80edo: Difference between revisions

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=== Potential for a general-purpose system ===

Though a strange tuning in lower prime limits, 80edo offers a very unique composite structure that can aid with familiarization/conceptualization by way of its subset edos of 2, 4, 5, 8, 10, 16 and 20. 80edo supports a plethora of multiperiod temperaments with accurate JI interval interpretations based on these edos; to see a fairly comprehensive list of these temperaments and of their most accurate JI interpretations with respect to integer multiples of their period, see [[#Consistent circles]]. These represent a large number of practically completely unexplored and novel high-limit temperaments with varying musical potential.

<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).

=== Prime harmonics ===

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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).

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).

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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 === Potential for a general-purpose system ===
 Though a strange tuning in lower prime limits, 80edo offers a very unique composite structure that can aid with familiarization/conceptualization by way of its subset edos of 2, 4, 5, 8, 10, 16 and 20. 80edo supports a plethora of multiperiod temperaments with accurate JI interval interpretations based on these edos; to see a fairly comprehensive list of these temperaments and of their most accurate JI interpretations with respect to integer multiples of their period, see [[#Consistent circles]]. These represent a large number of practically completely unexplored and novel high-limit temperaments with varying musical potential.
-<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).
 === Prime harmonics ===
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 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).
+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).
-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]].