89edo - Revision history

FloraC: Link & style

2026-04-18T08:35:29Z

Link & style

← Older revision		Revision as of 08:35, 18 April 2026
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	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
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	; [[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 ==

Hkm at 06:31, 12 January 2026

2026-01-12T06:31:35Z

← Older revision		Revision as of 06:31, 12 January 2026
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	== Theory ==		== Theory ==
	89edo has a [[3/1\|harmonic 3]] less than a cent flat and a [[5/1\|harmonic 5]] less than five cents sharp, with a [[7/1\|7]] two cents sharp and an [[11/1\|11]] 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas~~. On a related note, a notable characteristic of this edo is that it is the lowest in a series of four consecutive edos to temper out [[quartisma]]~~.		89edo has a [[3/1\|harmonic 3]] less than a cent flat and a [[5/1\|harmonic 5]] less than five cents sharp, with a [[7/1\|7]] two cents sharp and an [[11/1\|11]] 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas.

	It [[tempering out\|tempers out]] 32805/32768 ([[schisma]]) in the 5-limit; [[126/125]], [[1728/1715]], and [[2401/2400]] in the 7-limit; and [[176/175]], [[243/242]], [[441/440]] and [[540/539]] in the 11-limit. It is an especially good tuning for the [[myna]] temperament, both in the [[7-limit]], tempering out 126/125 and 1728/1715, and in the [[11-limit]], where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament [[thrush]], tempering out 126/125 and 176/175.		It [[tempering out\|tempers out]] 32805/32768 ([[schisma]]) in the 5-limit; [[126/125]], [[1728/1715]], and [[2401/2400]] in the 7-limit; and [[176/175]], [[243/242]], [[441/440]] and [[540/539]] in the 11-limit. It is an especially good tuning for the [[myna]] temperament, both in the [[7-limit]], tempering out 126/125 and 1728/1715, and in the [[11-limit]], where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament [[thrush]], tempering out 126/125 and 176/175.

ArrowHead294: /* Ups and downs notation */ Functionality added to template

2025-09-25T16:18:07Z

Ups and downs notation: Functionality added to template

← Older revision		Revision as of 16:18, 25 September 2025
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	=== Ups and downs notation ===		=== Ups and downs notation ===
	89edo can be notated with [[Kite's ups and downs notation]]:		89edo can be notated with [[Kite's ups and downs notation]]:
	{{Ups and downs sharpness~~\|89~~}}		{{Ups and downs sharpness}}

	Alternatively, ups and downs with [[Helmholtz–Ellis]] accidentals can be used:		Alternatively, ups and downs with [[Helmholtz–Ellis]] accidentals can be used:

Tristanbay: /* Ups and downs notation */ Added template for non-HEJI

2025-09-24T19:13:38Z

Ups and downs notation: Added template for non-HEJI

← Older revision		Revision as of 19:13, 24 September 2025
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	== Notation ==		== Notation ==
	=== Ups and downs notation ===		=== Ups and downs notation ===
	89edo can be notated ~~using~~ [[ups and downs notation]] ~~using [[Helmholtz–Ellis]] accidentals~~:		89edo can be notated with [[Kite's ups and downs notation]]:
			{{Ups and downs sharpness\|89}}

			Alternatively, ups and downs with [[Helmholtz–Ellis]] accidentals can be used:
	{{Sharpness-sharp8}}		{{Sharpness-sharp8}}

BudjarnLambeth: Remove the zeta peak table that had no explanation or context

2025-09-17T01:59:41Z

Remove the zeta peak table that had no explanation or context

← Older revision		Revision as of 01:59, 17 September 2025
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	{{Sharpness-sharp8}}		{{Sharpness-sharp8}}

	~~== Approximation to JI ==~~
	~~=== Zeta peak index ===~~
	~~{{ZPI~~
	~~\| zpi = 497~~
	~~\| steps = 89.0229355804124~~
	~~\| step size = 13.4796723133902~~
	~~\| tempered height = 7.567368~~
	~~\| pure height = 7.158697~~
	~~\| integral = 1.124501~~
	~~\| gap = 16.042570~~
	~~\| octave = 1199.69083589172~~
	~~\| consistent = 12~~
	~~\| distinct = 12~~
	}}

	== Regular temperament properties ==		== Regular temperament properties ==

← 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.