chlorine (named after the 17th element) tempers out the [[septendecima]] in the 5-limit, and {{monzo\| -49 4 22 -3 }} as well as the ragisma in the 7-limit. It has a 1/17-octave period, and can be described as {{nowrap\| 289 & 323 }} temperament. Not only the semitwelfth, but also the ~5/4 can be used as a generator.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

17th-octave temperaments

FloraC — Thu, 20 Aug 2026 09:34:43 GMT

Chlorine

← Older revision		Revision as of 09:34, 20 August 2026
Line 75:		Line 75:
	: ''For extensions, see [[Ragismic microtemperaments #Chlorine]].''		: ''For extensions, see [[Ragismic microtemperaments #Chlorine]].''

	Chlorine has a period of 1/17 octave. It tempers out {{monzo\| -52 -17 34 }}, the [[septendecima]], by which 17 classical chromatic semitones ((25/24)<sup>17</sup>) exceed an octave~~. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo\| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator~~.		Chlorine has a period of 1/17 octave. It tempers out {{monzo\| -52 -17 34 }}, the [[septendecima]], by which 17 classical chromatic semitones ((25/24)<sup>17</sup>) exceed an octave.

	The name of this temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].		The name of this temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].

Chlorine

FloraC — Thu, 20 Aug 2026 09:31:50 GMT

Update

← Older revision		Revision as of 09:31, 20 August 2026
Line 1:		Line 1:
	#redirect [[~~17th-octave temperaments~~#Chlorine]]		#redirect [[Ragismic microtemperaments #Chlorine]]

	[[Category:~~Temperaments~~]]		[[Category:Rank-2 temperaments]]
			[[Category:Microtemperaments]]
	[[Category:Ragismic microtemperaments]]		[[Category:Ragismic microtemperaments]]

17th-octave temperaments

FloraC — Thu, 20 Aug 2026 09:31:12 GMT

- septimal chlorine (properly addressed in Ragismic)

@@ Line 73: / Line 73: @@
 == Chlorine ==
 Chlorine has a period of 1/17 octave. It tempers out {{monzo| -52 -17 34 }}, the [[septendecima]], by which 17 classical chromatic semitones ((25/24)<sup>17</sup>) exceed an octave. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.
@@ Line 93: / Line 95: @@
 [[Badness]] (Sintel): 1.81
 {{Navbox fractional-octave}}
 [[Category:17edo]]

Ragismic microtemperaments

FloraC — Thu, 20 Aug 2026 09:30:07 GMT

Move chlorine back according to XW:TEMPCAT now that everything is sorted out

@@ Line 29: / Line 29: @@
 * ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
 * [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
 * ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]]
-Considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, octoid, seniority, monzismic, semidimfourth, acrokleismic, quasithird, quincy, deca, keenanose, counterkleismic, sfourth, aluminium, ragitritonic, quatracot, trideci, moulin, and palladium.
+Considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, chlorine, octoid, seniority, monzismic, semidimfourth, acrokleismic, quasithird, quincy, deca, keenanose, counterkleismic, sfourth, aluminium, ragitritonic, quatracot, trideci, moulin, and palladium.
 == Supermajor ==
@@ Line 499: / Line 498: @@
 Badness (Sintel): 0.899
 == Octoid ==

17th-octave temperaments

FloraC — Thu, 20 Aug 2026 09:26:01 GMT

Delete leaves

@@ Line 125: / Line 125: @@
 Badness (Sintel): 2.11
 {{Navbox fractional-octave}}
 [[Category:17edo]]

User:Eliora/Leaves

FloraC — Thu, 20 Aug 2026 09:25:36 GMT

As requested, deleted temp entries will be moved to the user page

New page

== Leaves ==
Defined as the {{nowrap|323 & 2023}} temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1729951171875/1727094849536, {{monzo| 10 39 -14 -14 }}

{{Mapping|legend=1| 17 10 31 9 | 0 14 7 32 }}
: mapping generators: ~25/24, ~6125/5832

[[Optimal tuning]]s:
* [[CTE]]: ~25/24 = 70.588{{c}}, ~6125/5832 = 85.427{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}}, ~6125/5832 = 85.426{{c}}

{{Optimal ET sequence|legend=1| 323, 1700d, 2023, 2346}}

[[Badness]] (Sintel): 32.032

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 160083/160000, 198359290368/198165034375, 1729951171875/1727094849536

Mapping: {{mapping| 17 10 31 9 106 | 0 14 7 32 -39 }}

Optimal tunings:
* CTE: ~25/24 = 70.588{{c}}, ~6125/5832 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}}, ~6125/5832 = 85.420{{c}}

{{Optimal ET sequence|legend=0| 323, 1700d, 2023, 2346e}}

Badness (Sintel): 20.456

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 160083/160000, 928125/927472, 1990656/1990625, 4831530/4826809

Mapping: {{mapping| 17 10 31 9 106 98 | 0 14 7 32 -39 -29 }}

Optimal tunings:
* CTE: ~25/24 = 70.588{{c}}, ~1024/975 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}}, ~1024/975 = 85.420{{c}}

{{Optimal ET sequence|legend=0| 323, 1700d, 2023, 2346e}}

Badness (Sintel): 10.096

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 39325/39304, 57375/57344, 71874/71825, 111537/111475, 140800/140777

Mapping: {{mapping| 17 10 31 9 106 98 107 | 0 14 7 32 -39 -29 -31 }}

Optimal tunings:
* CTE: ~25/24 = 70.588{{c}}, ~765/728 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}}, ~765/728 = 85.421{{c}}

{{Optimal ET sequence|legend=0| 323, 1700d, 2023, 2346e}}

Badness (Sintel): 6.678

2023edo

FloraC — Thu, 20 Aug 2026 09:22:21 GMT

- deleted temp

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is [[Enfactoring|enfactored]] in the 5-limit, with the same mapping as [[289edo]]. As such it maps the period to [[25/24]], which means [[septendecima]] is also tempered out. In the 17-limit on the [[patent val]], it is a tuning for the [[leaves]] temperament.
+edo is [[enfactoring|enfactored]] in the 5-limit, with the same mapping and commas as [[289edo]].
 If we impose a stricter harmonic approach, and require all errors to be below 25%, the subgroup consisting of first 7 such harmonics for 2023edo is 2.13.17.23.47.61.71.
@@ Line 14: / Line 14: @@
 === Subsets and supersets ===
 Since 2023 factors as {{factorization|2023}}, 2023edo has subset edos {{EDOs| 7, 17, 119, and 289 }}.
 == Music ==

1700edo

FloraC — Thu, 20 Aug 2026 09:22:13 GMT

- deleted temp

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is only [[consistent]] in the [[5-odd-limit]], and there is a large relative delta on the [[harmonic]] [[3/1|3]]. It has a reasonable approximation to the 2.9.15.21.11.13.17.23 [[subgroup]], or if the harmonic [[5/1|5]] is desired, the 2.9.5.21.11.23 subgroup. Otherwise, it can be considered in the 2.9.21.11.23.31 [[subgroup]] (not including either 5 or 15). Nonetheless, it tunes the {{nowrap|323 &amp; 2023}} temperament [[leaves]] in the 17-limit on the [[patent val]].
+edo is only [[consistent]] in the [[5-odd-limit]], and there is a large relative delta on the [[harmonic]] [[3/1|3]]. It has a reasonable approximation to the 2.9.15.21.11.13.17.23 [[subgroup]], or if the harmonic [[5/1|5]] is desired, the 2.9.5.21.11.23 subgroup. Otherwise, it can be considered in the 2.9.21.11.23.31 [[subgroup]] (not including either 5 or 15).
 === Odd harmonics ===
@@ Line 11: / Line 11: @@
 Since 1700 factors into {{factorization|1700}}, 1700edo has subset edos {{EDOs| 2, 4, 5, 10, 17, 20, 25, 34, 50, 68, 85, 100, 170, 340, 425, and 850 }}.
 One step of 1700edo is the [[relative cent]] for [[17edo]]. It has been named '''iota''' by [[Margo Schulter]] and [[George Secor]].

Xenharmonic Wiki:Deleted temperament entries

FloraC — Thu, 20 Aug 2026 09:21:46 GMT

+ leaves

← Older revision		Revision as of 09:21, 20 August 2026
Line 169:		Line 169:
	\|-		\|-
	\| French decimal		\| French decimal
	\| 2.5.7.11.13		\| 2.5.7(.11.13)
	\| {{Mapping\| 1 ~~2 54 177~~ -52 \| 0 1 -159 -539 173 }}		\| {{Mapping\| 1 0 372 1255 -398 \| 0 1 -159 -539 173 }}
	\| Pointless microtemperament: very high badness at effectively unusable complexity~~; it is~~ unusable ~~even by 1789edo idiosynchratic French revolution lore standards, notably it doesn't even temper out the jacobin comma 6656/6655~~		\| Pointless microtemperament: very high badness at effectively unusable complexity
			\|-
			\| Leaves
			\| 2.3.5.7(.11.13.17)
			\| {{Mapping\| 17 10 31 9 106 98 107 \| 0 14 7 32 -39 -29 -31 }}
			\| Pointless microtemperament: high badness at effectively unusable complexity
	\|}		\|}

	[[Category:Xenharmonic Wiki guidelines]]		[[Category:Xenharmonic Wiki guidelines]]

User:Eufalesio/Good scales

Eufalesio — Thu, 20 Aug 2026 09:12:21 GMT

More things, more expansions

Show changes

User:FloraC/Temperament name proposal

FloraC — Thu, 20 Aug 2026 09:08:34 GMT

← Older revision		Revision as of 09:08, 20 August 2026
Line 46:		Line 46:
	* Lifthrasirsma (2023)		* Lifthrasirsma (2023)
	* <s>Undecapyth comma (2024)</s>		* <s>Undecapyth comma (2024)</s>
			* Acroschisma (2026)

	== Variations on existing roots ==		== Variations on existing roots ==
	* Parakleismic extensions:		* Parakleismic extensions:
	** ~~Parakleismic~~ extensions:		** Septimal parakleismic extensions:
	*** Paralytic (11-limit 99e & 118) – analogous to catalytic		*** Paralytic (11-limit 99e & 118) – analogous to catalytic
	*** Semiparamint (13-limit 80 & 118)		*** Semiparamint (13-limit 80 & 118)

Valentine

FloraC — Thu, 20 Aug 2026 09:05:39 GMT

+ category

← Older revision		Revision as of 09:05, 20 August 2026
Line 384:		Line 384:
	[[Category:Valentine\| ]] <!-- main article -->		[[Category:Valentine\| ]] <!-- main article -->
	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]
			[[Category:Gamelismic clan]]
	[[Category:Starling temperaments]]		[[Category:Starling temperaments]]
	[[Category:~~Gamelismic clan~~]]		[[Category:Porwell temperaments]]

118edo

FloraC — Thu, 20 Aug 2026 09:03:20 GMT

Regular temperament properties: update

← Older revision		Revision as of 09:03, 20 August 2026
Line 1,152:		Line 1,152:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| -187 118 }}		\| {{Monzo\| -187 118 }}
	\| {{~~mapping~~\| 118 187 }}		\| {{Mapping\| 118 187 }}
	\| ~~−0~~.119		\| −0.119
	\| 0.082		\| 0.082
	\| 0.81		\| 0.81
Line 1,160:		Line 1,160:
	\| 2.3.5		\| 2.3.5
	\| 32805/32768, {{monzo\| 8 14 -13 }}		\| 32805/32768, {{monzo\| 8 14 -13 }}
	\| {{~~mapping~~\| 118 187 274 }}		\| {{Mapping\| 118 187 274 }}
	\| +0.036		\| +0.036
	\| 0.093		\| 0.093
Line 1,167:		Line 1,167:
	\| 2.3.5.7		\| 2.3.5.7
	\| 1029/1024, 3136/3125, 4375/4374		\| 1029/1024, 3136/3125, 4375/4374
	\| {{~~mapping~~\| 118 187 274 331 }}		\| {{Mapping\| 118 187 274 331 }}
	\| +0.270		\| +0.270
	\| 0.412		\| 0.412
Line 1,174:		Line 1,174:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 385/384, 441/440, 3136/3125, 4375/4374		\| 385/384, 441/440, 3136/3125, 4375/4374
	\| {{~~mapping~~\| 118 187 274 331 408 }}		\| {{Mapping\| 118 187 274 331 408 }}
	\| +0.341		\| +0.341
	\| 0.370		\| 0.370
Line 1,181:		Line 1,181:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 196/195, 352/351, 384/384, 625/624, 729/728		\| 196/195, 352/351, 384/384, 625/624, 729/728
	\| {{~~mapping~~\| 118 187 274 331 408 437 }} (118)		\| {{Mapping\| 118 187 274 331 408 437 }} (118)
	\| +0.125		\| +0.125
	\| 0.604		\| 0.604
Line 1,188:		Line 1,188:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 169/168, 325/324, 364/363, 385/384, 3136/3125		\| 169/168, 325/324, 364/363, 385/384, 3136/3125
	\| {{~~mapping~~\| 118 187 274 331 408 436 }} (118f)		\| {{Mapping\| 118 187 274 331 408 436 }} (118f)
	\| +0.583		\| +0.583
	\| 0.650		\| 0.650
Line 1,195:		Line 1,195:
	\| 2.3.5.7.11.17		\| 2.3.5.7.11.17
	\| 289/288, 385/384, 441/440, 561/560, 3136/3125		\| 289/288, 385/384, 441/440, 561/560, 3136/3125
	\| {{~~mapping~~\| 118 187 274 331 408 482 }}		\| {{Mapping\| 118 187 274 331 408 482 }}
	\| +0.417		\| +0.417
	\| 0.399		\| 0.399
Line 1,202:		Line 1,202:
	\| 2.3.5.7.11.17.19		\| 2.3.5.7.11.17.19
	\| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968		\| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
	\| {{~~mapping~~\| 118 187 274 331 408 482 501 }}		\| {{Mapping\| 118 187 274 331 408 482 501 }}
	\| +0.445		\| +0.445
	\| 0.376		\| 0.376
Line 1,229:		Line 1,229:
	\| 193.22		\| 193.22
	\| 28/25		\| 28/25
	\| [[~~Luna]] / [[hemithirds~~]] / [[lunatic]]		\| [[Hemithirds]] / [[lunatic]]
	\|-		\|-
	\| 1		\| 1
Line 1,235:		Line 1,235:
	\| 233.90		\| 233.90
	\| 8/7		\| 8/7
	\| [[~~Slendric]] / [[guiron~~]]		\| [[Guiron]]
	\|-		\|-
	\| 1		\| 1
Line 1,241:		Line 1,241:
	\| 315.25		\| 315.25
	\| 6/5		\| 6/5
	\| [[Parakleismic]] / ~~paralytic~~		\| [[Parakleismic]] / [[counterpara]]
	\|-		\|-
	\| 1		\| 1
Line 1,253:		Line 1,253:
	\| 498.31		\| 498.31
	\| 4/3		\| 4/3
	\| [[~~Helmholtz (temperament)\|Helmholtz]] / [[pontiac~~]] / helenoid / pontic		\| [[Pontiac]] / helenoid / pontic
	\|-		\|-
	\| 1		\| 1
Line 1,277:		Line 1,277:
	\| 71.19		\| 71.19
	\| 25/24		\| 25/24
	\| [[Vishnu]] / ~~ananta (118) / acyuta (118f)~~		\| [[Vishnu]] / [[narayana]]
	\|-		\|-
	\| 2		\| 2
Line 1,289:		Line 1,289:
	\| 162.71		\| 162.71
	\| 11/10		\| 11/10
	\| [[~~Kwazy]] / [[bisupermajor~~]]		\| [[Bisupermajor]]
	\|-		\|-
	\| 2		\| 2
Line 1,304:		Line 1,304:
	\|-		\|-
	\| 2		\| 2
	\| ~~31\118<br>(~~28\118)		\| 28\118
	\| ~~315.25<br>(~~284.75)		\| 284.75
	\| ~~6/5<br>(~~33/28)		\| 33/28
	\| [[Semiparakleismic]]		\| [[Semiparakleismic]]
	\|}		\|}
	<nowiki/>* ~~[[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and~~ [[normal forms #Minimal-generator form\|minimal form]] ~~in parentheses if distinct~~		<nowiki/>* In [[normal forms #Minimal-generator form\|minimal-generator form]]

	== Instruments ==		== Instruments ==

Counterpara

FloraC — Thu, 20 Aug 2026 08:54:48 GMT

Redirected page to Parakleismic family#Counterpara

New page

#redirect [[Parakleismic family #Counterpara]]

[[Category:Rank-2 temperaments]]
[[Category:Parakleismic family]]
[[Category:Horwell temperaments]]

Horwell temperaments

FloraC — Thu, 20 Aug 2026 08:53:29 GMT

+ link to counterpara

← Older revision		Revision as of 08:53, 20 August 2026
Line 11:		Line 11:
	* ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped\|Escapade family]]		* ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped\|Escapade family]]
	* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila\|Mabila family]]		* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila\|Mabila family]]
			* ''[[Counterpara]]'' (+100442349/100000000) → [[Parakleismic family #Counterpara\|Parakleismic family]]
	* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana\|Vishnu family]]		* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana\|Vishnu family]]
	* [[Hemithirds]] (+1029/1024) → [[Hemimean clan #Hemithirds\|Hemimean clan]]		* [[Hemithirds]] (+1029/1024) → [[Hemimean clan #Hemithirds\|Hemimean clan]]

373edo

FloraC — Thu, 20 Aug 2026 08:50:42 GMT

Regular temperament properties

← Older revision		Revision as of 08:50, 20 August 2026
Line 108:		Line 108:
	\| [[Untriton]] (5-limit)		\| [[Untriton]] (5-limit)
	\|}		\|}
	<nowiki/>* ~~[[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and~~ [[normal forms #Minimal-generator form\|minimal form]] ~~in parentheses if distinct~~		<nowiki/>* In [[normal forms #Minimal-generator form\|minimal-generator form]]

	== Music ==		== Music ==

137edo

FloraC — Thu, 20 Aug 2026 08:50:14 GMT

Regular temperament properties: update

← Older revision		Revision as of 08:50, 20 August 2026
Line 17:		Line 17:
	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal<br />8ve stretch (¢)		! rowspan="2" \| Optimal<br>8ve stretch (¢)
	! colspan="2" \| Tuning error		! colspan="2" \| Tuning error
	\|-		\|-
Line 24:		Line 24:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| -217 137 }}		\| {{Monzo\| -217 137 }}
	\| {{~~mapping~~\| 137 217 }}		\| {{Mapping\| 137 217 }}
	\| +0.3865		\| +0.3865
	\| 0.3866		\| 0.3866
Line 31:		Line 31:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| -21 3 7 }}, {{monzo\| -13 17 -6 }}		\| {{Monzo\| -21 3 7 }}, {{monzo\| -13 17 -6 }}
	\| {{~~mapping~~\| 137 217 318 }}		\| {{Mapping\| 137 217 318 }}
	\| +0.3887		\| +0.3887
	\| 0.3157		\| 0.3157
Line 42:		Line 42:
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperaments		! Temperaments
	\|-		\|-
Line 51:		Line 51:
	\| 3\137		\| 3\137
	\| 26.28		\| 26.28
	\| ~~1594323~~/~~1562500~~		\| 49/48
	\| [[Sfourth]] (~~5-limit~~)		\| [[Sfourth]] (137d…)
	\|-		\|-
	\| 1		\| 1
Line 70:		Line 70:
	\| 315.33		\| 315.33
	\| 6/5		\| 6/5
	\| [[Parakleismic]]		\| [[Parakleismic]] (137d) / [[counterpara]] (137)
	\|-		\|-
	\| 1		\| 1
Line 90:		Line 90:
	\| [[Emka]] (137d) / [[emkay]] (137)		\| [[Emka]] (137d) / [[emkay]] (137)
	\|}		\|}
	<nowiki />* ~~[[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and~~ [[normal forms #Minimal-generator form\|minimal form]] ~~in parentheses if distinct~~		<nowiki/>* In [[normal forms #Minimal-generator form\|minimal-generator form]]

	== Diagrams ==		== Diagrams ==

373edo

FloraC — Thu, 20 Aug 2026 08:45:42 GMT

Update

← Older revision		Revision as of 08:45, 20 August 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	373edo is [[consistency\|distinctly consistent]] to the [[15-odd-limit]]. It has a flat tendency, with [[harmonic]]s 3 through 13 all tuned flat. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 8 14 -13 }} ([[parakleisma]]) and {{monzo\| -51 19 9 }} (untriton comma) in the 5-limit; 2401/2400 ([[breedsma]]), 65625/65536 ([[horwell comma]]), and 43046721/42875000 in the 7-limit; [[3025/3024]], [[8019/8000]], 24057/24010, and 496125/495616 in the 11-limit; [[729/728]], [[1001/1000]], [[1716/1715]], [[4225/4224]], and [[10648/10647]] in the 13-limit, enabling [[squbemic chords]] and [[sinbadmic chords]]. It also [[support]]s the [[~~breedsmic temperaments #Hemitert\|~~hemitert ~~temperament~~]].		373edo is [[consistency\|distinctly consistent]] to the [[15-odd-limit]]. It has a flat tendency, with [[harmonic]]s 3 through 13 all tuned flat. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 8 14 -13 }} ([[parakleisma]]) and {{monzo\| -51 19 9 }} (untriton comma) in the [[5-limit]]; 2401/2400 ([[breedsma]]), 65625/65536 ([[horwell comma]]), and 43046721/42875000 in the [[7-limit]]; [[3025/3024]], [[8019/8000]], 24057/24010, and 496125/495616 in the [[11-limit]]; [[729/728]], [[1001/1000]], [[1716/1715]], [[4225/4224]], and [[10648/10647]] in the [[13-limit]], enabling [[squbemic chords]] and [[sinbadmic chords]]. It also [[support]]s the [[hemitert]] temperament.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 20:		Line 20:
	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal<br />8ve stretch (¢)		! rowspan="2" \| Optimal<br>8ve stretch (¢)
	! colspan="2" \| Tuning error		! colspan="2" \| Tuning error
	\|-		\|-
Line 27:		Line 27:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| -591 373 }}		\| {{Monzo\| -591 373 }}
	\| {{~~mapping~~\| 373 591 }}		\| {{Mapping\| 373 591 }}
	\| +0.1939		\| +0.1939
	\| 0.1939		\| 0.1939
Line 34:		Line 34:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| 8 14 -13 }}, {{monzo\| -51 19 9 }}		\| {{Monzo\| 8 14 -13 }}, {{monzo\| -51 19 9 }}
	\| {{~~mapping~~\| 373 591 866 }}		\| {{Mapping\| 373 591 866 }}
	\| +0.1658		\| +0.1658
	\| 0.1632		\| 0.1632
Line 42:		Line 42:
	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 65625/65536, 43046721/42875000		\| 2401/2400, 65625/65536, 43046721/42875000
	\| {{~~mapping~~\| 373 591 866 1047 }}		\| {{Mapping\| 373 591 866 1047 }}
	\| +0.1654		\| +0.1654
	\| 0.1413		\| 0.1413
Line 49:		Line 49:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 3025/3024, 8019/8000, 65625/65536		\| 2401/2400, 3025/3024, 8019/8000, 65625/65536
	\| {{~~mapping~~\| 373 591 866 1047 1290 }}		\| {{Mapping\| 373 591 866 1047 1290 }}
	\| +0.2008		\| +0.2008
	\| 0.1449		\| 0.1449
Line 56:		Line 56:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 729/728, 1001/1000, 1716/1715, 3025/3024, 4225/4224		\| 729/728, 1001/1000, 1716/1715, 3025/3024, 4225/4224
	\| {{~~mapping~~\| 373 591 866 1047 1290 1380 }}		\| {{Mapping\| 373 591 866 1047 1290 1380 }}
	\| +0.2056		\| +0.2056
	\| 0.1327		\| 0.1327
Line 66:		Line 66:
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperaments		! Temperaments
	\|-		\|-
Line 88:		Line 88:
	\| 315.28		\| 315.28
	\| 6/5		\| 6/5
	\| [[~~Parakleismic~~]] ~~(5-limit)~~		\| [[Counterpara]]
	\|-		\|-
	\| 1		\| 1
Line 108:		Line 108:
	\| [[Untriton]] (5-limit)		\| [[Untriton]] (5-limit)
	\|}		\|}
	<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

255edo

FloraC — Thu, 20 Aug 2026 08:42:28 GMT

Update

← Older revision		Revision as of 08:42, 20 August 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	The equal temperament [[tempering out\|tempers out]] the [[parakleisma]], {{monzo\| 8 14 -13 }}, and the [[septendecima]], {{monzo\| -52 -17 34 }}, in the 5-limit. In the 7-limit it tempers out ~~[[cataharry]],~~ 19683/19600, [[~~mirkwai~~]], 16875/16807 ~~and~~ [[~~horwell~~]], 65625/65536, so that it [[support]]s the [[mirkat]] temperament, and in fact provides the [[optimal patent val]]. It also gives the optimal patent val for mirkat in the 11-limit, tempering out [[540/539]], 1375/1372, [[3025/3024]] and [[8019/8000]]. In the 13-limit it tempers out [[847/845]], [[625/624]], [[1575/1573]] and [[1716/1715]].		The equal temperament [[tempering out\|tempers out]] the [[parakleisma]], {{monzo\| 8 14 -13 }}, and the [[septendecima]], {{monzo\| -52 -17 34 }}, in the [[5-limit]]. In the [[7-limit]] it tempers out 19683/19600 ([[cataharry comma]]), 16875/16807 ([[canopic comma]]), and 65625/65536 ([[horwell comma]]) so that it [[support]]s the [[mirkat]] temperament, and in fact provides the [[optimal patent val]]. It also gives the optimal patent val for mirkat in the [[11-limit]], tempering out [[540/539]], [[1375/1372]], [[3025/3024]] and [[8019/8000]]. In the [[13-limit]] it tempers out [[847/845]], [[625/624]], [[1575/1573]] and [[1716/1715]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 255 factors into {{~~factorization~~\|~~255~~}}, 255edo has subset edos {{EDOs\| 3, 5, 15, 17, 51, and 85 }}.		Since 255 factors into {{nowrap\| 3 × 5 × 17 }}, 255edo has subset edos {{EDOs\| 3, 5, 15, 17, 51, and 85 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 61:		Line 61:
	! Associated<br>ratio*		! Associated<br>ratio*
	! Temperaments		! Temperaments
	\|-
	~~\| 1~~
	~~\| 39\255~~
	~~\| 183.53~~
	~~\| 10/9~~
	~~\| [[Mirkat]] (255f)~~
	\|-		\|-
	\| 1		\| 1
Line 78:		Line 72:
	\| 315.29		\| 315.29
	\| 6/5		\| 6/5
	\| [[~~Parakleismic~~]] ~~(5-limit)~~		\| [[Counterpara]]
	\|-		\|-
	\| 1		\| 1
Line 87:		Line 81:
	\|-		\|-
	\| 3		\| 3
	\| 82\255~~<br>~~(3\255)		\| 39\255
	\| ~~385.88<br>(~~14.12)		\| 183.53
	\| ~~5/4<br>(~~126/125)		\| 10/9
			\| [[Mirkat]] (255f)
			\|-
			\| 3
			\| 3\255
			\| 14.12
			\| 126/125
	\| [[Mutt]] (7-limit)		\| [[Mutt]] (7-limit)
	\|-		\|-
	\| 5		\| 5
	\| ~~53\255<br>(~~2\255)		\| 2\255
	\| ~~249.41<br>(~~9.41)		\| 9.41
	\| ~~81/70<br>(~~176/175)		\| 176/175
	\| [[Hemiquintile]] / hemiquint (255) / hemiquintilis (255f)		\| [[Hemiquintile]] / hemiquint (255) / hemiquintilis (255f)
	\|-		\|-
	\| 5		\| 5
	\| ~~106\255<br>(~~4\255)		\| 4\255
	\| ~~498.82<br>(~~18.82)		\| 18.82
	\| ~~4/3<br>(~~81/80)		\| 81/80
	\| [[Quintile]]		\| [[Quintile]]
	\|-		\|-
	\| 17		\| 17
	\| ~~53\255<br>(~~7\255)		\| 7\255
	\| ~~249.41<br>(~~32.94)		\| 32.94
	\| ~~{{Monzo\| -25 -9 17 }}<br>(~~1990656/1953125)		\| 1990656/1953125
	\| [[Chlorine]] (5-limit)		\| [[Chlorine]] (5-limit)
	\|}		\|}
	<nowiki/>* ~~[[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and~~ [[normal forms #Minimal-generator form\|minimal form]] ~~in parentheses if distinct~~		<nowiki/>* In [[normal forms #Minimal-generator form\|minimal-generator form]]

	[[Category:Mirkat]]		[[Category:Mirkat]]

Maviloid

FloraC — Thu, 20 Aug 2026 08:35:53 GMT

Update

← Older revision		Revision as of 08:35, 20 August 2026
Line 1:		Line 1:
	#REDIRECT [[~~Breedsmic temperaments~~ #Maviloid]]		#REDIRECT [[Parakleismic family #Maviloid]]

	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]
	[[Category:Microtemperaments]]		[[Category:Microtemperaments]]
			[[Category:Parakleismic family]]
	[[Category:Breedsmic temperaments]]		[[Category:Breedsmic temperaments]]

Breedsmic temperaments

FloraC — Thu, 20 Aug 2026 08:35:06 GMT

- maviloid (moved to Parakleismic family)

@@ Line 26: / Line 26: @@
 * ''[[Neptune (temperament)|Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
 * ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
 * ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]
-Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, maviloid, lockerbie, unthirds, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].
+Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, lockerbie, unthirds, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].
 == Tertiaseptal ==
@@ Line 682: / Line 683: @@
 Badness (Sintel): 1.39
 == Lockerbie ==

Parakleismic family

FloraC — Thu, 20 Aug 2026 08:32:26 GMT

Move maviloid here

@@ Line 222: / Line 222: @@
 Badness (Sintel): 1.67
 == Paraguay ==
-Named by [[Xenllium]] in 2026, paraguay tempers out [[12005/11664]] and may be described as the {{nowrap| 19 & 61 }} temperament. It is a variant of septimal parakleismic, mapping 7th harmonic to 16 generators.
+Named by [[Xenllium]] in 2026, paraguay tempers out [[126/125]] and may be described as the {{nowrap| 19 & 61 }} temperament. It is a variant of septimal parakleismic, mapping 7th harmonic to 16 generators.
 [[Subgroup]]: 2.3.5.7
@@ Line 242: / Line 278: @@
 [[Badness]] (Sintel): 2.47
 [[Category:Parakleismic family| ]] <!-- main article -->
 [[Category:Temperament families]]
 [[Category:Catalogs of rank-2 temperaments]]

← Older revision		Revision as of 08:53, 20 August 2026
Line 11:		Line 11:
	* ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped\|Escapade family]]		* ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped\|Escapade family]]
	* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila\|Mabila family]]		* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila\|Mabila family]]
			* ''[[Counterpara]]'' (+100442349/100000000) → [[Parakleismic family #Counterpara\|Parakleismic family]]
	* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana\|Vishnu family]]		* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana\|Vishnu family]]
	* [[Hemithirds]] (+1029/1024) → [[Hemimean clan #Hemithirds\|Hemimean clan]]		* [[Hemithirds]] (+1029/1024) → [[Hemimean clan #Hemithirds\|Hemimean clan]]