80edo: Difference between revisions
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=== As a tuning of other temperaments === | === As a tuning of other temperaments === | ||
80et [[tempering out|tempers out]] [[2048/2025]] in the 5-limit; [[1728/1715]], [[3136/3125]], [[4000/3969]], and [[4375/4374]] in the [[7-limit]]; [[176/175]], [[540/539]] and [[4000/3993]] in the [[11-limit]]; [[169/168]], [[325/324]], [[351/350]], [[352/351]], [[364/363]] and [[1001/1000]] in the [[13-limit]]; [[136/135]], [[221/220]], [[256/255]], [[289/288]], [[561/560]], [[595/594]], [[715/714]], [[936/935]] and [[1275/1274]] in the [[17-limit]]; [[190/189]], [[286/285]], [[361/360]], [[400/399]], [[456/455]], [[476/475]], [[969/968]], [[1331/1330]], [[1445/1444]], [[1521/1520]], [[1540/1539]] and [[1729/1728]] in the [[19-limit]]; [[208/207]], [[253/252]], [[323/322]] and [[460/459]] in the [[23-limit]]; and 320/319 in the [[29-limit]]. The last comma is notable as it equates a sharp [[29/16]] with a near-perfect [[20/11]], although this equivalence begins to make more sense when you consider the error cancellations with other sharp harmonics and as a way to give more reasonable interpretations to otherwise questionably mapped intervals. It provides the [[optimal patent val]] for 5-limit [[diaschismic]], for 13-limit [[srutal]], and for 7-, 11- and 13-limit [[bidia]]. It is a good tuning for various temperaments in [[ | 80et [[tempering out|tempers out]] [[2048/2025]] in the 5-limit; [[1728/1715]], [[3136/3125]], [[4000/3969]], and [[4375/4374]] in the [[7-limit]]; [[176/175]], [[540/539]] and [[4000/3993]] in the [[11-limit]]; [[169/168]], [[325/324]], [[351/350]], [[352/351]], [[364/363]] and [[1001/1000]] in the [[13-limit]]; [[136/135]], [[221/220]], [[256/255]], [[289/288]], [[561/560]], [[595/594]], [[715/714]], [[936/935]] and [[1275/1274]] in the [[17-limit]]; [[190/189]], [[286/285]], [[361/360]], [[400/399]], [[456/455]], [[476/475]], [[969/968]], [[1331/1330]], [[1445/1444]], [[1521/1520]], [[1540/1539]] and [[1729/1728]] in the [[19-limit]]; [[208/207]], [[253/252]], [[323/322]] and [[460/459]] in the [[23-limit]]; and 320/319 in the [[29-limit]]. The last comma is notable as it equates a sharp [[29/16]] with a near-perfect [[20/11]], although this equivalence begins to make more sense when you consider the error cancellations with other sharp harmonics and as a way to give more reasonable interpretations to otherwise questionably mapped intervals. It provides the [[optimal patent val]] for 5-limit [[diaschismic]], for 13-limit [[srutal]], and for 7-, 11- and 13-limit [[bidia]]. It is a good tuning for various temperaments in [[cathartic family]], especially in higher limits. | ||
As an equal temperament, it is well-tuned for the important 11-limit and 17-limit half-octave-period temperament [[echidna]], the {{nowrap| 22 & 58 }} temperament, which affords great freedom in a 36-note mos and still many choices in a 22-note mos, offering a high-accuracy rank-2 detemper of [[22edo]], which in comparison conflates many important distinctions of the 11-limit. This is not insignificant as many abundant intervals of echidna, such as [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they form 80-note [[#Consistent circles|consistent circles]]. Echidna extends [[srutal archagall]], which is also tuned near-optimally for [[fiventeen]] – specifically, for the characteristic fiventeen pentad, 30:34:40:45:51:60, consisting of steps of [[20/17]] and [[9/8]]~[[17/15]], and is the smallest edo to improve on the tuning of srutal archagall plus fiventeen after [[34edo]]. In its representation of echidna, the least accurate tuning is that of [[7/4]], which is (relatively) very sharp in 80edo, for which [[58edo]] does better as a tuning of echidna, though much worse as a tuning for srutal archagall and especially fiventeen; one can reason this makes the 80edo tuning of echidna feel more like a detemper of 22edo, especially given the smaller step size between adjacent notes equated in 22edo. | As an equal temperament, it is well-tuned for the important 11-limit and 17-limit half-octave-period temperament [[echidna]], the {{nowrap| 22 & 58 }} temperament, which affords great freedom in a 36-note mos and still many choices in a 22-note mos, offering a high-accuracy rank-2 detemper of [[22edo]], which in comparison conflates many important distinctions of the 11-limit. This is not insignificant as many abundant intervals of echidna, such as [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they form 80-note [[#Consistent circles|consistent circles]]. Echidna extends [[srutal archagall]], which is also tuned near-optimally for [[fiventeen]] – specifically, for the characteristic fiventeen pentad, 30:34:40:45:51:60, consisting of steps of [[20/17]] and [[9/8]]~[[17/15]], and is the smallest edo to improve on the tuning of srutal archagall plus fiventeen after [[34edo]]. In its representation of echidna, the least accurate tuning is that of [[7/4]], which is (relatively) very sharp in 80edo, for which [[58edo]] does better as a tuning of echidna, though much worse as a tuning for srutal archagall and especially fiventeen; one can reason this makes the 80edo tuning of echidna feel more like a detemper of 22edo, especially given the smaller step size between adjacent notes equated in 22edo. | ||
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As a composite edo, the main subsets it lacks are subsets of [[3edo|3]] and [[9edo|9]], but 9\80 = 135{{cent}} offers a good approximation to 1\9 = 133.33…{{c}}, and instead of 1\3 = 400{{cent}}, it has 27\80 = 405{{cent}} as [[19/15]]~[[24/19]], thus serving a similar function to the [[nestoria]] major third. As a result, 80edo is in some sense uniquely tasked with approximating small edos because it will often share subsets that can help make the approximation feel more regular and consistent by interpreting it as a near-equal multiperiod mos. This has the benefit of offering a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI). | As a composite edo, the main subsets it lacks are subsets of [[3edo|3]] and [[9edo|9]], but 9\80 = 135{{cent}} offers a good approximation to 1\9 = 133.33…{{c}}, and instead of 1\3 = 400{{cent}}, it has 27\80 = 405{{cent}} as [[19/15]]~[[24/19]], thus serving a similar function to the [[nestoria]] major third. As a result, 80edo is in some sense uniquely tasked with approximating small edos because it will often share subsets that can help make the approximation feel more regular and consistent by interpreting it as a near-equal multiperiod mos. This has the benefit of offering a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI). | ||
80edo is notable in not only it is consistent in the 19-odd-limit, but a large number of its supersets are also consistent in at least 19-odd-limit, if not larger. These are {{EDOs| 320, 400, 1600, 1920, 2000, 2320, 3920, 4320 }}. | 80edo is notable in not only it is consistent in the 19-odd-limit, but a large number of its supersets are also consistent in at least 19-odd-limit, if not larger. These are {{EDOs| 320, 400, 1600, 1920, 2000, 2320, 3920, 4320 }}. This produces ample [[80th-octave temperaments]] that can be used for augmenting 80edo (see the dedicated page). | ||
== Intervals == | == Intervals == | ||
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| 24 | | 24 | ||
| 360 | | 360 | ||
| [[16/13]] | | [[16/13]], ''[[26/21]]'' | ||
|- | |- | ||
| 25 | | 25 | ||
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| 495 | | 495 | ||
| 4/3 | | 4/3 | ||
| [[ | | [[Leapmonth]] | ||
|- | |- | ||
| 1 | | 1 | ||
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| [[Degrees]] | | [[Degrees]] | ||
|} | |} | ||
<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|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 | ||
== Detemperaments == | == Detemperaments == | ||
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* [[Equiheptatonic]] (approximate): 11 12 11 12 11 12 11 | * [[Equiheptatonic]] (approximate): 11 12 11 12 11 12 11 | ||
* [[Maeve Gutierrez|Gutierrez Moonglade scale]] | * [[Maeve Gutierrez|Gutierrez Moonglade scale]] | ||
==Instruments== | |||
; Lumatone | |||
* [[Lumatone mapping for 80edo]] | |||
; Fretted instruments | |||
* [[Skip fretting system 80 7 4]] | |||
== Music == | == Music == | ||
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; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/shorts/H6DlCHKii-o ''microtonal improvisation in 80edo''] (2025) | * [https://www.youtube.com/shorts/H6DlCHKii-o ''microtonal improvisation in 80edo''] (2025) | ||
* [https://www.youtube.com/shorts/lWuqrdSr8pU ''80edo improv''] (2026) | |||
; [[Francium]] | ; [[Francium]] | ||
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; [[Xotla]] | ; [[Xotla]] | ||
* "Mollusc Merchant" from ''Jazzbeetle'' (2023) [https://xotla.bandcamp.com/track/mollusc-merchant-80edo Bandcamp] | [https://www.youtube.com/watch?v=5cb0WHAwVuM Original YouTube video] [https://www.youtube.com/watch?v=LnWJzffO7dY YouTube video without AI visuals] (2025) | * "Mollusc Merchant" from ''Jazzbeetle'' (2023) [https://xotla.bandcamp.com/track/mollusc-merchant-80edo Bandcamp] | [https://www.youtube.com/watch?v=5cb0WHAwVuM Original YouTube video] [https://www.youtube.com/watch?v=LnWJzffO7dY YouTube video without AI visuals] (2025) | ||
[[Category:19-limit]] | [[Category:19-limit]] | ||