89edo: Difference between revisions

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== 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.

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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~~ [[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 ~[[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.

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== 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]]:

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

~~== 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 ==

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| [[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 ==

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== Music ==

; [[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/shorts/mU4KcJd6E7U ''89edo improv''] (2026) (demonstrates 89edo as quarter-comma boethian temperament)

; [[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:Myna]]

[[Category:Thrush]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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]].
+edo 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.
@@ 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 [[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 ~[[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.
 edo 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 24: / Line 24: @@
 == Notation ==
 === Ups and downs notation ===
-edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:
+edo can be notated with [[Kite's ups and downs notation]]:
+{{Ups and downs sharpness}}
+Alternatively, ups and downs with [[Helmholtz–Ellis]] accidentals can be used:
 {{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 ==
@@ Line 124: / Line 111: @@
 | [[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 ==
@@ Line 136: / Line 123: @@
 == Music ==
+; [[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/shorts/mU4KcJd6E7U ''89edo improv''] (2026) (demonstrates 89edo as quarter-comma boethian temperament)
 ; [[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:Myna]]
 [[Category:Thrush]]