80edo: Difference between revisions

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=== Significance of echidna ===

As an equal temperament, it is well-tuned for the important 11-limit and 17-limit half-octave-period temperament [[echidna]] (the 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 (especially) [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they [[#Consistent circles|form 80-note consistent circles]]. Echidna supports [[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 + 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/diaschismic 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 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 (especially) [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they [[#Consistent circles|form 80-note consistent circles]]. Echidna supports [[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 + 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/diaschismic 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).

=== 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. Initially this doesn't seem very useful unless one is interested in learning those edos, but 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 w.r.t. (integer multiples of) their period, see [[#Consistent circles|the section on consistent circles]]. These represent a large number of practically completely unexplored and novel high-limit temperaments with varying musical potential.

<nowiki>*</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]]), 27edo tempers {[[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. 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 (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.

=== Commas ===

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 === Significance of echidna ===
-As an equal temperament, it is well-tuned for the important 11-limit and 17-limit half-octave-period temperament [[echidna]] (the 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 (especially) [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they [[#Consistent circles|form 80-note consistent circles]]. Echidna supports [[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 + 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/diaschismic 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 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 (especially) [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they [[#Consistent circles|form 80-note consistent circles]]. Echidna supports [[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 + 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/diaschismic 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).
+=== 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. Initially this doesn't seem very useful unless one is interested in learning those edos, but 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 w.r.t. (integer multiples of) their period, see [[#Consistent circles|the section on consistent circles]]. These represent a large number of practically completely unexplored and novel high-limit temperaments with varying musical potential.
+<nowiki>*</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]]), 27edo tempers {[[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. 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 (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.
 === Commas ===