89edo: Difference between revisions

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== Theory ==

89edo has a harmonic 3 less than a cent flat and a harmonic 5 less than five cents sharp, with a 7 two cents sharp and an 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.

~~89et~~ [[tempering out|tempers out]] the ~~commas~~ [[126/125]], [[1728/1715]], ~~[[32805/32768]],~~ [[2401/2400]], [[176/175]], [[243/242]], [[441/440]] and [[540/539]]. 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.

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.

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 ===

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=== Subsets and supersets ===

89edo is the 24th [[prime edo]], and ~~the 11th in the Fibonacci sequence~~, ~~which means its 55th step approximates logarithmic φ (i.e. (φ~~ - ~~1)×1200 cents) within a fraction of a cent.~~

89edo is the 24th [[prime edo]], following [[83edo]] and before [[97edo]].

== Intervals ==

== Notation ==

=== Ups and downs notation ===

89edo can be notated with [[Kite's ups and downs notation]]:

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

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 26:

Line 44:

| 2.3

| {{monzo| -141 89 }}

| [{{~~val~~| 89 141 }}]

| {{mapping| 89 141 }}

| +0.262

| 0.262

Line 33:

Line 51:

| 2.3.5

| 32805/32768, 10077696/9765625

| [{{~~val~~| 89 141 207 }}]

| {{mapping| 89 141 207 }}

| -0.500

| −0.500

| 1.098

| 8.15

Line 40:

Line 58:

| 2.3.5.7

| 126/125, 1728/1715, 32805/32768

| [{{~~val~~| 89 141 207 250 }}]

| {{mapping| 89 141 207 250 }}

| -0.550

| −0.550

| 0.955

| 7.08

Line 47:

Line 65:

| 2.3.5.7.11

| 126/125, 176/175, 243/242, 16384/16335

| [{{~~val~~| 89 141 207 250 308 }}]

| {{mapping| 89 141 207 250 308 }}

| -0.526

| −0.526

| 0.855

| 6.35

|}

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods<br />per 8ve

! Generator*

! Cents*

! Associated<br />ratio*

! Temperaments

|-

| 1

| 13\89

| 175.28

| 72/65

| [[Sesquiquartififths]] / [[sesquart]]

|-

| 1

| 21\89

| 283.15

| 13/11

| [[Neominor]]

|-

| 1

| 23\89

| 310.11

| 6/5

| [[Myna]]

|-

| 1

| 29\89

| 391.01

| 5/4

| [[Amigo]]

|-

| 1

| 37\89

| 498.87

| 4/3

| [[Grackle]]

|}

<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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* [[Myna15]]

== Instruments ==

; Lumatone

''See [[Lumatone mapping for 89edo]].''

== 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

[[Category:Listen]]

[[Category:Myna]]

[[Category:Thrush]]

~~[[Category:Quartismic]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|89}}
+{{ED intro}}
 == Theory ==
-edo has a harmonic 3 less than a cent flat and a harmonic 5 less than five cents sharp, with a 7 two cents sharp and an 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.
-et [[tempering out|tempers out]] the commas [[126/125]], [[1728/1715]], [[32805/32768]], [[2401/2400]], [[176/175]], [[243/242]], [[441/440]] and [[540/539]]. 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.
+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.
+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.
 === Prime harmonics ===
@@ Line 11: / Line 17: @@
 === Subsets and supersets ===
-edo is the 24th [[prime edo]], and the 11th in the Fibonacci sequence, which means its 55th step approximates logarithmic φ (i.e. (φ - 1)×1200 cents) within a fraction of a cent.
+edo is the 24th [[prime edo]], following [[83edo]] and before [[97edo]].
+== Intervals ==
+{{Interval table}}
+== Notation ==
+=== Ups and downs notation ===
+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}}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 26: / Line 44: @@
 | 2.3
 | {{monzo| -141 89 }}
-| [{{val| 89 141 }}]
+| {{mapping| 89 141 }}
 | +0.262
 | 0.262
@@ Line 33: / Line 51: @@
 | 2.3.5
 | 32805/32768, 10077696/9765625
-| [{{val| 89 141 207 }}]
+| {{mapping| 89 141 207 }}
-| -0.500
+| −0.500
 | 1.098
 | 8.15
@@ Line 40: / Line 58: @@
 | 2.3.5.7
 | 126/125, 1728/1715, 32805/32768
-| [{{val| 89 141 207 250 }}]
+| {{mapping| 89 141 207 250 }}
-| -0.550
+| −0.550
 | 0.955
 | 7.08
@@ Line 47: / Line 65: @@
 | 2.3.5.7.11
 | 126/125, 176/175, 243/242, 16384/16335
-| [{{val| 89 141 207 250 308 }}]
+| {{mapping| 89 141 207 250 308 }}
-| -0.526
+| −0.526
 | 0.855
 | 6.35
 |}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 13\89
+| 175.28
+| 72/65
+| [[Sesquiquartififths]] / [[sesquart]]
+|-
+| 1
+| 21\89
+| 283.15
+| 13/11
+| [[Neominor]]
+|-
+| 1
+| 23\89
+| 310.11
+| 6/5
+| [[Myna]]
+|-
+| 1
+| 29\89
+| 391.01
+| 5/4
+| [[Amigo]]
+|-
+| 1
+| 37\89
+| 498.87
+| 4/3
+| [[Grackle]]
+|}
+<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 58: / Line 118: @@
 * [[Myna15]]
+== Instruments ==
+; Lumatone
+''See [[Lumatone mapping for 89edo]].''
+== 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
+[[Category:Listen]]
 [[Category:Myna]]
 [[Category:Thrush]]
-[[Category:Quartismic]]