User:Eufalesio/EDO impressions

My main reason to use edos is to "buy" the entire gamut and be able to do JIoid stuff in it with the most accuracy, the least amount of pitch classes, and the most conceptualization ease within a chain-of-fifths framework.

I thus value edos that have a manageable grain, approximate a lot of stuff, and allow easy chain-of-fifths frameworks; It's easier for me to think in tempered commas along the chain of fifths. I care about the ya (ya), yaza (yaza), yatha (2.3.5.13), yazatha (2.3.5.7.13), and yazalatha (13-limit) JI subgroups, liking my error to be balanced across primes, but the error on 3 to be minimal so that I can work within a chain-of-fifths framework. I can't not have a fifth, and it can't be badly out of tune. If there is no diatonic, it's outright useless to me. How out of tune do I allow it to be? Between 24\41 and 38\65 both inclusive, including multiples. From primes 17 onwards (and specially 19) I only care about the octave-reduced harmonics, possibly up a fourth or a fifth, and nothing else. Generally though, I only do yazalatha.

I will make an exception to meantone and compton; I do like meantone as it is arguably one of the best ya temperaments, and it is obviously the theoretical backbone of Western music theory still today. Compton because 12edo is very powerful, and some of its supersets are arguably among the best edos.

I'm not a strict octave purist; I tolerate tempering the octave to achieve a better harmonic palette, and often do so to achieve higher consistency or better intonation, sacrificing a tiny bit of error on the octave to approximate all the harmonics I care about better. I don't mind having the subharmonics being worsely tuned, as they are not mirror images of harmonics, they are their own thing.

EDOS I have things to talk about (and it's good)

5edo

The first usable edo and the first edo to have any semblance of a perfect fifth. It features an extremely simplified za, forming a consistent circle of 8/7 and 3/2. Due to its extremely coarse grain, it is extremely simple to use, as each step is large enough that no cluttering will ever occur. It is horrible in all other limits... well... except... zathisa, but who here cares about that?

Also due to its extremely coarse grain, you can play anything, short of bashing keys and sitting on the keyboard, and it will sound good. This is because the edostep is so large that it doesn't cause audible cluttering... unless you're playing too low. The sonic profile of this edo is immediately recognizable.

It greatly benefits from non-harmonic timbres, or bell-like sounds, much like that of slendro. Its melodic capabilities are basically the same as that of all pentatonic scales, which is to say: great! It would be wrong to only call 5edo innacurate. A better descriptor would be: coarse. The coarsest, in fact. And due to the fact that it is so coarse, but it is still quite relatively accurate, it is a great edo. A

7edo

The second usable edo. Its fifth is a tad flat, but usable, and it is also the first edo to feature a heptatonic scale, obviously... an equalized diatonic. The edostep is now small enough so that cluttering can occur, but still somewhat bashable. The sonic profile is also immediately recognizable. While the ya is not there, the melodic coolness you can pull of with this coarse edo are nothing to scoff at. C, not for accuracy, but for cool.

12edo

It's easily one of the best edos. 12edo is many, many things all at once, and I would say that its place in mainstream music is well earned. It's the first edo that can deal with the ya at all, and it also has an incredibly accurate fifth for its size, making it the only temperament that is both meantone and pythagorean at the same time... Super practical, and very easy to conceptualize. S

17edo

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 ya 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+

19edo

The next most easily accessible edo, and one that offers a change in perspective. It is meantone just like 12edo, but it has a completely different sonic profile, due to the flatter ya, which I find very enjoyable. The near just minor thirds are definitely something to remark, though not as discernible as it is in its supersets. Great stuff! A

22edo

While the change in perspective that 19edo offers is mixed with familiarity, 22edo is an entirely different beast. It features a very exaggerated non-meantone ya, making it the ultimate porcupine, which is not a temperament known for its accuracy, but it's cool! It also supports magic, featuring a flatter 5, which I enjoy, though the incredibly sharp 6/5 is a tad excessive.

The yaza structure inside the diatonic scale is something very sui generis, though it's yazala is kinda meh, but what can I say, it's the first edo to be consistent in the 11-odd-limit! C, not for accuracy, but for cool.

24edo

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+

31edo

The best meantone edo. Manageable grain, incredible yazala. You can't get more juice out of meantone without diminishing returns. From this point on, it becomes hard to justify using a finer meantone gamut. SSS

34edo

17edo, but good. By splitting the edo in two, we get a great ya, the best so far. Fails at 7 and 11, which is sad, but at least is a very good yazatha, 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+

41edo

The first usable schismic edo (29edo and 17edo don't count because their 5/4's are wack). Still manageable grain, hyperaccurate fifths and the non-meantoneness is definitely welcome. It is the first edo to introduce a comma accidental framework, which in my opinion is one of the best frameworks for composition. The yazala is marvelous (pun intended) but the yazalatha is... lacking. However, since it tempers so many things together, it is extremely useful.

Still, even if the ya is not that accurate, since the innacuracy is flatwards, I think it's much more enjoyable, as I like wide minor thirds. Also supports Bohlen Pierce, which is also incredibly cool. AC, not for air conditioner, but for accuracy and cool.

46edo

The best diaschismic in my opinion. Yazalatha 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-

50edo

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-

62edo

Keeps all the yazala 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-

53edo

Pythagorean tuning incarnate, and astounding ya. yathana is especially potent, but the .7.23 is still very much usable, even the .11! It doesn't temper as many things together as 41edo, so it feels like a less compromised system, still, I feel bad for the rest of the edos near this one, because this trumps a lot of the competition. But what can I say? Suck it losers! SS

72edo

The first compton edo that achieves any semblance of JIoid goodness. This was one of the first finer edos I've composed in. It has an astounding yazala, and decent 19-limit! It's also a multiple of 12, so it is very transposing-friendly and building it is trivial! It's a miracle, and it also supports it! SSS

80edo

Ultimate diaschismic. Extremely surprising that this "coarse" edo can be used all the way up to the 29-odd-limit. I ran some tests on it once, and while the monotone approximations are definitely interesting, the diaschismic framework is not one I'm too comfortable with, and its approximations are a tad sharp, requiring octave compression for a better otonal result. Despite that, I can recognize and appreciate the power of this edo. C+

84edo

I haven't composed anything in it, but theory tells me that it's a really good compton edo. The bad tuning of the 11 is a bit sad, but it can be useful all the way up to the 31-limit. The yazatha here is instead a great subgroup, which is a good selling point for me. Had I known about it, I could have probably used this instead of 72edo, but I'm now not that interested in compton anymore. A-

87edo

Theory says that it is a really strong yazalatha 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

GOAT. The combination of the two smallest schismic edos, which are both incredibly solid choices. However, 41edo is a tad too tempered and 53edo is a tad too untempered and too close to using pythagorean tuning as if it were yazalatha. However, 94edo is tempered just enough, into a neatly optimized cassandra package. I am heavily biased towards this, as it represents the ultimate cassandra, the simplest extended chain-of-fifths framework which I find extremely easy to work with.

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 ya, but since it is flat and not sharp, I find it much more palatable, as I like slightly wide minor thirds. I really only use it for the yazalathana, but the 23-limit goodness is no joke. Perfect score.

130edo

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 a very accurate yazalatha, and a schismic chain-of-fifths framework? Cool. Only problem... It's rastmic. Rastmic is a tad innacurate of a framework for an edo in the hundreds, but oh well. S

159edo

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 ila, as I care for the entire yazalatha, 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

Ennealimmal, personified. I haven't composed in it directly, but since I did compose in yaza JI, it'd sound the same. It features a dead-accurate yaza that I cannot distinguish from just. It's that good. The great innacuracy of the prime 11 is a bit sad, though it still has a usable yazatha, which has that going for it. A

217edo

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's a simple high-accuracy system with very useful subsets. It introduces the cassaschismic framework in which you have to use 3-5 cent steps apart from the pyth spine, but apart from that, it's still within the realms of manageability. Also, it has an incredible yazatha, which I really respect. S+

224edo

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 yazalatha is extremely accurate, even more than the 217edo, and for that I think it deserves praise. However, 217edo has 31edo as a subset, whilst 224edo only has powers of 2 which suck ass as subsets. B+

270edo

Ultimate low complexity JIoid edo. Though a tad large now, consistency and accuracy within its yazalathana is insane. I've done some tests with this and this is straight up unbeatable. For practically all musical purposes, this sounds like JI. There is literally no edo better than this within reason, as its step size is still discernible (by me (in mid frequency range)). Perfect score.

311edo

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. Since its yazalathana mappings are the same as 270edo, it can be used to retune pieces for much more in tune otonality, but I'm not one to do that. This is my boundary of practicality. From this point on, edosteps begin to blend with each other and using JI is most surely a better option. SS

665edo

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 and sell it to connoisseurs disenchanted with the mainstream.

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 think this is good. A-

7315edo

Undecupling 665edo results in what I believe to be one of the potentially theoretically most robust yet precise JI-oid systems beyond practicality. 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

If 311edo was my boundary of practicality, 8539edo is my boundary of sanity. The precision of this behemoth is unfathomable. 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

EDOS I have things to talk about (and it's bad)

0edo

Rhythm personified. As an edo, it is horrible. There is nothing. Everything is tempered out. To an extent, it not only is useless, it's also ontologically terrifying. The end of pitch. However, going back to the real world, this is just glorified rhythm, and so useless from a tuning standpoint. FFF

1edo

Octaves. Extremely boring to use still, as octaves are hyperconsonant, so there is no inertia. You really have to get creative to make something cool in this. Ligeti pulled it off. But I won't care to try. F

2,3,4,6edo

Only ever good as subsets of other edos such as 12edo. Basically just compton. Anywhere else, they stand out, and not positively, though 3edo has a surprisingly accurate 5. D (D+ for 3edo)

9edo

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.

16edo

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

26edo

The ultimate flattone. I respect that this edo can moderately deal with the yazalatha, as it is the smallest edo that is consistent in such. However, its intonation is too out of wack for my taste. Interesting as it simplifies the yazalatha a lot, potentially useful to those who are interested in meantone, but this is a very extreme meantone. C-

29edo

It's the next edo which has a fifth that's better than 12edo's... and that's it? It's worse everywhere else! By itself, it's really only a slightly worse Pythagorean tuning, which to me is a bad selling point. The fact that it's consistent in the 15-odd-limit is useless to me in this instance. The supersets 58 and 87 are decent, but I think there are better alternatives. E

43edo

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

81, 131... very fine edos that support golden meantone

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

55edo and other fine very sharp meantonoid edos

Even worse than 43edo. In fact, it's a zeta valley edo, which means that it does a bad job at approximating JI, and that in my eyes is a failed edo. I don't know what the Mozarts were on while they suggested a meantone this sharp... it's not good at all! FF

282edo

It is surprisingly 94*3, taking all the worse mappings of 94edo and improving them to reach 29-limit consistency. However, 311edo which is only a bit larger is a much better alternative for high-limit, and 270edo is a way better choice for yazalatha. There is no reason to use this unless to improve the intonation of a fixed, non-negotiably 94edo piece. D

190537edo

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 even once, at f=440.

If you say that you need accuracy to this precision to do any semblance of a musical work, I am, beyond a reasonable doubt, wholly confident that you are joking or neck deep in bullshit. And if for some god-removed reason you aren't... you are literally 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.

EDOS I don't have much to talk about

I'm just going to say they are useless because they have horribly tuned fifths. 8,11,13,14,18,20,21,23,25,30,35,40,42,45,47. FFF

If an edo is not anywhere in this article is because I believe there are better options, or that I haven't even thought about it.