Collection of EDO impressions: Difference between revisions

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: '''Budjarn Lambeth:''' Offers exciting melodic shapes, but requires careful attention to timbre to prevent it sounding "out of tune".

: '''Zhenlige:''' A stack of [[7/6]]. A subset of [[ennealimmal]].

: '''Eufalesio:''' ~~Potentially useful~~, ~~but~~ I ~~don~~'~~t really like it~~. D

: '''Eufalesio:''' Because this system does not support diatonic, I consider it useless. On its own. However, as a subset of other edos, it absolutely rules, as it is '''the''' basis for ennealimmal. A great deal of ''nineven'' edos are top-tier, because of this. Alone, F. As a subset, A.

== [[10edo]] ==

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: '''Vector:''' Definitive proof that a fifth doesn't need to be a 3/2. (TBA)

: '''Budjarn Lambeth:''' Offers exciting melodic shapes, but requires careful attention to [[timbre]] to prevent it sounding "out of tune".

: '''Eufalesio:''' ~~Potentially useful~~, but I don't ~~really like it~~. D

: '''Eufalesio:''' Supposedly, it would be one of the best antidiatonic systems, next to 9edo, but if an edo does not have diatonic, I consider it useless. Unlike 9edo which forms the basis for ennealimmal, I don't know of any good temperaments with a 1/16 octave period. 2^n edos suck in general. D

== [[17edo]] ==

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: '''Budjarn Lambeth:''' Offers exciting melodic shapes, but requires careful attention to [[timbre]] to prevent it sounding "out of tune".

: '''Zhenlige:''' A circle of fifths in [[34edo]]. Interesting sharp fifths. The smallest [[5L 2s|diatonic]] EDO with neutral intervals. The boundary between neogothic and superpyth. Like [[12edo]], its diatonic thirds do not approximate any simple ratios well, and a [[well temperament]] may help. Its [[13/1|13]] is good, and [[11/1|11]] and [[7/1|7]] have a similar precision to 12edo's [[5/1|5]]. It benefits from compression.

: '''Eufalesio''': Despite being the next edo with a usable fifth, the fact that it tempers the interval whose edostep best approximates it is the ultimate irony. I like the slightly sharp fifths and neo-gothic feel, but the lack of 5-limit is a hole I can't easily live without, and no matter how good it is on other limits (and it is ''great''), the lack of 5 is sad. C

: '''Eufalesio''': Despite being the next edo with a usable fifth, the fact that it tempers out the interval whose edostep best approximates it is the ultimate irony. I like the slightly sharp fifths and neo-gothic feel, but the lack of 5-limit is a hole I can't easily live without, and no matter how good it is on other limits (and it is ''great''), the lack of 5 is sad. C+

== [[18edo]] ==

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: '''Budjarn Lambeth:''' Along with [[36edo]], it is one of the two possible ways to extend 12edo while preserving equal spacing, and keeping the number of notes somewhat manageable. 36edo is ideal if you want to add intervals involving the 7th harmonic into 12edo, while 24edo is ideal if you want to add intervals involving the 11th harmonic. Comparing and contrasting 24edo and 36edo can help you get a feel for the difference between the "vibe" of the 11th harmonic, and the "vibe" of the 7th harmonic. I recommend dipping your toes into each of the two. — Try using familar 12edo intervals in lower registers of your instrument(s)/mix, while mixing in some of the strange new 24edo intervals in the higher registers. Thus will mimic the shape of the [[harmonic series]] and sound nice and glittery.

: '''Zhenlige:''' What some non-microtonalists think microtonality is. 12edo with neutrals. Good for prime [[11/1|11]]. Accurate in subgroup 2.3.11.17.19.

: '''Eufalesio:''' Entry-level xenharmonic edo. A huge improvement to the 2.3.5.11, but nothing much more to remark. Probably the most common xenharmonic edo among non xen spaces, and for good reason. We've all used it. It's trivial to build it. – Still, some ensembles fail at playing quartertones accurately (singers are the worst, some can even fail to sing 12edo accurately, which is a feat...) C

: '''Eufalesio:''' Entry-level xenharmonic edo. A huge improvement to the 2.3.5.11, but nothing much more to remark. Probably the most common xenharmonic edo among non xen spaces, and for good reason. We've all used it. It's trivial to build it. – Still, some ensembles fail at playing quartertones accurately (singers are the worst, some can even fail to sing 12edo accurately, which is a feat...) C+

== [[25edo]] ==

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: '''Fumica:''' This is to 17edo what [[24edo]] is to [[12edo]]. While 17edo is often good enough, this offers some more sophisticated solutions such as [[tetracot]]. Even the [[harmonic]]s 7 and 11, which come from 17edo and are commonly cited as relatively poor in this edo, are convincing enough to me, since when I worked with [[modus]] I never had a problem with the intonation at all, unlike with [[porcupine]]. The sound is better than the structure. B-tier.

: '''Zhenlige:''' [[17edo]] with [[5/1|5]] and [[17/1|17]] added, making a good 2.3.5.13.17 system. A slightly stretched [[Carlos Gamma]] scale.

: '''Eufalesio:''' 17edo, but good. By splitting the edo in two, we get a great 5-limit, the best so far. Fails at 7 and 11, which is sad, but at least is a very good 2.3.5.13, as it also supports kleismic. And diaschismic. I've used this to retune some classical pieces. It's good for its grain, but it's not the best. B+

== [[35edo]] ==

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: '''Fumica:''' Potentially useful as every other step of [[70edo]]. D-tier.

: '''Budjarn Lambeth:''' A very good [[dual-fifth]] edo.

: ''Zhenlige:''' The largest non-[[5L 2s|diatonic]] EDO.

: '''Zhenlige:''' The largest non-[[5L 2s|diatonic]] EDO.

: '''Eufalesio:''' Useless. FF

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: '''Budjarn Lambeth''': Better than [[12edo]] for most pop or earlier classical music that doesn't have lots of key changes in the one piece. The fifths are still pretty good, but the thirds and sixths sound so much warmer and more expressively. But, it is unsuitable if you want to use lots of key changes (like in jazz, later classical, or prog rock). Japanese pentatonic scales with semitones in them sound gorgeous in 43edo. I recommend the [[meantone]][19] [[MOS scale]] in 43edo to composers who want to dip their toes into [[microtonal|microtonality]] without getting in too deep.

: '''Zhenlige:''' Close to 1/5-comma [[meantone]] which gives pure [[15/8]]. Not very notable besides that. Its fifth is too sharp for [[septimal meantone]].

: '''Eufalesio:''' This meantone edo may have a seemingly good val to approximate higher limits, but doing so from a meantone framework is dumb. Apart from that, the lower limits, ones that I hold to high standards, are worsely tuned than in 31edo. In my opinion, the best meantones are the golden meantones, and from 31edo on, the peak has already been reached. E

== [[44edo]] ==

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: '''Fumica:''' The second essential comma-level edo. Five more notes than [[41edo]], offering the distinction of two types of [[neutral]] intervals at the cost of a narrower [[septimal diesis]]. As an eighth-tone system, it has a true [[quartertone]]. With that and all the accurate approximations, the expressive possibilities are endless. Best as a 2.3.5.7.11.17.23-[[subgroup]] temperament. A-tier.

: '''Zhenlige:''' [[13-limit]] [[diaschismic]] and [[valentine]]. Near pure [[11/7]]. It has quartertones similar to [[22edo]] but approximates JI intervals more accurately. Its 30::36 are all 2 steps apart.

: '''Eufalesio:''' The best diaschismic. 13-limit stuff, though a bit sharp and not as accurate as 41-edo, it is good. I haven't composed anything with it, however, as I think diaschismic is kinda hard to conceptualize, and the sharpness of the 5 is something that I find less desirable. B

: '''Eufalesio:''' The best diaschismic in my opinion. 13-limit stuff, though a bit sharp and not as accurate as 41-edo, it is good. I haven't composed anything with it, however, as I think diaschismic is kinda hard to conceptualize, and the sharpness of the 5 is something that I find less desirable. B-

== [[47edo]] ==

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: '''Fumica:''' Close to [[2/7-comma meantone]] so it has a niche. Has the same problem as [[45edo]], though less severe. C-tier.

: '''Zhenlige:''' [[Meantone]] with a flatter fifth than [[31edo]], but I usually use [[golden meantone]] (with slight octave stretching) for this range.

: '''Eufalesio:''' Still a good meantone edo, and though it is a much better approximant for golden meantone, I prefer using golden meantone as a rank-2, and not buying the entire gamut. The 19-limit usability is surprising, still. However, having all those new intervals inside a meantone edo feels in my opinion strangely unnatural, as we're stretching the meantone chain-of-fifths beyond what's supposed to. – For ~~bigger~~ edos in this range, meantone ceases to do it for me, but I respect it. C

: '''Eufalesio:''' Still a good meantone edo, and though it is a much better approximant for golden meantone, I prefer using golden meantone as a rank-2, and not buying the entire gamut. The 19-limit usability is very surprising, still. However, having all those new intervals inside a meantone edo feels in my opinion strangely unnatural, as we're stretching the meantone chain-of-fifths beyond what's supposed to. The meantone chain of fifths already hits its apex with 31edo. – For finer edos in this range, meantone ceases to do it for me, but I respect this one. C-

== [[51edo]] ==

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== [[62edo]] ==

: '''Fumica:''' The ultimate [[23-limit]] [[meantone]] tuning. It re-tunes [[harmonic]]s 13, 17, and 19, and paves the path to the 23 from [[31edo]]. I find these additions to 31edo's [[11-limit]] very favorable. A-tier.

: '''Eufalesio:''' Keeps all the 11-limit goodness from 31edo and greatly improves on primes from 13 and beyond. It can be used all the way to the 23-limit, with monotonic error. Interesting, but approaching higher limits from a meantone framework is dumb. Like 50edo, I still give it my respect, but mainly because it's a multiple of 31edo. C-

== [[63edo]] ==

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== [[81edo]] ==

: '''Zhenlige:''' The [[optimal patent val]] for [[meantone]] and some of its higher-limit extentions, but I won't use such a large EDO for a temperament with relatively low accuracy, and rather use [[golden meantone]] instead, which is simpler and more elegant mathematically.

: '''Eufalesio:''' ~~Interesting~~, ~~but~~ rank-2 golden meantone ~~is basically the same~~. D

: '''Eufalesio:''' 81edo is already the absolute maximum for golden meantone, as anything finer and the patent val fifth stops supporting it. If 50edo was already a bit too much, 81edo and beyond are definitely too much. At that point, it's better to not buy the entire gamut and just use rank-2 golden meantone. D

== [[84edo]] ==

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: '''Bozu:''' [[29edo]] with each interval sliced into three. You can do some nifty stuff with it, but the number of notes is too crazy to cover much with midi unless you choose a subset. Pushing a continuum beyond this.

: '''Zhenlige:''' Good [[mystery]] EDO. Useful for high limit JI. Playable by using three [[29edo]] instruments.

: '''Eufalesio:''' Theory says that it is a really strong 13-limit edo. So much so, that it is the first edo with distinct consistency and pure consistency in the 13-odd-limit, and normal consistency in the 15-odd-limit, and for that, I give it my respects. However, I like my fifths to have minimal error, and being a subset of 29edo, the fifths are good, but not as good. C+

== [[94edo]] ==

: '''Aura:''' Surprisingly, I have attempted to use this EDO before, and it is the first EDO I've attempted to use that wasn't some kind of superset of [[12edo]]. I've noticed just from working out the [[JI]] intervals that this EDO approximates that the [[7-limit]] for this EDO is really good- better than what this EDO has to offer in the [[5-limit]]. Furthermore, all of the pitches in this EDO are connected by a single, complicated circle of fifths. It is from working with this EDO that I learned the ways that the [[paradiatonic]] prime-limits (that would be the [[7-limit]], the [[11-limit]], and the [[13-limit]]) are connected with each other.

: '''Zhenlige:''' Good for high-limit JI with the [[garibaldi]] structure similar to [[41edo]] and [[53edo]]. Containing [[Carlos Beta]].

: '''Eufalesio:''' GOAT. The combination of the two smallest schismic edos, which are both incredibly solid choices, into one neatly rounded package that is very optimized. I am heavily '''biased''' towards this, as it represents the ultimate cassandra, and a chain-of-fifths framework that I find extremely easy to work with. It also tempers a lot of things together, much like 41edo, – Naturals for prime 3 or 19. ±1 for 17 or 23. ∓2 for 5 or 7. ±4 for 11 or 13. Throughout many different peer-reviewed experiments and in many on my compositions, I've found that this edo is good enough for most xen purposes. Still a tiny smidge innacurate in the 5-limit, but since it is flat and not sharp, I find it much more palatable, as I like wide minor thirds. I really only use it for the 2.3.5.7.11.13.19, but the 23-limit goodness is no joke. SSS

: '''Eufalesio:''' GOAT. The combination of the two smallest schismic edos, which are both incredibly solid choices, into one neatly rounded package that is very optimized. '''I am heavily''' '''biased''' '''towards this''', as it represents the ultimate cassandra, and a chain-of-fifths framework that I find extremely easy to work with. It also tempers a lot of things together, much like 41edo, – Naturals for prime 3 or 19. ±1 for 17 or 23. ∓2 for 5 or 7. ±4 for 11 or 13. Throughout many different peer-reviewed experiments and in many on my compositions, I've found that this edo is good enough for most xen purposes. Still a tiny smidge innacurate in the 5-limit, but since it is flat and not sharp, I find it much more palatable, as I like wide minor thirds. I really only use it for the 2.3.5.7.11.13.19, but the 23-limit goodness is no joke. SSS

== [[99edo]] ==

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== [[130edo]] ==

: '''Eufalesio:''' I haven't composed in it, but theory screams to me that this edo is a beast. I like to think of it as 65edo, but good. It has an extremely accurate 13-limit, and a schismic chain-of-fifths framework? Count me in! S

: '''Eufalesio:''' I haven't composed in it, but theory screams to me that this edo is a beast. I like to think of it as 65edo, but good. It has an extremely accurate 13-limit, and a schismic chain-of-fifths framework? Count me in! S+

== [[159edo]] ==

: '''Aura:''' This is the main system I use in writing [[microtonal]] music. After finishing the list of [[JI]] equivalents of the various steps of this EDO, I have since found that not only is 159edo very good for those who like to make more just versions of the [[quartertone|quartertone-based intervals]] you see in [[24edo]], but is also very capable of approximating the steps of many lower EDOs within five [[cents]], making for some decent retunings of some of the more commonly used EDOs such as {{EDOs|22edo, 31edo, and even 41edo}}, which was part of the premise of "[[:File:Space Tour.mp3|Space Tour]]". Based on this discovery alone, I'd have to say that 159edo is not just a superset of [[53edo]], but rather, an EDO that is quite full of potential. However, the fact is that this EDO is [[consistent]] all the way up to the [[17-limit]], and has a good 23-[[prime]], and, should you skip the 17-prime, you have access to a decent 19-prime and 29-prime. This, and the fact that one has access to a bunch of [[microtemperament]]s in this EDO, all for a step-size that's slightly above the average [[JND]], means I can also perform other tricks in composition. I imagine at this point that some would ask me why I don't just use JI, and the answer is that even an EDO in the hundreds like 159edo is considerably more simple than JI, as you have to account for a lot of [[unnoticeable comma]]s in JI- a near-pointless endeavor as virtually nobody can hear such small differences in pitch.

: '''Eufalesio:''' Aura's favorite tuning. He does have a point, it takes an extremely good edo, and tripling it makes it even better! 29-limit goodness! I don't care as much for the insanely accurate 2.3.11, as I care for the entirety of the 2.3.5.7.11.13.19(.29)~~. It really~~ is ~~that good~~. I've composed stuff with it, and it isn't as easy to do as in other edos, but the result is still ~~worth it~~. SS

: '''Eufalesio:''' Aura's favorite tuning. He does have a point, it takes an extremely good edo, and tripling it makes it even better! 29-limit goodness! I don't care as much for the insanely accurate 2.3.11, as I care for the entirety of the 2.3.5.7.11.13.19(.29), on which it is worse than other alternatives, as primes 7 and 13 are relatively innacurate. I've composed stuff with it, and it isn't as easy to do as in other edos, but the result is still decent. SS-

== [[171edo]] ==

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== [[217edo]] ==

: '''Eufalesio:''' It's the septuple of 31edo, and that is nothing less than a miracle (though it doesn't support miracle). I've done some tests on it, and it's 31-limit is incredible. It introduces an unfamiliar and slightly tedious gari-vulture-esque framework in which you have to use schisma-function steps apart from pythcommas, but apart from that, it's still within the realms of manageability. Also, it has an incredible 2.3.5.13, which I really respect. Also important to remark, from this point onward, edos start to sound more or less the same, as the absolute error gets lower and lower, and the difference between edosteps becomes harder to reliably hear. S

: '''Eufalesio:''' It's the septuple of 31edo, and that is nothing less than a miracle (though it doesn't support miracle). I've done some tests on it, and it's 31-limit is incredible. It introduces an unfamiliar and slightly tedious gari-vulture-esque framework in which you have to use schisma-function steps apart from pythcommas, but apart from that, it's still within the realms of manageability. Also, it has an incredible 2.3.5.13, which I really respect. Also important to remark, from this point onward, edos start to sound more or less the same, as the absolute error gets lower and lower, and the difference between edosteps becomes harder to reliably hear.–Apart from that, 217edo's 2.3.5.7.11.13.19 mappings can be easily converted to 270edo or 311edo if need be, S+

== [[224edo]] ==

: '''Zhenlige:''' Like [[171edo]] but with a slightly sharper (and closer to just) fifth, worse [[7-limit]] but better [[13-limit]].

: '''Eufalesio:''' A cousin to 217edo which is still schismic, dare I say the ultimate schismic edo, though still harder to conceptualize. Theory tells me that the 13-limit is extremely accurate, even more than the 217edo, and for that I think it deserves ~~some attention~~. But 217edo is smaller, and ~~it contains 31edo, so... I think I'll stick with the other one~~. B

: '''Eufalesio:''' A cousin to 217edo which is still schismic, dare I say the ultimate schismic edo, though still harder to conceptualize. Theory tells me that the 13-limit is extremely accurate, even more than the 217edo, and for that I think it deserves praise. But 217edo is smaller and its mappings can be easily expanded to more accurate edos. B+

== [[270edo]] ==

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: '''Zhenlige:''' Good for very high limit JI.

: '''Eufalesio:''' Ultimate ultra-high-limit JI. Absolute error is a smidge worse than 270edo, but it makes up by being consistent to the goddamn 41-odd-limit. Serendipity personified. Very hard to justify using anything else other than this, as the difference between edosteps from this point on is definitely nigh impossible to hear. I see it as an ultimate tuning of sorts for practicality's sake. SSS

== [[665edo]] ==

: '''Eufalesio:''' Ultimate pyth. It has an unfathomably perfect 2.3, and I say that in an almost literal sense. It is very much fathomable, obviously: the beat period of 665edo's fifth is 5077906.80060 s*Hz with two sawtooth waves in perfect sync, which would be around 3 hours, 12 minutes 21 seconds at f=440. 3 fucking hours. That's what it would take you to hear the beating of 665edo. It is, for all intents and purposes, unfathomable to focused human perception. Or, you could make a 3-hour track out of this. –However, this is not why you would use 665edo, as this essentially allows you to extend the precision limit of the chain of fifths from very good to ''extreme,'' by adding the mercator (+53 fifths) and an equalized qian comma (+306/-359 fifths) into the mix, also working as a schisma. Yes, it has a bad prime 11, but it is surprisingly good in the rest of primes up to the 27-odd-limit, which is very surprising for a convergent. I will likely never use this, but since I do greatly care about the chain of fifths as a theoretical construct, I care about this ''theoretically''. A-

== [[1600edo]] ==

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== [[2460edo]] ==

:'''Eufalesio:''' The only reason I've put this one here is because it is a 12n edo, and that makes it ''slightly'' easier to work with, and very transposing friendly. It's astonishingly accurate, though dividing the semitone into 205ths is reasonably excessive. C

== [[7315edo]] ==

:'''Eufalesio:''' Undecupling 665edo results in what I believe to be one of the potentially theoretically most robust yet precise JI-oid systems. Splitting the equalized qian comma in 11s greatly amplifies the accuracy of this edo and allows you to keep the unfathomably accurate chain of fifths as a strong backbone, and thirteenths of a qian comma serving as nanoalterations. I will likely never use this due to the insane precision it demands, but I have nothing other than respect for this behemoth of an edo. S

== [[8539edo]] ==

:'''Eufalesio:''' This level of fineness is at the bleeding edge of insanity. The precision of this behemoth is astounding. I firmly believe no sane person would compose anything requiring a tuning precision higher than what this offers. And I'm one to ogle at impossibly gargantuan edos, I'll admit, but that ogling is only theoretical. Beyond here... there be monsters... and hot sauce. C

== [[190537edo]] ==

:'''Eufalesio:''' This is the next edo in the list of record k-strong telicity, and it starts to get scary from this point. This is '''unadulterated cosmic horror''' disguised as math in the 3-limit, forget about the rest of primes. The beat period of this fifth is 984 572 779 224.54 s*Hz with two sawtooth waves in perfect sync, which would be around 70 years and 331 days at f=440. This edo has a fifth that is accurate to a level that is quite possibly beyond the scope of a human lifetime. Think about it. There is a chance you'll die before listening to 190537edo's fifth beat, at f=440. – If you say that you need accuracy to this precision, I am, beyond a reasonable doubt, wholly confident you are not human, or not anymore. Perhaps in a couple centuries, posthumanity will be able to comprehend this near-perfection, but as we stand right now, it is impossible to comprehend. Unrankable.

== Sources ==

@@ Line 134: / Line 134: @@
 : '''Budjarn Lambeth:''' Offers exciting melodic shapes, but requires careful attention to timbre to prevent it sounding "out of tune".
 : '''Zhenlige:''' A stack of [[7/6]]. A subset of [[ennealimmal]].
-: '''Eufalesio:''' Potentially useful, but I don't really like it. D
+: '''Eufalesio:''' <span data-darkreader-inline-color="">Because this system does not support diatonic, I consider it useless. On its own. However, as a subset of other edos, it absolutely rules, as it is</span> '''the''' <span data-darkreader-inline-color="">basis for ennealimmal. A great deal of</span> ''nineven'' <span data-darkreader-inline-color="">edos are top-tier, because of this. Alone, F. As a subset, A.</span>
 == [[10edo]] ==
@@ Line 242: / Line 242: @@
 : '''Vector:''' Definitive proof that a fifth doesn't need to be a 3/2. (TBA)
 : '''Budjarn Lambeth:''' Offers exciting melodic shapes, but requires careful attention to [[timbre]] to prevent it sounding "out of tune".
-: '''Eufalesio:''' Potentially useful, but I don't really like it. D
+: '''Eufalesio:''' <span data-darkreader-inline-color="">Supposedly, it would be one of the best antidiatonic systems, next to 9edo, but if an edo does not have diatonic, I consider it useless. Unlike 9edo which forms the basis for ennealimmal, I don't know of any good temperaments with a 1/16 octave period. 2^n edos suck in general. D</span>
 == [[17edo]] ==
@@ Line 256: / Line 256: @@
 : '''Budjarn Lambeth:''' Offers exciting melodic shapes, but requires careful attention to [[timbre]] to prevent it sounding "out of tune".
 : '''Zhenlige:''' A circle of fifths in [[34edo]]. Interesting sharp fifths. The smallest [[5L 2s|diatonic]] EDO with neutral intervals. The boundary between neogothic and superpyth. Like [[12edo]], its diatonic thirds do not approximate any simple ratios well, and a [[well temperament]] may help. Its [[13/1|13]] is good, and [[11/1|11]] and [[7/1|7]] have a similar precision to 12edo's [[5/1|5]]. It benefits from compression.
-: '''Eufalesio''': Despite being the next edo with a usable fifth, the fact that it tempers the interval whose edostep best approximates it is the ultimate irony. I like the slightly sharp fifths and neo-gothic feel, but the lack of 5-limit is a hole I can't easily live without, and no matter how good it is on other limits (and it is ''great''), the lack of 5 is sad. C
+: '''Eufalesio''': Despite being the next edo with a usable fifth, the fact that it tempers out the interval whose edostep best approximates it is the ultimate irony. I like the slightly sharp fifths and neo-gothic feel, but the lack of 5-limit is a hole I can't easily live without, and no matter how good it is on other limits (and it is ''great''), the lack of 5 is sad. C+
 == [[18edo]] ==
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 : '''Budjarn Lambeth:''' Along with [[36edo]], it is one of the two possible ways to extend 12edo while preserving equal spacing, and keeping the number of notes somewhat manageable. 36edo is ideal if you want to add intervals involving the 7th harmonic into 12edo, while 24edo is ideal if you want to add intervals involving the 11th harmonic. Comparing and contrasting 24edo and 36edo can help you get a feel for the difference between the "vibe" of the 11th harmonic, and the "vibe" of the 7th harmonic. I recommend dipping your toes into each of the two. — Try using familar 12edo intervals in lower registers of your instrument(s)/mix, while mixing in some of the strange new 24edo intervals in the higher registers. Thus will mimic the shape of the [[harmonic series]] and sound nice and glittery.
 : '''Zhenlige:''' What some non-microtonalists think microtonality is. 12edo with neutrals. Good for prime [[11/1|11]]. Accurate in subgroup 2.3.11.17.19.
-: '''Eufalesio:''' Entry-level xenharmonic edo. A huge improvement to the 2.3.5.11, but nothing much more to remark. Probably the most common xenharmonic edo among non xen spaces, and for good reason. We've all used it. It's trivial to build it. – Still, some ensembles fail at playing quartertones accurately (singers are the worst, some can even fail to sing 12edo accurately, which is a feat...) C
+: '''Eufalesio:''' Entry-level xenharmonic edo. A huge improvement to the 2.3.5.11, but nothing much more to remark. Probably the most common xenharmonic edo among non xen spaces, and for good reason. We've all used it. It's trivial to build it. – Still, some ensembles fail at playing quartertones accurately (singers are the worst, some can even fail to sing 12edo accurately, which is a feat...) C+
 == [[25edo]] ==
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 : '''Fumica:''' This is to 17edo what [[24edo]] is to [[12edo]]. While 17edo is often good enough, this offers some more sophisticated solutions such as [[tetracot]]. Even the [[harmonic]]s 7 and 11, which come from 17edo and are commonly cited as relatively poor in this edo, are convincing enough to me, since when I worked with [[modus]] I never had a problem with the intonation at all, unlike with [[porcupine]]. The sound is better than the structure. B-tier.
 : '''Zhenlige:''' [[17edo]] with [[5/1|5]] and [[17/1|17]] added, making a good 2.3.5.13.17 system. A slightly stretched [[Carlos Gamma]] scale.
+: '''Eufalesio:''' 17edo, but good. By splitting the edo in two, we get a great 5-limit, the best so far. Fails at 7 and 11, which is sad, but at least is a very good 2.3.5.13, as it also supports kleismic. And diaschismic. I've used this to retune some classical pieces. It's good for its grain, but it's not the best. B+
 == [[35edo]] ==
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 : '''Fumica:''' Potentially useful as every other step of [[70edo]]. D-tier.
 : '''Budjarn Lambeth:''' A very good [[dual-fifth]] edo.
-: ''Zhenlige:''' The largest non-[[5L 2s|diatonic]] EDO.
+: '''Zhenlige:''' The largest non-[[5L 2s|diatonic]] EDO.
 : '''Eufalesio:''' Useless. FF
@@ Line 499: / Line 500: @@
 : '''Budjarn Lambeth''': Better than [[12edo]] for most pop or earlier classical music that doesn't have lots of key changes in the one piece. The fifths are still pretty good, but the thirds and sixths sound so much warmer and more expressively. But, it is unsuitable if you want to use lots of key changes (like in jazz, later classical, or prog rock). Japanese pentatonic scales with semitones in them sound gorgeous in 43edo. I recommend the [[meantone]][19] [[MOS scale]] in 43edo to composers who want to dip their toes into [[microtonal|microtonality]] without getting in too deep.
 : '''Zhenlige:''' Close to 1/5-comma [[meantone]] which gives pure [[15/8]]. Not very notable besides that. Its fifth is too sharp for [[septimal meantone]].
+: '''Eufalesio:''' This meantone edo may have a seemingly good val to approximate higher limits, but doing so from a meantone framework is dumb. Apart from that, the lower limits, ones that I hold to high standards, are worsely tuned than in 31edo. In my opinion, the best meantones are the golden meantones, and from 31edo on, the peak has already been reached. E
 == [[44edo]] ==
@@ Line 516: / Line 518: @@
 : '''Fumica:''' The second essential comma-level edo. Five more notes than [[41edo]], offering the distinction of two types of [[neutral]] intervals at the cost of a narrower [[septimal diesis]]. As an eighth-tone system, it has a true [[quartertone]]. With that and all the accurate approximations, the expressive possibilities are endless. Best as a 2.3.5.7.11.17.23-[[subgroup]] temperament. A-tier.
 : '''Zhenlige:''' [[13-limit]] [[diaschismic]] and [[valentine]]. Near pure [[11/7]]. It has quartertones similar to [[22edo]] but approximates JI intervals more accurately. Its 30::36 are all 2 steps apart.
-: '''Eufalesio:''' The best diaschismic. 13-limit stuff, though a bit sharp and not as accurate as 41-edo, it is good. I haven't composed anything with it, however, as I think diaschismic is kinda hard to conceptualize, and the sharpness of the 5 is something that I find less desirable. B
+: '''Eufalesio:''' The best diaschismic in my opinion. 13-limit stuff, though a bit sharp and not as accurate as 41-edo, it is good. I haven't composed anything with it, however, as I think diaschismic is kinda hard to conceptualize, and the sharpness of the 5 is something that I find less desirable. B-
 == [[47edo]] ==
@@ Line 539: / Line 541: @@
 : '''Fumica:''' Close to [[2/7-comma meantone]] so it has a niche. Has the same problem as [[45edo]], though less severe. C-tier.
 : '''Zhenlige:''' [[Meantone]] with a flatter fifth than [[31edo]], but I usually use [[golden meantone]] (with slight octave stretching) for this range.
-: '''Eufalesio:''' Still a good meantone edo, and though it is a much better approximant for golden meantone, I prefer using golden meantone as a rank-2, and not buying the entire gamut. The 19-limit usability is surprising, still. However, having all those new intervals inside a meantone edo feels in my opinion strangely unnatural, as we're stretching the meantone chain-of-fifths beyond what's supposed to. – For bigger edos in this range, meantone ceases to do it for me, but I respect it. C
+: '''Eufalesio:''' Still a good meantone edo, and though it is a much better approximant for golden meantone, I prefer using golden meantone as a rank-2, and not buying the entire gamut. The 19-limit usability is very surprising, still. However, having all those new intervals inside a meantone edo feels in my opinion strangely unnatural, as we're stretching the meantone chain-of-fifths beyond what's supposed to. The meantone chain of fifths already hits its apex with 31edo. – For finer edos in this range, meantone ceases to do it for me, but I respect this one. C-
 == [[51edo]] ==
@@ Line 579: / Line 581: @@
 == [[62edo]] ==
 : '''Fumica:''' The ultimate [[23-limit]] [[meantone]] tuning. It re-tunes [[harmonic]]s 13, 17, and 19, and paves the path to the 23 from [[31edo]]. I find these additions to 31edo's [[11-limit]] very favorable. A-tier.
+: '''Eufalesio:''' Keeps all the 11-limit goodness from 31edo and greatly improves on primes from 13 and beyond. It can be used all the way to the 23-limit, with monotonic error. Interesting, but approaching higher limits from a meantone framework is dumb. Like 50edo, I still give it my respect, but mainly because it's a multiple of 31edo. C-
 == [[63edo]] ==
@@ Line 622: / Line 625: @@
 == [[81edo]] ==
 : '''Zhenlige:''' The [[optimal patent val]] for [[meantone]] and some of its higher-limit extentions, but I won't use such a large EDO for a temperament with relatively low accuracy, and rather use [[golden meantone]] instead, which is simpler and more elegant mathematically.
-: '''Eufalesio:''' Interesting, but rank-2 golden meantone is basically the same. D
+: '''Eufalesio:''' 81edo is already the absolute maximum for golden meantone, as anything finer and the patent val fifth stops supporting it. If 50edo was already a bit too much, 81edo and beyond are definitely too much. At that point, it's better to not buy the entire gamut and just use rank-2 golden meantone. D
 == [[84edo]] ==
@@ Line 632: / Line 635: @@
 : '''Bozu:''' [[29edo]] with each interval sliced into three. You can do some nifty stuff with it, but the number of notes is too crazy to cover much with midi unless you choose a subset. Pushing a continuum beyond this.
 : '''Zhenlige:''' Good [[mystery]] EDO. Useful for high limit JI. Playable by using three [[29edo]] instruments.
+: '''Eufalesio:''' Theory says that it is a really strong 13-limit edo. So much so, that it is the first edo with distinct consistency and pure consistency in the 13-odd-limit, and normal consistency in the 15-odd-limit, and for that, I give it my respects. However, I like my fifths to have minimal error, and being a subset of 29edo, the fifths are good, but not as good. C+
 == [[94edo]] ==
 : '''Aura:''' Surprisingly, I have attempted to use this EDO before, and it is the first EDO I've attempted to use that wasn't some kind of superset of [[12edo]].  I've noticed just from working out the [[JI]] intervals that this EDO approximates that the [[7-limit]] for this EDO is really good- better than what this EDO has to offer in the [[5-limit]].  Furthermore, all of the pitches in this EDO are connected by a single, complicated circle of fifths.  It is from working with this EDO that I learned the ways that the [[paradiatonic]] prime-limits (that would be the [[7-limit]], the [[11-limit]], and the [[13-limit]]) are connected with each other.
 : '''Zhenlige:''' Good for high-limit JI with the [[garibaldi]] structure similar to [[41edo]] and [[53edo]]. Containing [[Carlos Beta]].
-: '''Eufalesio:''' GOAT. The combination of the two smallest schismic edos, which are both incredibly solid choices, into one neatly rounded package that is very optimized. I am heavily '''biased''' towards this, as it represents the ultimate cassandra, and a chain-of-fifths framework that I find extremely easy to work with. It also tempers a lot of things together, much like 41edo, – Naturals for prime 3 or 19. ±1 for 17 or 23. ∓2 for 5 or 7. ±4 for 11 or 13. Throughout many different peer-reviewed experiments and in many on my compositions, I've found that this edo is good enough for most xen purposes. Still a tiny smidge innacurate in the 5-limit, but since it is flat and not sharp, I find it much more palatable, as I like wide minor thirds. I really only use it for the 2.3.5.7.11.13.19, but the 23-limit goodness is no joke. SSS
+: '''Eufalesio:''' GOAT. The combination of the two smallest schismic edos, which are both incredibly solid choices, into one neatly rounded package that is very optimized. '''I am heavily''' '''biased''' '''towards this''', as it represents the ultimate cassandra, and a chain-of-fifths framework that I find extremely easy to work with. It also tempers a lot of things together, much like 41edo, – Naturals for prime 3 or 19. ±1 for 17 or 23. ∓2 for 5 or 7. ±4 for 11 or 13. Throughout many different peer-reviewed experiments and in many on my compositions, I've found that this edo is good enough for most xen purposes. Still a tiny smidge innacurate in the 5-limit, but since it is flat and not sharp, I find it much more palatable, as I like wide minor thirds. I really only use it for the 2.3.5.7.11.13.19, but the 23-limit goodness is no joke. SSS
 == [[99edo]] ==
@@ Line 651: / Line 655: @@
 == [[130edo]] ==
-: '''Eufalesio:''' I haven't composed in it, but theory screams to me that this edo is a beast. I like to think of it as 65edo, but good. It has an extremely accurate 13-limit, and a schismic chain-of-fifths framework? Count me in! S
+: '''Eufalesio:''' I haven't composed in it, but theory screams to me that this edo is a beast. I like to think of it as 65edo, but good. It has an extremely accurate 13-limit, and a schismic chain-of-fifths framework? Count me in! S+
 == [[159edo]] ==
 : '''Aura:''' This is the main system I use in writing [[microtonal]] music.  After finishing the list of [[JI]] equivalents of the various steps of this EDO, I have since found that not only is 159edo very good for those who like to make more just versions of the [[quartertone|quartertone-based intervals]] you see in [[24edo]], but is also very capable of approximating the steps of many lower EDOs within five [[cents]], making for some decent retunings of some of the more commonly used EDOs such as {{EDOs|22edo, 31edo, and even 41edo}}, which was part of the premise of "[[:File:Space Tour.mp3|Space Tour]]".  Based on this discovery alone, I'd have to say that 159edo is not just a superset of [[53edo]], but rather, an EDO that is quite full of potential.  However, the fact is that this EDO is [[consistent]] all the way up to the [[17-limit]], and has a good 23-[[prime]], and, should you skip the 17-prime, you have access to a decent 19-prime and 29-prime.  This, and the fact that one has access to a bunch of [[microtemperament]]s in this EDO, all for a step-size that's slightly above the average [[JND]], means I can also perform other tricks in composition.  I imagine at this point that some would ask me why I don't just use JI, and the answer is that even an EDO in the hundreds like 159edo is considerably more simple than JI, as you have to account for a lot of [[unnoticeable comma]]s in JI- a near-pointless endeavor as virtually nobody can hear such small differences in pitch.
-: '''Eufalesio:''' Aura's favorite tuning. He does have a point, it takes an extremely good edo, and tripling it makes it even better! 29-limit goodness! I don't care as much for the insanely accurate 2.3.11, as I care for the entirety of the 2.3.5.7.11.13.19(.29). It really is that good. I've composed stuff with it, and it isn't as easy to do as in other edos, but the result is still worth it. SS
+: '''Eufalesio:''' Aura's favorite tuning. He does have a point, it takes an extremely good edo, and tripling it makes it even better! 29-limit goodness! I don't care as much for the insanely accurate 2.3.11, as I care for the entirety of the 2.3.5.7.11.13.19(.29), on which it is worse than other alternatives, as primes 7 and 13 are relatively innacurate. I've composed stuff with it, and it isn't as easy to do as in other edos, but the result is still decent. SS-
 == [[171edo]] ==
@@ Line 662: / Line 666: @@
 == [[217edo]] ==
-: '''Eufalesio:''' It's the septuple of 31edo, and that is nothing less than a miracle (though it doesn't support miracle). I've done some tests on it, and it's 31-limit is incredible. It introduces an unfamiliar and slightly tedious gari-vulture-esque framework in which you have to use schisma-function steps apart from pythcommas, but apart from that, it's still within the realms of manageability. Also, it has an incredible 2.3.5.13, which I really respect. Also important to remark, from this point onward, edos start to sound more or less the same, as the absolute error gets lower and lower, and the difference between edosteps becomes harder to reliably hear. S
+: '''Eufalesio:''' It's the septuple of 31edo, and that is nothing less than a miracle (though it doesn't support miracle). I've done some tests on it, and it's 31-limit is incredible. It introduces an unfamiliar and slightly tedious gari-vulture-esque framework in which you have to use schisma-function steps apart from pythcommas, but apart from that, it's still within the realms of manageability. Also, it has an incredible 2.3.5.13, which I really respect. Also important to remark, from this point onward, edos start to sound more or less the same, as the absolute error gets lower and lower, and the difference between edosteps becomes harder to reliably hear.–Apart from that, 217edo's 2.3.5.7.11.13.19 mappings can be easily converted to 270edo or 311edo if need be, S+
 == [[224edo]] ==
 : '''Zhenlige:''' Like [[171edo]] but with a slightly sharper (and closer to just) fifth, worse [[7-limit]] but better [[13-limit]].
-: '''Eufalesio:''' A cousin to 217edo which is still schismic, dare I say the ultimate schismic edo, though still harder to conceptualize. Theory tells me that the 13-limit is extremely accurate, even more than the 217edo, and for that I think it deserves some attention. But 217edo is smaller, and it contains 31edo, so... I think I'll stick with the other one. B
+: '''Eufalesio:''' A cousin to 217edo which is still schismic, dare I say the ultimate schismic edo, though still harder to conceptualize. Theory tells me that the 13-limit is extremely accurate, even more than the 217edo, and for that I think it deserves praise. But 217edo is smaller and its mappings can be easily expanded to more accurate edos. B+
 == [[270edo]] ==
@@ Line 675: / Line 679: @@
 : '''Zhenlige:''' Good for very high limit JI.
 : '''Eufalesio:''' Ultimate ultra-high-limit JI. Absolute error is a smidge worse than 270edo, but it makes up by being consistent to the goddamn 41-odd-limit. Serendipity personified. Very hard to justify using anything else other than this, as the difference between edosteps from this point on is definitely nigh impossible to hear. I see it as an ultimate tuning of sorts for practicality's sake. SSS
+== [[665edo]] ==
+: '''Eufalesio:''' Ultimate pyth. It has an unfathomably perfect 2.3, and I say that in an almost literal sense. It is very much fathomable, obviously: the beat period of 665edo's fifth is 5077906.80060 s*Hz with two sawtooth waves in perfect sync, which would be around 3 hours, 12 minutes 21 seconds at f=440. 3 fucking hours. That's what it would take you to hear the beating of 665edo. It is, for all intents and purposes, unfathomable to focused human perception. Or, you could make a 3-hour track out of this. –However, this is not why you would use 665edo, as this essentially allows you to extend the precision limit of the chain of fifths from very good to ''extreme,'' by adding the mercator (+53 fifths) and an equalized qian comma (+306/-359 fifths) into the mix, also working as a schisma. Yes, it has a bad prime 11, but it is surprisingly good in the rest of primes up to the 27-odd-limit, which is very surprising for a convergent. I will likely never use this, but since I do greatly care about the chain of fifths as a theoretical construct, I care about this ''theoretically''. A-
 == [[1600edo]] ==
@@ Line 681: / Line 688: @@
 == [[2460edo]] ==
 :'''Eufalesio:''' The only reason I've put this one here is because it is a 12n edo, and that makes it ''slightly'' easier to work with, and very transposing friendly. It's astonishingly accurate, though dividing the semitone into 205ths is reasonably excessive. C
+== [[7315edo]] ==
+:'''Eufalesio:''' Undecupling 665edo results in what I believe to be one of the potentially theoretically most robust yet precise JI-oid systems. Splitting the equalized qian comma in 11s greatly amplifies the accuracy of this edo and allows you to keep the unfathomably accurate chain of fifths as a strong backbone, and thirteenths of a qian comma serving as nanoalterations. I will likely never use this due to the insane precision it demands, but I have nothing other than respect for this behemoth of an edo. S
 == [[8539edo]] ==
 :'''Eufalesio:''' This level of fineness is at the bleeding edge of insanity. The precision of this behemoth is astounding. I firmly believe no sane person would compose anything requiring a tuning precision higher than what this offers. And I'm one to ogle at impossibly gargantuan edos, I'll admit, but that ogling is only theoretical. Beyond here... there be monsters... and hot sauce. C
+== [[190537edo]] ==
+:'''Eufalesio:''' This is the next edo in the list of record k-strong telicity, and it starts to get scary from this point. This is '''unadulterated cosmic horror''' disguised as math in the 3-limit, forget about the rest of primes. The beat period of this fifth is 984 572 779 224.54 s*Hz with two sawtooth waves in perfect sync, which would be around 70 years and 331 days at f=440. This edo has a fifth that is accurate to a level that is quite possibly beyond the scope of a human lifetime. Think about it. There is a chance you'll die before listening to 190537edo's fifth beat, at f=440. – If you say that you need accuracy to this precision, I am, beyond a reasonable doubt, wholly confident you are not human, or not anymore. Perhaps in a couple centuries, posthumanity will be able to comprehend this near-perfection, but as we stand right now, it is impossible to comprehend. Unrankable.
 == Sources ==