89edo - Revision history

Lucius Chiaraviglio: /* Music */ Bryan Deister's ''89edo improv'' (2026-10-28) (demonstrates 89edo as a tuning actually not of myna, but of a sensipent-related tuning favoring minor thirds)

2026-11-02T04:44:23Z

Music: Bryan Deister's ''89edo improv'' (2026-10-28) (demonstrates 89edo as a tuning actually not of myna, but of a sensipent-related tuning favoring minor thirds)

← Older revision		Revision as of 04:44, 2 November 2026
Line 126:		Line 126:
	* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025) (demonstrates a highly xenharmonic take on 89edo)		* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025) (demonstrates a highly xenharmonic take on 89edo)
	* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026-04-15) (demonstrates 89edo as quarter-comma boethian temperament)		* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026-04-15) (demonstrates 89edo as quarter-comma boethian temperament)
	* [https://www.youtube.com/watch?v=Rjx1zr3HIE0 ''89edo improv''] (2026-10-28) (demonstrates 89edo as a tuning of ~~myna~~)		* [https://www.youtube.com/watch?v=Rjx1zr3HIE0 ''89edo improv''] (2026-10-28) (demonstrates 89edo as a tuning of a sensipent-related tuning favoring minor thirds)

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

Lucius Chiaraviglio: /* Music */ Bryan Deister's ''89edo improv'' (2026-10-28): demonstrates 89edo as a tuning of Myna

2026-10-31T22:43:59Z

Music: Bryan Deister's ''89edo improv'' (2026-10-28): demonstrates 89edo as a tuning of Myna

← Older revision		Revision as of 22:43, 31 October 2026
Line 126:		Line 126:
	* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025) (demonstrates a highly xenharmonic take on 89edo)		* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025) (demonstrates a highly xenharmonic take on 89edo)
	* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026-04-15) (demonstrates 89edo as quarter-comma boethian temperament)		* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026-04-15) (demonstrates 89edo as quarter-comma boethian temperament)
	* [https://www.youtube.com/watch?v=Rjx1zr3HIE0 ''89edo improv''] (2026-10-28)		* [https://www.youtube.com/watch?v=Rjx1zr3HIE0 ''89edo improv''] (2026-10-28) (demonstrates 89edo as a tuning of myna)

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

Lucius Chiaraviglio: /* Music */ Add Bryan Deister's ''89edo improv'' (2026-10-28); add a more specific date to the previous one

2026-10-29T07:07:01Z

Music: Add Bryan Deister's ''89edo improv'' (2026-10-28); add a more specific date to the previous one

← Older revision		Revision as of 07:07, 29 October 2026
Line 125:		Line 125:
	; [[Bryan Deister]]		; [[Bryan Deister]]
	* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025) (demonstrates a highly xenharmonic take on 89edo)		* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025) (demonstrates a highly xenharmonic take on 89edo)
	* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026) (demonstrates 89edo as quarter-comma boethian temperament)		* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026-04-15) (demonstrates 89edo as quarter-comma boethian temperament)
			* [https://www.youtube.com/watch?v=Rjx1zr3HIE0 ''89edo improv''] (2026-10-28)

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

Lucius Chiaraviglio: /* Theory */ Fix mismatched '('.

2026-10-14T07:53:15Z

Theory: Fix mismatched '('.

← Older revision		Revision as of 07:53, 14 October 2026
Line 11:		Line 11:
	Harmonic [[17/1\|17]] is tuned fairly well, and harmonic [[19/1\|19]] is tuned very well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]] (Boethius' comma), and [[1216/1215]] in the 19-limit. Not only this, but the small tuning error in harmonic 19 is almost the same as the small error in harmonic 3, the error being extremely close to one quarter of Boethius' comma flat (and since the comma is small to begin with, that is also not very far from one third of this comma flat), enabling the use of the extremely accurate Boethius' major third ~[[24/19]] as a substitute for the less accurate classic major third ~[[5/4]], and the still very accurate Boethius' minor third ~[[19/16]] to substitute for the less accurate classic minor third ~[[6/5]], thereby obtaining [[meantone]] ease of use with a near-just (slightly flat) harmonic 3, as demonstrated in some of the music below (see [[#Music]]). Thus, 89edo is audibly indistinguishable from quarter-comma [[boethian]] temperament.		Harmonic [[17/1\|17]] is tuned fairly well, and harmonic [[19/1\|19]] is tuned very well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]] (Boethius' comma), and [[1216/1215]] in the 19-limit. Not only this, but the small tuning error in harmonic 19 is almost the same as the small error in harmonic 3, the error being extremely close to one quarter of Boethius' comma flat (and since the comma is small to begin with, that is also not very far from one third of this comma flat), enabling the use of the extremely accurate Boethius' major third ~[[24/19]] as a substitute for the less accurate classic major third ~[[5/4]], and the still very accurate Boethius' minor third ~[[19/16]] to substitute for the less accurate classic minor third ~[[6/5]], thereby obtaining [[meantone]] ease of use with a near-just (slightly flat) harmonic 3, as demonstrated in some of the music below (see [[#Music]]). Thus, 89edo is audibly indistinguishable from quarter-comma [[boethian]] temperament.

	89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.		89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ, i.e. 1200{{nowrap\|(φ − 1)}}{{c}}, within a fraction of a cent.

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

FloraC: Link & style

2026-04-18T08:35:29Z

Link & style

← Older revision		Revision as of 08:35, 18 April 2026
Line 9:		Line 9:
	The [[13-limit]] is a little tricky as [[13/1\|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes.		The [[13-limit]] is a little tricky as [[13/1\|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes.

	Harmonic [[17/1\|17]] is tuned fairly well, and harmonic [[19/1\|19]] is tuned very well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]] (Boethius' comma), and [[1216/1215]] in the 19-limit. Not only this, but the small tuning error in harmonic 19 is almost the same as the small error in harmonic 3, the error being extremely close to one quarter of Boethius' comma flat (and since the comma is small to begin with, that is also not very far from one third of this comma flat), enabling the use of the extremely accurate Boethius' major third ~[[24/19]] as a substitute for the less accurate classic major third ~[[5/4]], and the still very accurate Boethius' minor third ~[[19/16]] to substitute for the less accurate classic minor third ~[[6/5]], thereby obtaining [[meantone]] ease of use with a near-just (slightly flat) harmonic 3, as demonstrated in some of the music below (see [[#Music]]). Thus, 89edo is audibly indistinguishable from quarter-comma ~~Boethian~~ temperament.		Harmonic [[17/1\|17]] is tuned fairly well, and harmonic [[19/1\|19]] is tuned very well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]] (Boethius' comma), and [[1216/1215]] in the 19-limit. Not only this, but the small tuning error in harmonic 19 is almost the same as the small error in harmonic 3, the error being extremely close to one quarter of Boethius' comma flat (and since the comma is small to begin with, that is also not very far from one third of this comma flat), enabling the use of the extremely accurate Boethius' major third ~[[24/19]] as a substitute for the less accurate classic major third ~[[5/4]], and the still very accurate Boethius' minor third ~[[19/16]] to substitute for the less accurate classic minor third ~[[6/5]], thereby obtaining [[meantone]] ease of use with a near-just (slightly flat) harmonic 3, as demonstrated in some of the music below (see [[#Music]]). Thus, 89edo is audibly indistinguishable from quarter-comma [[boethian]] temperament.

	89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.		89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.
Line 125:		Line 125:
	; [[Bryan Deister]]		; [[Bryan Deister]]
	* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025) (demonstrates a highly xenharmonic take on 89edo)		* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025) (demonstrates a highly xenharmonic take on 89edo)
	* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026) (demonstrates 89edo as quarter-comma ~~Boethian~~ temperament)		* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026) (demonstrates 89edo as quarter-comma boethian temperament)

	; [[Francium]]		; [[Francium]]
	* [https://www.youtube.com/watch?v=5Du9RfDUqCs ''Singing Golden Myna''] (2022) – ~~myna~~[11] in 89edo		* [https://www.youtube.com/watch?v=5Du9RfDUqCs ''Singing Golden Myna''] (2022) – Myna[11] in 89edo

	[[Category:Listen]]		[[Category:Listen]]
	[[Category:Myna]]		[[Category:Myna]]
	[[Category:Thrush]]		[[Category:Thrush]]

Lucius Chiaraviglio: /* Music */ Add a bit of description for 2 Bryan Deister videos that sound extremely different with respect to xenharmonic content

2026-04-17T00:16:59Z

Music: Add a bit of description for 2 Bryan Deister videos that sound extremely different with respect to xenharmonic content

← Older revision		Revision as of 00:16, 17 April 2026
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	== Music ==		== Music ==
	; [[Bryan Deister]]		; [[Bryan Deister]]
	* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025)		* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025) (demonstrates a highly xenharmonic take on 89edo)
	* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026)		* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026) (demonstrates 89edo as quarter-comma Boethian temperament)

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

Lucius Chiaraviglio: /* Theory */ Fix typo

2026-04-17T00:14:31Z

Theory: Fix typo

← Older revision		Revision as of 00:14, 17 April 2026
Line 9:		Line 9:
	The [[13-limit]] is a little tricky as [[13/1\|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes.		The [[13-limit]] is a little tricky as [[13/1\|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes.

	Harmonic [[17/1\|17]] is tuned fairly well, and harmonic [[19/1\|19]] is tuned very well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]] (Boethius' comma), and [[1216/1215]] in the 19-limit. Not only this, but the small tuning error in harmonic 19 is almost the same as the small error in harmonic 3, the error being extremely close to one quarter of Boethius' comma flat (and since the comma is small to begin with, that is also not very far from one third of this comma flat), enabling the use of the extremely accurate Boethius' major third ~[[24/19]] as a substitute for the less accurate classic major third ~[[5/4]], and the still very accurate Boethius' minor third ~[[~~198~~/16]] to substitute for the less accurate classic minor third ~[[6/5]], thereby obtaining [[meantone]] ease of use with a near-just (slightly flat) harmonic 3, as demonstrated in some of the music below (see [[#Music]]). Thus, 89edo is audibly indistinguishable from quarter-comma Boethian temperament.		Harmonic [[17/1\|17]] is tuned fairly well, and harmonic [[19/1\|19]] is tuned very well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]] (Boethius' comma), and [[1216/1215]] in the 19-limit. Not only this, but the small tuning error in harmonic 19 is almost the same as the small error in harmonic 3, the error being extremely close to one quarter of Boethius' comma flat (and since the comma is small to begin with, that is also not very far from one third of this comma flat), enabling the use of the extremely accurate Boethius' major third ~[[24/19]] as a substitute for the less accurate classic major third ~[[5/4]], and the still very accurate Boethius' minor third ~[[19/16]] to substitute for the less accurate classic minor third ~[[6/5]], thereby obtaining [[meantone]] ease of use with a near-just (slightly flat) harmonic 3, as demonstrated in some of the music below (see [[#Music]]). Thus, 89edo is audibly indistinguishable from quarter-comma Boethian temperament.

	89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.		89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.

Lucius Chiaraviglio: /* Theory */ 89edo is extremely close to quarter-comma Boethian meantone; added a video to demonstrate it

2026-04-17T00:13:52Z

Theory: 89edo is extremely close to quarter-comma Boethian meantone; added a video to demonstrate it

← Older revision		Revision as of 00:13, 17 April 2026
Line 9:		Line 9:
	The [[13-limit]] is a little tricky as [[13/1\|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes.		The [[13-limit]] is a little tricky as [[13/1\|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes.

	~~The~~ [[17/1\|17]] and [[19/1\|19]] ~~are~~ tuned ~~fairly~~ well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]], and [[1216/1215]] in the 19-limit.		Harmonic [[17/1\|17]] is tuned fairly well, and harmonic [[19/1\|19]] is tuned very well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]] (Boethius' comma), and [[1216/1215]] in the 19-limit. Not only this, but the small tuning error in harmonic 19 is almost the same as the small error in harmonic 3, the error being extremely close to one quarter of Boethius' comma flat (and since the comma is small to begin with, that is also not very far from one third of this comma flat), enabling the use of the extremely accurate Boethius' major third ~[[24/19]] as a substitute for the less accurate classic major third ~[[5/4]], and the still very accurate Boethius' minor third ~[[198/16]] to substitute for the less accurate classic minor third ~[[6/5]], thereby obtaining [[meantone]] ease of use with a near-just (slightly flat) harmonic 3, as demonstrated in some of the music below (see [[#Music]]). Thus, 89edo is audibly indistinguishable from quarter-comma Boethian temperament.

	89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.		89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.

Lucius Chiaraviglio: /* Music */ Add Bryan Deister's ''89edo improv'' (2026)

2026-04-16T23:01:36Z

Music: Add Bryan Deister's ''89edo improv'' (2026)

← Older revision		Revision as of 23:01, 16 April 2026
Line 125:		Line 125:
	; [[Bryan Deister]]		; [[Bryan Deister]]
	* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025)		* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025)
			* [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026)

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

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"

2026-03-13T13:31:39Z

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 13:31, 13 March 2026
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	\| [[Grackle]]		\| [[Grackle]]
	\|}		\|}
	<nowiki/>* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|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

	== Scales ==		== Scales ==

← Older revision		Revision as of 00:14, 17 April 2026
Line 9:		Line 9:
	The [[13-limit]] is a little tricky as [[13/1\|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes.		The [[13-limit]] is a little tricky as [[13/1\|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes.

	Harmonic [[17/1\|17]] is tuned fairly well, and harmonic [[19/1\|19]] is tuned very well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]] (Boethius' comma), and [[1216/1215]] in the 19-limit. Not only this, but the small tuning error in harmonic 19 is almost the same as the small error in harmonic 3, the error being extremely close to one quarter of Boethius' comma flat (and since the comma is small to begin with, that is also not very far from one third of this comma flat), enabling the use of the extremely accurate Boethius' major third ~[[24/19]] as a substitute for the less accurate classic major third ~[[5/4]], and the still very accurate Boethius' minor third ~[[~~198~~/16]] to substitute for the less accurate classic minor third ~[[6/5]], thereby obtaining [[meantone]] ease of use with a near-just (slightly flat) harmonic 3, as demonstrated in some of the music below (see [[#Music]]). Thus, 89edo is audibly indistinguishable from quarter-comma Boethian temperament.		Harmonic [[17/1\|17]] is tuned fairly well, and harmonic [[19/1\|19]] is tuned very well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]] (Boethius' comma), and [[1216/1215]] in the 19-limit. Not only this, but the small tuning error in harmonic 19 is almost the same as the small error in harmonic 3, the error being extremely close to one quarter of Boethius' comma flat (and since the comma is small to begin with, that is also not very far from one third of this comma flat), enabling the use of the extremely accurate Boethius' major third ~[[24/19]] as a substitute for the less accurate classic major third ~[[5/4]], and the still very accurate Boethius' minor third ~[[19/16]] to substitute for the less accurate classic minor third ~[[6/5]], thereby obtaining [[meantone]] ease of use with a near-just (slightly flat) harmonic 3, as demonstrated in some of the music below (see [[#Music]]). Thus, 89edo is audibly indistinguishable from quarter-comma Boethian temperament.

	89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.		89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.

← Older revision		Revision as of 00:13, 17 April 2026
Line 9:		Line 9:
	The [[13-limit]] is a little tricky as [[13/1\|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes.		The [[13-limit]] is a little tricky as [[13/1\|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes.

	~~The~~ [[17/1\|17]] and [[19/1\|19]] ~~are~~ tuned ~~fairly~~ well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]], and [[1216/1215]] in the 19-limit.		Harmonic [[17/1\|17]] is tuned fairly well, and harmonic [[19/1\|19]] is tuned very well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]] (Boethius' comma), and [[1216/1215]] in the 19-limit. Not only this, but the small tuning error in harmonic 19 is almost the same as the small error in harmonic 3, the error being extremely close to one quarter of Boethius' comma flat (and since the comma is small to begin with, that is also not very far from one third of this comma flat), enabling the use of the extremely accurate Boethius' major third ~[[24/19]] as a substitute for the less accurate classic major third ~[[5/4]], and the still very accurate Boethius' minor third ~[[198/16]] to substitute for the less accurate classic minor third ~[[6/5]], thereby obtaining [[meantone]] ease of use with a near-just (slightly flat) harmonic 3, as demonstrated in some of the music below (see [[#Music]]). Thus, 89edo is audibly indistinguishable from quarter-comma Boethian temperament.

	89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.		89edo is the 11th in the {{w\|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap\|(φ − 1)}}{{c}} within a fraction of a cent.