80edo: Difference between revisions

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=== 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 [[canou 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.

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<nowiki />* The strangeness of its tuning can largely be explained by the addition of vals [[80edo]] = [[53edo]] + [[27edo]], where [[27edo]] exaggerates the idiosyncratic mapping of the 2.3.5.7.13 subgroup, as while 53edo tempers {[[625/624|S25]], [[676/675|S26]], [[729/728|S27]]} (supporting [[catakleismic]]) and {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]} (supporting [[buzzard]]), 27edo tempers {[[1728/1715|S6/S7]], [[64/63|S8]], [[325/324|S25*S26]], [[351/350|S26*S27]]} [[Square superparticular|implying]] {[[4375/4374|S25/S27]], [[169/168|S13]]} but maps S25~S27 positively and S26 negatively, which 80et thus inherits though with less damage. This is not insignificant, because this plays a special role (as we'll see in the next section on subsets).

~~==== Based on subsets ====~~

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..{{cent}}, and one could argue that 1\3 = 400{{cent}} is the most difficult small edo interval to interpret (assuming interpreting it as [[5/4]] is not convincing or pleasing enough) in that its interpretations tend to be a large variety of high-complexity intervals, though if one wants a similar sound there is 27\80 = 405{{cent}} as ~[[24/19]]~[[19/15]] (though 24/19 is more accurate), 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).

Even if one finds this reasoning about not having subsets of 3 and 9 unconvincing, there is the fact that the idiosyncracies in the tuning profile of 80edo is intimately related to those of 27edo, so that it shares a deep logic with it through the 13-limit {{nowrap| 27e & 53 }} temperament [[quartonic]]. Even the sharp 7 is explained by 27edo being a sharp [[superpyth]] system. More mysterious is that the approximation of 1\9 at 9\80 = 135{{cent}}, when taken as a generator, is related to the shared [[41-limit]] structure between 80edo and the ultimate general purpose system, [[311edo]], through the {{nowrap|80 & 231}} temperament [[superlimmal]], where it represents [[27/25]]~[[40/37]], implying a slightly sharp tuning for 27/25, which is characteristic.

~~=== Commas ===~~

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 [[canou family]], especially in higher limits.

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 80 factors into primes as 2<sup>4</sup> × 5, 80edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, and 40 }}.

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 one could argue that 1\3 = 400{{cent}} is the most difficult small edo interval to interpret (assuming interpreting it as [[5/4]] is not convincing or pleasing enough) in that its interpretations tend to be a large variety of high-complexity intervals, though if one wants a similar sound there is 27\80 = 405{{cent}} as ~[[24/19]]~[[19/15]] (though 24/19 is more accurate), 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).

Even if one finds this reasoning about not having subsets of 3 and 9 unconvincing, there is the fact that the idiosyncracies in the tuning profile of 80edo is intimately related to those of 27edo, so that it shares a deep logic with it through the 13-limit {{nowrap| 27e & 53 }} temperament [[quartonic]]. Even the sharp 7 is explained by 27edo being a sharp [[superpyth]] system. More mysterious is that the approximation of 1\9 at 9\80 = 135{{cent}}, when taken as a generator, is related to the shared [[41-limit]] structure between 80edo and the ultimate general purpose system, [[311edo]], through the {{nowrap|80 & 231}} temperament [[superlimmal]], where it represents [[27/25]]~[[40/37]], implying a slightly sharp tuning for 27/25, which is characteristic.

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 }}. Temperament mergers of these produce various [[80th-octave temperaments]].

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 === As a tuning of other temperaments ===
+et [[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 [[canou 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.
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 <nowiki />* The strangeness of its tuning can largely be explained by the addition of vals [[80edo]] = [[53edo]] + [[27edo]], where [[27edo]] exaggerates the idiosyncratic mapping of the 2.3.5.7.13 subgroup, as while 53edo tempers {[[625/624|S25]], [[676/675|S26]], [[729/728|S27]]} (supporting [[catakleismic]]) and {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]} (supporting [[buzzard]]), 27edo tempers {[[1728/1715|S6/S7]], [[64/63|S8]], [[325/324|S25*S26]], [[351/350|S26*S27]]} [[Square superparticular|implying]] {[[4375/4374|S25/S27]], [[169/168|S13]]} but maps S25~S27 positively and S26 negatively, which 80et thus inherits though with less damage. This is not insignificant, because this plays a special role (as we'll see in the next section on subsets).
-==== Based on subsets ====
-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..{{cent}}, and one could argue that 1\3 = 400{{cent}} is the most difficult small edo interval to interpret (assuming interpreting it as [[5/4]] is not convincing or pleasing enough) in that its interpretations tend to be a large variety of high-complexity intervals, though if one wants a similar sound there is 27\80 = 405{{cent}} as ~[[24/19]]~[[19/15]] (though 24/19 is more accurate), 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).
-Even if one finds this reasoning about not having subsets of 3 and 9 unconvincing, there is the fact that the idiosyncracies in the tuning profile of 80edo is intimately related to those of 27edo, so that it shares a deep logic with it through the 13-limit {{nowrap| 27e & 53 }} temperament [[quartonic]]. Even the sharp 7 is explained by 27edo being a sharp [[superpyth]] system. More mysterious is that the approximation of 1\9 at 9\80 = 135{{cent}}, when taken as a generator, is related to the shared [[41-limit]] structure between 80edo and the ultimate general purpose system, [[311edo]], through the {{nowrap|80 &amp; 231}} temperament [[superlimmal]], where it represents [[27/25]]~[[40/37]], implying a slightly sharp tuning for 27/25, which is characteristic.
-=== Commas ===
-et [[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 [[canou family]], especially in higher limits.
 === Prime harmonics ===
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 === Subsets and supersets ===
 Since 80 factors into primes as 2<sup>4</sup> × 5, 80edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, and 40 }}.
+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 one could argue that 1\3 = 400{{cent}} is the most difficult small edo interval to interpret (assuming interpreting it as [[5/4]] is not convincing or pleasing enough) in that its interpretations tend to be a large variety of high-complexity intervals, though if one wants a similar sound there is 27\80 = 405{{cent}} as ~[[24/19]]~[[19/15]] (though 24/19 is more accurate), 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).
+Even if one finds this reasoning about not having subsets of 3 and 9 unconvincing, there is the fact that the idiosyncracies in the tuning profile of 80edo is intimately related to those of 27edo, so that it shares a deep logic with it through the 13-limit {{nowrap| 27e & 53 }} temperament [[quartonic]]. Even the sharp 7 is explained by 27edo being a sharp [[superpyth]] system. More mysterious is that the approximation of 1\9 at 9\80 = 135{{cent}}, when taken as a generator, is related to the shared [[41-limit]] structure between 80edo and the ultimate general purpose system, [[311edo]], through the {{nowrap|80 &amp; 231}} temperament [[superlimmal]], where it represents [[27/25]]~[[40/37]], implying a slightly sharp tuning for 27/25, which is characteristic.
 edo 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 }}. Temperament mergers of these produce various [[80th-octave temperaments]].