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== Theory ==
== 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]].
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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=== Miscellany ===
=== Miscellany ===
If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].
If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].
== 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 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"
|-
! Degree
! Cents
! Approximate ratios*
|-
| 1
| 8.57
| [[256/255|S15]], [[225/224|S14]], [[196/195|S14]], [[169/168|S13]], ''[[121/120|S11]]''
|-
| 2
| 17.14
| ''[[144/143|S12]]'', [[105/104]], [[100/99]], [[99/98]], [[91/90]], [[85/84]]
|-
| 3
| 25.71
| [[78/77]], [[66/65]], [[65/64]], [[64/63]], ''[[55/54]]''
|-
| 4
| 34.29
| [[56/55]], 54/53, 53/52, [[52/51]], [[51/50]], [[50/49]], [[49/48]], 46/45
|-
| 5
| 42.86
| [[45/44]], 44/43, 43/42, [[128/125]], 42/41, 41/40, [[40/39]], 39/38
|-
| 6
| 51.43
| [[36/35]], [[35/34]], [[34/33]], [[33/32]]
|-
| 7
| 60.0
| [[30/29]], [[29/28]], [[28/27]]
|-
| 8
| 68.57
| [[27/26]], 53/51, [[26/25]], 51/49, [[25/24]]
|-
| 9
| 77.14
| [[24/23]], [[23/22]], [[68/65]], [[22/21]]
|-
| 10
| 85.71
| [[21/20]], 104/99, 41/39, [[20/19]]
|-
| 11
| 94.29
| 96/91, 19/18, 56/53, 37/35, [[55/52]]
|-
| 12
| 102.86
| [[18/17]], [[35/33]], [[52/49]], [[17/16]]
|-
| 13
| 111.43
| [[16/15]]
|-
| 14
| 120.0
| [[15/14]]
|-
| 15
| 128.57
| [[14/13]]
|-
| 16
| 137.14
| [[27/25]], 40/37, 53/49, [[13/12]]
|-
| 17
| 145.71
| [[38/35]], [[25/23]], 99/91, 37/34, [[49/45]]
|-
| 18
| 154.29
| [[12/11]], [[35/32]], 58/53, [[23/21]]
|-
| 19
| 162.86
| 45/41, [[56/51]], [[11/10]]
|-
| 20
| 171.43
| [[54/49]], 75/68, [[32/29]], 85/77, 53/48, [[21/19]]
|-steps
| 21
| 180.0
| 72/65, 41/37, 51/46, [[10/9]]
|-
| 22
| 188.57
| [[49/44]], [[39/35]], [[29/26]]
|-
| 23
| 197.14
| ''[[19/17]]'', [[28/25]], 102/91, 46/41, 37/33, [[55/49]]
|-
| 24
| 205.71
| [[9/8]], [[44/39]]
|-
| 25
| 214.29
| [[26/23]], 60/53, 43/38, [[17/15]]
|-
| 26
| 222.86
| [[25/22]], 58/51, [[33/29]], 41/36
|-
| 27
| 231.43
| [[8/7]], [[55/48]]
|-
| 28
| 240.0
| ''[[63/55]]'', [[39/34]], [[147/128]], 85/74, [[23/20]]
|-
| 29
| 248.57
| ''[[38/33]]'', 53/46, [[15/13]], [[52/45]], ''[[22/19]]''
|-
| 30
| 257.14
| [[51/44]], [[29/25]], 65/56
|-
| 31
| 265.71
| [[64/55]], 99/85, [[7/6]]
|-
| 32
| 274.29
| [[90/77]], 48/41, 41/35, [[75/64]], [[34/29]], 88/75
|-
| 33
| 282.86
| ''[[27/23]]'', [[20/17]], 53/45, [[33/28]], 46/39
|-
| 34
| 291.43
| [[13/11]], 58/49, 45/38, [[32/27]]
|-
| 35
| 300.0
| [[19/16]], [[44/37]], [[25/21]]
|-
| 36
| 308.57
| 43/36, 49/41, 55/46
|-
| 37
| 317.14
| [[6/5]], [[77/64]]
|-
| 38
| 325.71
| 53/44, 41/34, 35/29, 64/53, 29/24, ''[[23/19]]''
|-
| 39
| 334.29
| [[40/33]], 91/75, [[17/14]]
|-
| 40
| 342.86
| [[28/23]], [[39/32]], 50/41, [[128/105]], ''[[11/9]]''
|-
| 41
| 351.43
| [[60/49]], [[49/40]]
|-
| 42
| 360.0
| ''[[27/22]]'', [[16/13]]
|-
| 43
| 368.57
| [[21/17]], 68/55, [[99/80]], [[26/21]]
|-
| 44
| 377.14
| [[36/29]], 41/33, 46/37, 51/41, [[56/45]], 66/53
|-
| 45
| 385.71
| [[96/77]], [[5/4]]
|-
| 46
| 394.29
| [[64/51]], 54/43, [[49/39]], [[44/35]], ''[[34/27]]''
|-
| 47
| 402.86
| [[63/50]], [[29/23]], 53/42, [[24/19]]
|-
| 48
| 411.43
| [[19/15]], 52/41, [[33/26]], [[80/63]]
|-
| 49
| 420.0
| ''[[108/85]]'', [[14/11]], [[65/51]], [[51/40]], ''[[23/18]]''
|-
| 50
| 428.57
| [[32/25]], 41/32, [[50/39]], 68/53, [[77/60]]
|-
| 51
| 437.14
| [[9/7]], 85/66, 58/45, 49/38
|-
| 52
| 445.71
| 84/65, 53/41, [[128/99]], 75/58, [[22/17]], [[136/105]], [[35/27]]
|-
| 53
| 454.29
| 100/77, [[13/10]]
|-
| 54
| 462.86
| [[30/23]], [[64/49]], [[98/75]], [[17/13]], [[55/42]]
|-
| 55
| 471.43
| ''[[72/55]]'', 38/29, [[21/16]], 130/99, 46/35, [[25/19]]
|-
| 56
| 480.0
| [[29/22]], [[33/25]], 70/53
|-
| 57
| 488.57
| [[45/34]], 53/40, 65/49, [[85/64]]
|-
| 58
| 497.14
| [[4/3]]
|-
| 59
| 505.71
| 91/68, [[75/56]], 55/41, 51/38
|-
| 60
| 514.29
| [[35/26]], [[66/49]], [[85/63]]
|-
| 61
| 522.86
| [[27/20]], [[104/77]], [[88/65]], [[65/48]]
|-
| 62
| 531.43
| [[19/14]], 72/53, 53/39, [[34/25]], [[49/36]]
|-
| 63
| 540.0
| [[15/11]], 56/41, 41/30, [[26/19]]
|-
| 64
| 548.57
| [[48/35]], [[70/51]], 136/99, [[11/8]]
|-
| 65
| 557.14
| 51/37, [[40/29]], [[29/21]]
|-
| 66
| 565.71
| [[18/13]], 104/75, 68/49, [[25/18]]
|-
| 67
| 574.29
| [[32/23]], [[39/28]], 46/33, 53/38, 60/53
|-
| 68
| 582.86
| [[7/5]]
|-
| 69
| 591.43
| [[45/32]], [[128/91]], [[55/39]]
|-
| 70
| 600.0
| [[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:
{{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 ==
== Approximation to JI ==
=== Higher-limit JI ===
=== Higher-limit JI ===
140edo is very strong as a high-limit system, performing generally very well for its size in extremely high [[odd-limit]]s like 125 but also being a good choice for all odd limits 41 through 51. The main flaw is inconsistency; the cost of getting so much right is that there is a lot of things it maps inconsistently (so with more than {{nowrap|0.5\140 {{=}} ~4.28{{c}}}} of error, but almost always less than {{nowrap|1\140 {{=}} 8.57{{c}} of error}}), even though there is far more that it gets right. It is especially notable as a tuning of [[degrees]] (with 1\20 period), [[decoid]] (with 1\10 period) and [[thunderclysmic]] (with 1\5 period), all extending to high limits (largely) through the wealth of interpretations of intervals of [[5edo]].
140edo is very strong as a high-limit system, performing generally very well for its size in extremely high [[odd-limit]]s like 125 but also being a good choice for all odd limits 41 through 53. The main flaw is inconsistency; the cost of getting so much right is that there is a lot of things it maps inconsistently (so with more than {{nowrap|0.5\140 {{=}} ~4.28{{c}}}} of error, but almost always less than {{nowrap|1\140 {{=}} 8.57{{c}} of error}}), even though there is far more that it gets right. It is especially notable as a tuning of [[degrees]] (with 1\20 period), [[decoid]] (with 1\10 period) and [[thunderclysmic]] (with 1\5 period), all extending to high limits (largely) through the wealth of interpretations of intervals of [[5edo]].


In fact, in the full 125-odd-limit, according to the 113-limit [[patent val]], there is only 10 interval pairs that are mapped with more than 1\140 of error, and they are all intervals of {{nowrap| 11<sup>2</sup> {{=}} 121 }}, due to 11 being relatively flat. They are: 121/118, 121/114, 121/109, 121/93, 121/89, 121/83, 121/81, 121/79, 121/73, 121/62 (and their octave-complements). This is remarkable because there is an astounding 1600 interval pairs in the 125-odd-limit. As for inconsistencies, there are 374 inconsistent interval pairs in the 125-odd-limit out of 1600, or around 23%. If we omit intervals of 121, it drops to 329/1543, or around 21%.
In fact, in the full 125-odd-limit, according to the 113-limit [[patent val]], there is only 10 interval pairs that are mapped with more than 1\140 of error, and they are all intervals of {{nowrap| 11<sup>2</sup> {{=}} 121 }}, due to 11 being relatively flat. They are: 121/118, 121/114, 121/109, 121/93, 121/89, 121/83, 121/81, 121/79, 121/73, 121/62 (and their octave-complements). This is remarkable because there is an astounding 1600 interval pairs in the 125-odd-limit. As for inconsistencies, there are 374 inconsistent interval pairs in the 125-odd-limit out of 1600, or around 23%. If we omit intervals of 121, it drops to 329/1543, or around 21%.
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=== Zeta peak index ===
=== 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 ==
== Regular temperament properties ==
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| 351.43
| 351.43
| 49/40
| 49/40
| [[Hemififths]]
| [[Hemififths]] (7-limit)
|-
|-
| 1
| 1
Line 220: Line 500:
| [[Oquatonic]]
| [[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 ==
== Instruments ==
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