80edo: Difference between revisions

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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..{{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 ~~(no-11's)~~ 13-limit 27&53 temperament [[quartonic]]. ~~In fact, the generator for quartonic, ~45{{cent}}, is approximated well by [[159edo]] too — another candidate general-purpose system~~. Even the [[#Significance of echidna|aforementioned]] sharp 7 is explained by 27edo being a sharp [[superpyth]] system.

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 27e&53 temperament [[quartonic]].. Even the [[#Significance of echidna|aforementioned]] 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 80&231 temperament [[superlimmal]], where it represents [[27/25]]~[[40/37]], implying a slightly sharp tuning for 27/25, which is characteristic.

=== Commas ===

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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..{{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 (no-11's) 13-limit 27&53 temperament [[quartonic]]. In fact, the generator for quartonic, ~45{{cent}}, is approximated well by [[159edo]] too — another candidate general-purpose system. Even the [[#Significance of echidna|aforementioned]] sharp 7 is explained by 27edo being a sharp [[superpyth]] system.
+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 27e&53 temperament [[quartonic]].. Even the [[#Significance of echidna|aforementioned]] 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 80&231 temperament [[superlimmal]], where it represents [[27/25]]~[[40/37]], implying a slightly sharp tuning for 27/25, which is characteristic.
 === Commas ===