140edo: Difference between revisions

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== Theory ==

140edo is related to [[70edo]], from which it inherits the slightly sharp tuning of the [[3/1|3rd]] [[harmonic]] and the slightly flat tunings of the [[11/1|11th]], [[13/1|13th]] and [[17/1|17th]] harmonics, but the [[5/1|5th]] and [[7/1|7th]] harmonics are much improved, enabling it to approximate [[JI]] of various [[prime limit|limits]]. Its peak form is in the [[17-limit|17-]], [[19-limit|19-]] and [[23-limit]], despite the obvious lack of [[consistency]] in the corresponding [[odd limit]]s. In fact, the only inconsistently mapped intervals in the [[17-odd-limit]] are [[11/9]] and its [[octave complement]], though with the [[23-odd-limit]], [[19/11]], [[19/17]], [[23/18]], [[23/19]] and their octave complements are also added to that list.

140edo is related to [[70edo]], from which it inherits the slightly sharp tuning of the [[3/1|3rd]] [[harmonic]] and the slightly flat tunings of the [[11/1|11th]], [[13/1|13th]] and [[17/1|17th]] harmonics, but the [[5/1|5th]] and [[7/1|7th]] harmonics are much improved, enabling it to approximate [[JI]] of various [[prime limit|limits]]. Its peak form is in the [[17-limit|17-]], [[19-limit|19-]], [[23-limit|23-]], and [[29-limit]], despite the obvious lack of [[consistency]] in the corresponding [[odd limit]]s. In fact, the only inconsistently mapped intervals in the [[17-odd-limit]] are [[11/9]] and its [[octave complement]], though with the [[23-odd-limit]], [[19/11]], [[19/17]], [[23/18]], [[23/19]] and their octave complements are also added to that list.

In the 5-limit, 140et [[tempering out|tempers out]] [[15625/15552]], making it a kleismic system, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the {{nowrap| 53 & 87 }} temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].

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<nowiki>*</nowiki> As a no-31's no-47's 53-limit temperament.

== Notation ==

=== Ups and downs notation ===

140edo can be notated using [[Kite's ups and downs notation|ups and downs]] with quarter-tone accidentals:

Mapping an arrow to 3\140 rather than 1\140 is an alternative approach which takes advantage of 140edo being a tuning of akea temperament. This way, one arrow is equivalent to 81/80~64/63, and two arrows are equivalent to 33/32~1053/1024. This notation style (without quarter-tone accidentals) was used by [[User:Tristanbay|Tristan Bay]] to compose the song ''Interpolate Me'' in the music tracker [[Osctet]].

== Approximation to JI ==

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| 351.43

| 49/40

| [[Hemififths]]

| [[Hemififths]] (7-limit)

|-

| 1

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| [[Oquatonic]]

|}

<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 ==

* [[7L 7s in 140edo]]: 13 7 13 7 13 7 13 7 13 7 13 7 13 7 (''[[whitewood]][14]'')

== Music ==

[[Art Esploro]]

* ''Falling Stars'' (2026) ([https://www.patreon.com/posts/falling-stars-159221000 preview])

[[User:Tristanbay|'''Tristan Bay''']]

* [https://youtu.be/RLcZe7vlR5c ''Interpolate Me''] (2026)

== Instruments ==

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 == Theory ==
-edo is related to [[70edo]], from which it inherits the slightly sharp tuning of the [[3/1|3rd]] [[harmonic]] and the slightly flat tunings of the [[11/1|11th]], [[13/1|13th]] and [[17/1|17th]] harmonics, but the [[5/1|5th]] and [[7/1|7th]] harmonics are much improved, enabling it to approximate [[JI]] of various [[prime limit|limits]]. Its peak form is in the [[17-limit|17-]], [[19-limit|19-]] and [[23-limit]], despite the obvious lack of [[consistency]] in the corresponding [[odd limit]]s. In fact, the only inconsistently mapped intervals in the [[17-odd-limit]] are [[11/9]] and its [[octave complement]], though with the [[23-odd-limit]], [[19/11]], [[19/17]], [[23/18]], [[23/19]] and their octave complements are also added to that list.
+edo is related to [[70edo]], from which it inherits the slightly sharp tuning of the [[3/1|3rd]] [[harmonic]] and the slightly flat tunings of the [[11/1|11th]], [[13/1|13th]] and [[17/1|17th]] harmonics, but the [[5/1|5th]] and [[7/1|7th]] harmonics are much improved, enabling it to approximate [[JI]] of various [[prime limit|limits]]. Its peak form is in the [[17-limit|17-]], [[19-limit|19-]], [[23-limit|23-]], and [[29-limit]], despite the obvious lack of [[consistency]] in the corresponding [[odd limit]]s. In fact, the only inconsistently mapped intervals in the [[17-odd-limit]] are [[11/9]] and its [[octave complement]], though with the [[23-odd-limit]], [[19/11]], [[19/17]], [[23/18]], [[23/19]] and their octave complements are also added to that list.
 In the 5-limit, 140et [[tempering out|tempers out]] [[15625/15552]], making it a kleismic system, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the {{nowrap| 53 & 87 }} temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].
@@ Line 311: / Line 311: @@
 |}
 <nowiki>*</nowiki> As a no-31's no-47's 53-limit temperament.
+== Notation ==
+=== Ups and downs notation ===
+edo can be notated using [[Kite's ups and downs notation|ups and downs]] with quarter-tone accidentals:
+{{Ups and downs sharpness|140|true}}
+Mapping an arrow to 3\140 rather than 1\140 is an alternative approach which takes advantage of 140edo being a tuning of akea temperament. This way, one arrow is equivalent to 81/80~64/63, and two arrows are equivalent to 33/32~1053/1024. This notation style (without quarter-tone accidentals) was used by [[User:Tristanbay|Tristan Bay]] to compose the song ''Interpolate Me'' in the music tracker [[Osctet]].
 == Approximation to JI ==
@@ Line 420: / Line 426: @@
 | 351.43
 | 49/40
-| [[Hemififths]]
+| [[Hemififths]] (7-limit)
 |-
 | 1
@@ Line 494: / Line 500: @@
 | [[Oquatonic]]
 |}
-<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 ==
+* [[7L 7s in 140edo]]: 13 7 13 7 13 7 13 7 13 7 13 7 13 7 (''[[whitewood]][14]'')
+== Music ==
+[[Art Esploro]]
+* ''Falling Stars'' (2026) ([https://www.patreon.com/posts/falling-stars-159221000 preview])
+[[User:Tristanbay|'''Tristan Bay''']]
+* [https://youtu.be/RLcZe7vlR5c ''Interpolate Me''] (2026)
 == Instruments ==