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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== Interval table ==

Due to its inconsistencies in higher limits (as discussed in [[#Higher-limit JI]]), this table focuses primarily on the 29-limit add-41 add-53, with some more accurate intervals of 37 and 43 included. It was generated partially algorithmically with [[User:Godtone#My Python 3 code|Godtone's code]]. Many additions to this interpretation from higher limits are possible; specifically, by observing omitted odds early on in the table, it's possible to guess where and why the inconsistencies arise.

Due to its inconsistencies in higher limits (as discussed in [[#Higher-limit JI]]), this table focuses primarily on the 29-limit add-41 add-53 and especially the 17-limit, with some more accurate intervals of 37 and 43 included. It was generated partially algorithmically with [[User:Godtone#My Python 3 code|Godtone's code]]. Many additions to this interpretation from higher limits are possible; specifically, by observing omitted odds early on in the table, it's possible to guess where and why the inconsistencies arise.

{| class="wikitable center-all right-2 left-3"

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| 1

| 8.57

| [[256/255|S15]], [[225/224|S14]] [[196/195|S14]], [[169/168|S13]], ''[[121/120|S11]]''

| [[256/255|S15]], [[225/224|S14]], [[196/195|S14]], [[169/168|S13]], ''[[121/120|S11]]''

|-

| 2

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| 9

| 77.14

| [[24/23]], [[68/65]], [[23/22]], [[22/21]]

| [[24/23]], [[23/22]], [[68/65]], [[22/21]]

|-

| 10

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| 11

| 94.29

| 96/91, 19/18, [[55/52]]

| 96/91, 19/18, 56/53, 37/35, [[55/52]]

|-

| 12

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| 28

| 240.0

| ''[[63/55]]'', [[39/34]], [[147/128]], [[23/20]]

| ''[[63/55]]'', [[39/34]], [[147/128]], 85/74, [[23/20]]

|-

| 29

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| 33

| 282.86

| ''27/23'', [[20/17]], 53/45, [[33/28]], 46/39

| ''[[27/23]]'', [[20/17]], 53/45, [[33/28]], 46/39

|-

| 34

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| 64

| 548.57

| [[48/35]], [[70/51]], [[136/99]], [[11/8]]

| [[48/35]], [[70/51]], 136/99, [[11/8]]

|-

| 65

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| 66

| 565.71

| [[18/13]], [[104/75]], 68/49, [[25/18]]

| [[18/13]], 104/75, 68/49, [[25/18]]

|-

| 67

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| 70

| 600.0

| [[24/17]], 41/29, [[140/99]], [[99/70]], 58/41 [[17/12]]

| [[24/17]], 41/29, [[140/99]], [[99/70]], 58/41, [[17/12]]

|}

<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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=== Zeta peak index ===

{~~| class="wikitable center-all"~~

{{ZPI

|-

| zpi = 872

~~! colspan="3"~~ | ~~Tuning~~

| steps = 139.990541024216

~~! colspan~~=~~"3" | Strength~~

| step size = 8.57200773152536

~~! colspan="2"~~ | ~~Closest edo~~

| tempered height = 10.076688

~~! colspan~~=~~"2" | Integer limit~~

| pure height = 9.983474

|-

| integral = 1.548424

~~! ZPI~~

| gap = 19.514765

~~! Steps per octave~~

| octave = 1200.08108241355

~~! Step size (cents)~~

| consistent = 10

~~! Height~~

| distinct = 10

~~! Integral~~

}}

~~! Gap~~

~~! Edo~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[872zpi]]~~

| 139.990541024216

| 8.57200773152536

| 10.076688

| 1.548424

| 19.514765

| ~~140edo~~

| 1200.08108241355

| 10

|}

== Regular temperament properties ==

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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 ==

@@ Line 3: / Line 3: @@
 == 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 22: / Line 22: @@
 == Interval table ==
-Due to its inconsistencies in higher limits (as discussed in [[#Higher-limit JI]]), this table focuses primarily on the 29-limit add-41 add-53, with some more accurate intervals of 37 and 43 included. It was generated partially algorithmically with [[User:Godtone#My Python 3 code|Godtone's code]]. Many additions to this interpretation from higher limits are possible; specifically, by observing omitted odds early on in the table, it's possible to guess where and why the inconsistencies arise.
+Due to its inconsistencies in higher limits (as discussed in [[#Higher-limit JI]]), this table focuses primarily on the 29-limit add-41 add-53 and especially the 17-limit, with some more accurate intervals of 37 and 43 included. It was generated partially algorithmically with [[User:Godtone#My Python 3 code|Godtone's code]]. Many additions to this interpretation from higher limits are possible; specifically, by observing omitted odds early on in the table, it's possible to guess where and why the inconsistencies arise.
 {| class="wikitable center-all right-2 left-3"
@@ Line 32: / Line 32: @@
 | 1
 | 8.57
-| [[256/255|S15]], [[225/224|S14]] [[196/195|S14]], [[169/168|S13]], ''[[121/120|S11]]''
+| [[256/255|S15]], [[225/224|S14]], [[196/195|S14]], [[169/168|S13]], ''[[121/120|S11]]''
 |-
 | 2
@@ Line 64: / Line 64: @@
 | 9
 | 77.14
-| [[24/23]], [[68/65]], [[23/22]], [[22/21]]
+| [[24/23]], [[23/22]], [[68/65]], [[22/21]]
 |-
 | 10
@@ Line 72: / Line 72: @@
 | 11
 | 94.29
-| 96/91, 19/18, [[55/52]]
+| 96/91, 19/18, 56/53, 37/35, [[55/52]]
 |-
 | 12
@@ Line 140: / Line 140: @@
 | 28
 | 240.0
-| ''[[63/55]]'', [[39/34]], [[147/128]], [[23/20]]
+| ''[[63/55]]'', [[39/34]], [[147/128]], 85/74, [[23/20]]
 |-
 | 29
@@ Line 160: / Line 160: @@
 | 33
 | 282.86
-| ''27/23'', [[20/17]], 53/45, [[33/28]], 46/39
+| ''[[27/23]]'', [[20/17]], 53/45, [[33/28]], 46/39
 |-
 | 34
@@ Line 284: / Line 284: @@
 | 64
 | 548.57
-| [[48/35]], [[70/51]], [[136/99]], [[11/8]]
+| [[48/35]], [[70/51]], 136/99, [[11/8]]
 |-
 | 65
@@ Line 292: / Line 292: @@
 | 66
 | 565.71
-| [[18/13]], [[104/75]], 68/49, [[25/18]]
+| [[18/13]], 104/75, 68/49, [[25/18]]
 |-
 | 67
@@ Line 308: / Line 308: @@
 | 70
 | 600.0
-| [[24/17]], 41/29, [[140/99]], [[99/70]], 58/41 [[17/12]]
+| [[24/17]], 41/29, [[140/99]], [[99/70]], 58/41, [[17/12]]
 |}
 <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 326: / Line 332: @@
 === Zeta peak index ===
-{| class="wikitable center-all"
+{{ZPI
-|-
+| zpi = 872
-! colspan="3" | Tuning
+| steps = 139.990541024216
-! colspan="3" | Strength
+| step size = 8.57200773152536
-! colspan="2" | Closest edo
+| tempered height = 10.076688
-! colspan="2" | Integer limit
+| pure height = 9.983474
-|-
+| integral = 1.548424
-! ZPI
+| gap = 19.514765
-! Steps per octave
+| octave = 1200.08108241355
-! Step size (cents)
+| consistent = 10
-! Height
+| distinct = 10
-! Integral
+}}
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[872zpi]]
-| 139.990541024216
-| 8.57200773152536
-| 10.076688
-| 1.548424
-| 19.514765
-| 140edo
-| 1200.08108241355
-| 10
-| 10
-|}
 == Regular temperament properties ==
@@ Line 437: / Line 426: @@
 | 351.43
 | 49/40
-| [[Hemififths]]
+| [[Hemififths]] (7-limit)
 |-
 | 1
@@ Line 511: / 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 ==