80edo - Revision history

Eliora: /* Subsets and supersets */ wording

2026-09-03T17:05:07Z

Subsets and supersets: wording

← Older revision		Revision as of 17:05, 3 September 2026
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	As a composite edo, the main subsets it lacks are subsets of [[3edo\|3]] and [[9edo\|9]], but 9\80 = 135{{cent}} offers a good approximation to 1\9 = 133.33…{{c}}, and instead of 1\3 = 400{{cent}}, it has 27\80 = 405{{cent}} as [[19/15]]~[[24/19]], thus serving a similar function to the [[nestoria]] major third. As a result, 80edo is in some sense uniquely tasked with approximating small edos because it will often share subsets that can help make the approximation feel more regular and consistent by interpreting it as a near-equal multiperiod mos. This has the benefit of offering a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI).		As a composite edo, the main subsets it lacks are subsets of [[3edo\|3]] and [[9edo\|9]], but 9\80 = 135{{cent}} offers a good approximation to 1\9 = 133.33…{{c}}, and instead of 1\3 = 400{{cent}}, it has 27\80 = 405{{cent}} as [[19/15]]~[[24/19]], thus serving a similar function to the [[nestoria]] major third. As a result, 80edo is in some sense uniquely tasked with approximating small edos because it will often share subsets that can help make the approximation feel more regular and consistent by interpreting it as a near-equal multiperiod mos. This has the benefit of offering a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI).

	80edo is notable in not only it is consistent in the 19-odd-limit, but a large number of its supersets are also consistent in at least 19-odd-limit, if not larger. These are {{EDOs\| 320, 400, 1600, 1920, 2000, 2320, 3920, 4320 }}. ~~Temperament mergers of these produce various~~ [[80th-octave temperaments]].		80edo is notable in not only it is consistent in the 19-odd-limit, but a large number of its supersets are also consistent in at least 19-odd-limit, if not larger. These are {{EDOs\| 320, 400, 1600, 1920, 2000, 2320, 3920, 4320 }}. This produces ample [[80th-octave temperaments]] that can be used for augmenting 80edo (see the dedicated page).

	== Intervals ==		== Intervals ==

FloraC: Text replacement - "canou family" to "cathartic family"

2026-08-10T14:35:43Z

Text replacement - "canou family" to "cathartic family"

← Older revision		Revision as of 14:35, 10 August 2026
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	=== As a tuning of other temperaments ===		=== As a tuning of other temperaments ===
	80et [[tempering out\|tempers out]] [[2048/2025]] in the 5-limit; [[1728/1715]], [[3136/3125]], [[4000/3969]], and [[4375/4374]] in the [[7-limit]]; [[176/175]], [[540/539]] and [[4000/3993]] in the [[11-limit]]; [[169/168]], [[325/324]], [[351/350]], [[352/351]], [[364/363]] and [[1001/1000]] in the [[13-limit]]; [[136/135]], [[221/220]], [[256/255]], [[289/288]], [[561/560]], [[595/594]], [[715/714]], [[936/935]] and [[1275/1274]] in the [[17-limit]]; [[190/189]], [[286/285]], [[361/360]], [[400/399]], [[456/455]], [[476/475]], [[969/968]], [[1331/1330]], [[1445/1444]], [[1521/1520]], [[1540/1539]] and [[1729/1728]] in the [[19-limit]]; [[208/207]], [[253/252]], [[323/322]] and [[460/459]] in the [[23-limit]]; and 320/319 in the [[29-limit]]. The last comma is notable as it equates a sharp [[29/16]] with a near-perfect [[20/11]], although this equivalence begins to make more sense when you consider the error cancellations with other sharp harmonics and as a way to give more reasonable interpretations to otherwise questionably mapped intervals. It provides the [[optimal patent val]] for 5-limit [[diaschismic]], for 13-limit [[srutal]], and for 7-, 11- and 13-limit [[bidia]]. It is a good tuning for various temperaments in [[~~canou~~ family]], especially in higher limits.		80et [[tempering out\|tempers out]] [[2048/2025]] in the 5-limit; [[1728/1715]], [[3136/3125]], [[4000/3969]], and [[4375/4374]] in the [[7-limit]]; [[176/175]], [[540/539]] and [[4000/3993]] in the [[11-limit]]; [[169/168]], [[325/324]], [[351/350]], [[352/351]], [[364/363]] and [[1001/1000]] in the [[13-limit]]; [[136/135]], [[221/220]], [[256/255]], [[289/288]], [[561/560]], [[595/594]], [[715/714]], [[936/935]] and [[1275/1274]] in the [[17-limit]]; [[190/189]], [[286/285]], [[361/360]], [[400/399]], [[456/455]], [[476/475]], [[969/968]], [[1331/1330]], [[1445/1444]], [[1521/1520]], [[1540/1539]] and [[1729/1728]] in the [[19-limit]]; [[208/207]], [[253/252]], [[323/322]] and [[460/459]] in the [[23-limit]]; and 320/319 in the [[29-limit]]. The last comma is notable as it equates a sharp [[29/16]] with a near-perfect [[20/11]], although this equivalence begins to make more sense when you consider the error cancellations with other sharp harmonics and as a way to give more reasonable interpretations to otherwise questionably mapped intervals. It provides the [[optimal patent val]] for 5-limit [[diaschismic]], for 13-limit [[srutal]], and for 7-, 11- and 13-limit [[bidia]]. It is a good tuning for various temperaments in [[cathartic family]], especially in higher limits.

	As an equal temperament, it is well-tuned for the important 11-limit and 17-limit half-octave-period temperament [[echidna]], the {{nowrap\| 22 & 58 }} temperament, which affords great freedom in a 36-note mos and still many choices in a 22-note mos, offering a high-accuracy rank-2 detemper of [[22edo]], which in comparison conflates many important distinctions of the 11-limit. This is not insignificant as many abundant intervals of echidna, such as [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they form 80-note [[#Consistent circles\|consistent circles]]. Echidna extends [[srutal archagall]], which is also tuned near-optimally for [[fiventeen]] – specifically, for the characteristic fiventeen pentad, 30:34:40:45:51:60, consisting of steps of [[20/17]] and [[9/8]]~[[17/15]], and is the smallest edo to improve on the tuning of srutal archagall plus fiventeen after [[34edo]]. In its representation of echidna, the least accurate tuning is that of [[7/4]], which is (relatively) very sharp in 80edo, for which [[58edo]] does better as a tuning of echidna, though much worse as a tuning for srutal archagall and especially fiventeen; one can reason this makes the 80edo tuning of echidna feel more like a detemper of 22edo, especially given the smaller step size between adjacent notes equated in 22edo.		As an equal temperament, it is well-tuned for the important 11-limit and 17-limit half-octave-period temperament [[echidna]], the {{nowrap\| 22 & 58 }} temperament, which affords great freedom in a 36-note mos and still many choices in a 22-note mos, offering a high-accuracy rank-2 detemper of [[22edo]], which in comparison conflates many important distinctions of the 11-limit. This is not insignificant as many abundant intervals of echidna, such as [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they form 80-note [[#Consistent circles\|consistent circles]]. Echidna extends [[srutal archagall]], which is also tuned near-optimally for [[fiventeen]] – specifically, for the characteristic fiventeen pentad, 30:34:40:45:51:60, consisting of steps of [[20/17]] and [[9/8]]~[[17/15]], and is the smallest edo to improve on the tuning of srutal archagall plus fiventeen after [[34edo]]. In its representation of echidna, the least accurate tuning is that of [[7/4]], which is (relatively) very sharp in 80edo, for which [[58edo]] does better as a tuning of echidna, though much worse as a tuning for srutal archagall and especially fiventeen; one can reason this makes the 80edo tuning of echidna feel more like a detemper of 22edo, especially given the smaller step size between adjacent notes equated in 22edo.

Lucius Chiaraviglio: Move Instruments above Music to be consistent with most other EDO pages

2026-07-04T04:24:18Z

Move Instruments above Music to be consistent with most other EDO pages

← Older revision		Revision as of 04:24, 4 July 2026
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	* [[Equiheptatonic]] (approximate): 11 12 11 12 11 12 11		* [[Equiheptatonic]] (approximate): 11 12 11 12 11 12 11
	* [[Maeve Gutierrez\|Gutierrez Moonglade scale]]		* [[Maeve Gutierrez\|Gutierrez Moonglade scale]]

			==Instruments==
			; Lumatone
			* [[Lumatone mapping for 80edo]]
			; Fretted instruments
			* [[Skip fretting system 80 7 4]]

	== Music ==		== Music ==
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	; [[Xotla]]		; [[Xotla]]
	* "Mollusc Merchant" from ''Jazzbeetle'' (2023) [https://xotla.bandcamp.com/track/mollusc-merchant-80edo Bandcamp] \| [https://www.youtube.com/watch?v=5cb0WHAwVuM Original YouTube video] [https://www.youtube.com/watch?v=LnWJzffO7dY YouTube video without AI visuals] (2025)		* "Mollusc Merchant" from ''Jazzbeetle'' (2023) [https://xotla.bandcamp.com/track/mollusc-merchant-80edo Bandcamp] \| [https://www.youtube.com/watch?v=5cb0WHAwVuM Original YouTube video] [https://www.youtube.com/watch?v=LnWJzffO7dY YouTube video without AI visuals] (2025)

	~~==Instruments==~~
	~~; Lumatone~~
	* [[Lumatone mapping for 80edo]]
	~~; Fretted instruments~~
	* [[Skip fretting system 80 7 4]]

	[[Category:19-limit]]		[[Category:19-limit]]

Lucius Chiaraviglio: /* Music */ Add Bryan Deister's ''80edo improv'' (2026)

2026-07-04T04:22:30Z

Music: Add Bryan Deister's ''80edo improv'' (2026)

← Older revision		Revision as of 04:22, 4 July 2026
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	; [[Bryan Deister]]		; [[Bryan Deister]]
	* [https://www.youtube.com/shorts/H6DlCHKii-o ''microtonal improvisation in 80edo''] (2025)		* [https://www.youtube.com/shorts/H6DlCHKii-o ''microtonal improvisation in 80edo''] (2025)
			* [https://www.youtube.com/shorts/lWuqrdSr8pU ''80edo improv''] (2026)

	; [[Francium]]		; [[Francium]]

FloraC: /* Regular temperament properties */ leapfrog -> leapmonth

2026-04-29T18:16:08Z

Regular temperament properties: leapfrog -> leapmonth

← Older revision		Revision as of 18:16, 29 April 2026
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	\| 495		\| 495
	\| 4/3		\| 4/3
	\| [[~~Leapfrog~~]]		\| [[Leapmonth]]
	\|-		\|-
	\| 1		\| 1

Overthink: /* Intervals */ add 26/21

2026-03-26T03:25:06Z

Intervals: add 26/21

← Older revision		Revision as of 03:25, 26 March 2026
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	\| 24		\| 24
	\| 360		\| 360
	\| [[16/13]]		\| [[16/13]], ''[[26/21]]''
	\|-		\|-
	\| 25		\| 25

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:30:00Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:30, 13 March 2026
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	\| [[Degrees]]		\| [[Degrees]]
	\|}		\|}
	<nowiki/>* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms\|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

	== Detemperaments ==		== Detemperaments ==

Godtone at 20:42, 20 February 2026

2026-02-20T20:42:59Z

← Older revision		Revision as of 20:42, 20 February 2026
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	== Theory ==		== Theory ==
	80edo is the first edo that represents the [[19-odd-limit]] [[tonality diamond]] [[consistent]]ly, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits are consistent, and its [[patent val]] generally does well at approximating the [[29-limit\|29-prime-limited]] [[harmonic series]] segments, such as modes 16 through 30 but especially modes 8 through 15. It achieves this much consistency with all primes in the 29-limit except 13 being sharp of just; the inconsistencies usually arise through not cancelling the over-sharpness of compound harmonics [[21/1\|21]], [[27/1\|27]], [[35/1\|35]], [[45/1\|45]], [[49/1\|49]], and their octave-equivalents, which may be seen as an interesting limitation. This means it can be used as a general-purpose approximate 29-limit system with a relatively manageable number of tones, with some care taken around inconsistency. In fact, it is almost consistent to the no-21 no-27 [[29-odd-limit]], with the exception of [[25/13]] and its octave complement, meaning it makes a surprisingly reasonable 25-odd-limit system, with only [[26/21]], [[21/17]], [[21/16]] and their [[octave complement]]s as extra inconsistencies, which a theorist might find various justifications for. Possible additions to this include [[33/1\|33]], [[37/1\|37]], [[39/1\|39]], and [[41/1\|41]]. Thus, it can also model larger primes if one is willing to accept their sharpness, and for this purpose, it does well for its size at the no-31's [[41-limit]], or even the [[43-limit]] if you are fine with [[43/32]] being slightly flat causing more inconsistencies.		80edo is the first edo that represents the [[19-odd-limit]] [[tonality diamond]] [[consistent]]ly, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits are consistent, and its [[patent val]] generally does well at approximating the [[29-limit\|29-prime-limited]] [[harmonic series]] segments, such as modes 16 through 30 but especially modes 8 through 15. It achieves this much consistency with all primes in the 29-limit except 13 being sharp of just; the inconsistencies usually arise through not cancelling the over-sharpness of compound harmonics [[21/1\|21]], [[27/1\|27]], [[35/1\|35]], [[45/1\|45]], [[49/1\|49]], and their octave-equivalents, which may be seen as an interesting limitation. This means it can be used as a general-purpose approximate 29-limit system with a relatively manageable number of tones, with some care taken around inconsistency. In fact, it is almost consistent to the no-21 no-27 [[29-odd-limit]], with the exception of [[25/13]] and its octave complement, meaning it makes a surprisingly reasonable [[25-odd-limit]] system, with only [[26/21]], [[21/17]], [[21/16]] and their [[octave complement]]s as extra inconsistencies, which a theorist might find various justifications for. Possible additions to this include [[33/1\|33]], [[37/1\|37]], [[39/1\|39]], and [[41/1\|41]]. Thus, it can also model larger primes if one is willing to accept their sharpness, and for this purpose, it does well for its size at the no-31's [[41-limit]], or even the [[43-limit]] if you are fine with [[43/32]] being slightly flat causing more inconsistencies.

	If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80\|Ringer 80 section of this article]].		If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80\|Ringer 80 section of this article]].

Godtone: note 25-odd-limit capability

2026-02-20T20:42:14Z

note 25-odd-limit capability

← Older revision		Revision as of 20:42, 20 February 2026
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	== Theory ==		== Theory ==
	80edo is the first edo that represents the [[19-odd-limit]] [[tonality diamond]] [[consistent]]ly, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits are consistent, and its [[patent val]] generally does well at approximating the [[29-limit\|29-prime-limited]] [[harmonic series]] segments, such as modes 16 through 30 but especially modes 8 through 15. It achieves this much consistency with all primes in the 29-limit except 13 being sharp of just; the inconsistencies usually arise through not cancelling the over-sharpness of compound harmonics [[21/1\|21]], [[27/1\|27]], [[35/1\|35]], [[45/1\|45]], [[49/1\|49]], and their octave-equivalents, which may be seen as an interesting limitation. This means it can be used as a general-purpose approximate 29-limit system with a relatively manageable number of tones, with some care taken around inconsistency. In fact, it is almost consistent to the no-21 no-27 [[29-odd-limit]], with the exception of [[25/13]] and its octave complement. Possible additions to this include [[33/1\|33]], [[37/1\|37]], [[39/1\|39]], and [[41/1\|41]]. Thus, it can also model larger primes if one is willing to accept their sharpness, and for this purpose, it does well for its size at the no-31's [[41-limit]], or even the [[43-limit]] if you are fine with [[43/32]] being slightly flat causing more inconsistencies.		80edo is the first edo that represents the [[19-odd-limit]] [[tonality diamond]] [[consistent]]ly, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits are consistent, and its [[patent val]] generally does well at approximating the [[29-limit\|29-prime-limited]] [[harmonic series]] segments, such as modes 16 through 30 but especially modes 8 through 15. It achieves this much consistency with all primes in the 29-limit except 13 being sharp of just; the inconsistencies usually arise through not cancelling the over-sharpness of compound harmonics [[21/1\|21]], [[27/1\|27]], [[35/1\|35]], [[45/1\|45]], [[49/1\|49]], and their octave-equivalents, which may be seen as an interesting limitation. This means it can be used as a general-purpose approximate 29-limit system with a relatively manageable number of tones, with some care taken around inconsistency. In fact, it is almost consistent to the no-21 no-27 [[29-odd-limit]], with the exception of [[25/13]] and its octave complement, meaning it makes a surprisingly reasonable 25-odd-limit system, with only [[26/21]], [[21/17]], [[21/16]] and their [[octave complement]]s as extra inconsistencies, which a theorist might find various justifications for. Possible additions to this include [[33/1\|33]], [[37/1\|37]], [[39/1\|39]], and [[41/1\|41]]. Thus, it can also model larger primes if one is willing to accept their sharpness, and for this purpose, it does well for its size at the no-31's [[41-limit]], or even the [[43-limit]] if you are fine with [[43/32]] being slightly flat causing more inconsistencies.

	If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80\|Ringer 80 section of this article]].		If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80\|Ringer 80 section of this article]].

Overthink: /* Intervals */ fixed prime limit

2025-11-07T07:11:29Z

Intervals: fixed prime limit

← Older revision		Revision as of 07:11, 7 November 2025
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	\| …		\| …
	\|}		\|}
	<nowiki>*</nowiki> {{sg\|no-31's [[37-limit]]}} Inconsistent interpretations in ''italic''.		<nowiki>*</nowiki> {{sg\|80edo\|limit=no-31's [[37-limit]]}} Inconsistent interpretations in ''italic''.

	== Notation ==		== Notation ==