User:Eboone/EDO Impressions

I, at least currently, work strictly in edos. I don't tend to go any higher than 72edo in my own works, so here are my impressions of all the positive integer edos up to 72. This page is designed to be read wholly from top to bottom, but you can still read specific entries in isolation. Additionally, each edo category also has its own tier list and recap.

Of course, all tier lists and impressions on this page are subject to change in the future.

All demos are short loops at 120bpm created by me in MuseScore 3.

Categorization

The 8 Categories of EDOs 1-72

I have split the edos 1-72 into 8 distinct categories based on step size:

1-4edo — Trivial (300-1200¢)
5-9edo — Macrotonal (133.33-240¢)
10-14edo — Macro-Semitonal (85.71-120¢)
15-19edo — Micro-Semitonal (63.16-80¢)
20-29edo — Macro-Diesitonal (41.38-60¢)
30-39edo — Micro-Diesitonal (30.77-40¢)
40-55edo — Super-Syntonic (21.82-30¢)
56-72edo — Sub-Syntonic (16.67-21.43¢)

These terms are not standard in any way. I just made them up. Also, I am aware that "macrotonal" generally refers to any temperament whose step size is larger than 100¢. That is not necessarily the meaning here.

Why Categorize EDOs?

The main reason I wanted to split these edos into categories was so that this wiki page wouldn't be an absolutely mammoth wall of text on mobile. Plus, it just felt like a fun challenge.

But why by step size? EDOs are generally distinguished by their intervals, so wouldn't it make sense to group edos by similar interval content? Sure, but how would you actually do that? There would be so much overlap that any and all meaningful distinction between categories would be lost. 29edo, 41edo, and 53edo would be grouped together as Pythagorean systems, but 41edo and 53edo would also be grouped with 31edo and 50edo as 5-limit systems, and 31edo and 50edo would be grouped with 36edo and 72edo as septimal systems, and 72edo would be grouped with 24edo and 26edo as undecimal systems...

As you can see, that falls apart quickly (as does attempting to group edos by factors or mosses or anything else, for the same reason), meaning the only other option is to categorize edos by step size. This is a great option because the size of an edo's step is what dictates its voice leading capabilities. For example, chromatic motion in 12edo is limited to semitones, whereas 31edo allows for movement by diesis. Also, no more extreme overlap between categories; 12 is similar to 11 and 13, and 31 is similar to 30 and 32. The borders between categories may be fuzzy, but that's to be expected when categorizing tuning systems anyway, since harmony itself is fuzzy by nature.

In terms of the actual categories themselves, I essentially divided them by significantly different interval regions, and then into smaller subcategories. There are really 4 overarching categories: macrotonal, semitonal, diesitonal, and syntonic, but these still feel too large and contain vastly different edos within themselves. So, I basically just split each category down the middle.

Trivial

1edo

No prime factorization, 1° = 1200¢, No fifth

Tier: F

1edo metal on E

Honestly, can this one even be called an edo? I mean, "equal divisions of the octave" implies that the octave is divided. Anyway, this edo is pretty lame because the whole point of using different temperaments is to access different melodic and harmonic opportunities, and this edo doesn't allow any of that. The only way to make this edo musical is through rhythm, dynamics, form, etc... stuff that isn't related to tuning.

Although, technically there is some room for "melodic" expression if you allow yourself to use devices such as pitch bend. Or, perhaps you could use timbres that don't exactly settle on one pitch but clearly aren't unpitched either, like an old guitar string with wobbly harmonics. These "cheat codes" are partially what make one-note jazz solos so interesting, but of course the rest of the interest comes from the novelty of playing only one note amidst an electron cloud of dissonance. On its own, there really isn't much this temperament offers.

At least it's better than 0edo.

2edo

Prime, 1° = 600¢, No fifth

Tier: F

2edo mischievous theme on E and B♭

This edo is literally just a tritone. And, in this case, it's a rather interesting one. It's the 600¢ hemioctave tritone, the only one that is its own octave complement. So, while its melodic and harmonic capabilities in isolation are pretty bare, its symmetrical nature allows for some cool gimmicks. If instead it were, say, 11/8, that would open the possibility of using both 11/8 and 16/11, its octave complement. This would allow for greater melodic and harmonic expression, but at the expense of having a neat symmetrical tritone.

To illustrate this tritone's nature, the demo for this edo is completely flipped on its head. If you listen carefully, you will notice that the pizzicato violins and celli play each other's parts upside down in the second half of the loop. The gimmick is that this only affects contour, not the harmony. So yeah, that's pretty cool.

Anyway, this edo isn't much. In fact, I can't honestly say it's any better than 1edo. In theory, there should be no reason to choose 1edo over 2edo, but how much does that one extra note really add in practice? At the end of the day, both edos are mere novelties.

3edo

Prime, 1° = 400¢, No fifth

Tier: F+

3edo foggy ambience on D augmented

This edo is just an augmented triad, but specifically one that is derived by stacking three 400¢ major thirds. This means it closes the octave and, as such, is its own inversion. So, once again, the melodic and harmonic capabilities of this triad in isolation are pretty slim compared to, say, one derived by stacking 5/4. Yet, the symmetrical nature of this chord allows for quite a "sturdy" sound that helps to keep the temperament together.

The major third itself is 13.79¢ sharp of 5/4, enough for noticeable inharmonicity to occasionally take place in timbres with a prominent 5th harmonic (such as a piano). This contributes to the "foggy" sound of the temperament, as the demo illustrates.

The uninvertible nature of the augmented triad is illustrated in the synth, as it oscillates between two inversions of the triad yet the quality of the chord remains utterly identical. If it were any other type of augmented triad, the inversions would still sound quite similar but they would each have a different vibe.

Anyway, I'd say this edo is definitely more interesting than 1edo and 2edo, mostly because it is the smallest edo with a triad.

4edo

2², 1° = 300¢, No fifth

Tier: F+

4edo fanfare on B diminished

While I wouldn't necessarily say 2edo is a better version of 1edo, I will say 4edo is a better version of 2edo. It's a diminished tetrad, but specifically one that is derived by stacking four 300¢ minor thirds. This means it closes the octave and, as such, is its own inversion. So, as always, the melodic and harmonic capabilities of this tetrad in isolation are pretty slim compared to, say, one derived by stacking 6/5. Yet, the symmetrical nature of this chord allows for quite a, dare I say, "sturdy" sound that helps to keep the temperament together... how familiar.

The minor third is sat between 6/5 and 32/27, giving it a mildly dark sound that is quite apt for the diminished tetrad. It's a powerful minor third, but without the 5-limit resonance of 6/5. Overall, a great minor third.

I'd say this edo is on equal footing with 3edo, maybe slightly behind. You could certainly argue that it's better than 3edo, considering it contains 2 interlocked instances of 2edo, as well as an extra note. This seems like a convincing case for 4edo, but I personally think augmented triads are more interesting than diminished tetrads in isolation. It's really all about taste, as is everything in the world of music.

Trivial EDO Recap

Tier list of edos 1-4

These first four edos are honestly just memes, so no amount of good things I say about them can eke them out of F tier. I can't really hear them as anything other than subsets of 12edo, which really makes them seem pointless. They work best as additional colors in larger edos that contain them, not as temperaments on their own.

Macrotonal

5edo

Prime, 1° = 240¢, Fifth = 720¢ (3\5)

Tier: B

5edo tribal groove in A equipentatonic

This is the first edo with a perfect fifth, coming in 18.04¢ sharp of 3/2. Because of how sharp the fifth is, 5edo works best in timbres with quick decay (like a marimba), so that the dissonance of the fifth is softened. Although, the sharp fifth when used melodically has quite a desirable "uplifting" quality that brings it out amidst the texture of the rest of the scale.

This edo also features interseptimal intervals, a very interesting class of intervals characterized by toeing the lines between extreme septimal intervals. There's the 240¢ "second-third," which lies between 8/7 and 7/6, and the 960¢ "seventh-sixth," which lies between 7/4 and 12/7. I'd say 240¢ definitely sounds more like a supermajor second than an inframinor third in most cases, and I find it almost impossible to hear 960¢ as a sixth since it's so close to the 7th harmonic, but it is interesting how these intervals are technically ambiguous.

Oh yeah, did I just mention it has a decent approximation of the 7th harmonic? It's only 8.83¢ flat of 7/4, which is cool because this approximation extends to all 5n-edos. Anyway, the equipentatonic scale that this edo sports is a bit reminiscent of the minor pentatonic scale of 12edo (or, dare I say, the dorian pentatonic scale), making it quite familiar to unaccustomed ears, but still with a tang of xen. In many cases, it is rather difficult to notice if a piece is in this edo if you aren't familiar with it. Because of this, I'd say 5edo is a good starting edo when venturing into xen territory.

Overall, this is pretty fun edo to work in. It is fairly limiting harmonically, since it has no third, but the melodies always have a really fun sound no matter how hard you may try to make something sinister. Plus, the fact that this is the only edo with a scale but no thirds is a rather interesting property. It forces you to hone in on nontonal musical factors such as rhythm, dynamics, and form, which I find quite valuable considering how easy it is to hyperfocus on harmony in larger edos. With all that being said, however, I really can't put this edo any higher than B tier because of how non-versatile it is.

6edo

2×3, 1° = 200¢, No fifth

Tier: C

6edo dream sequence in F whole-tone

Of the subsets of 12edo, I'd say this is the most interesting because it contains a whole tone, and it's the only one that isn't just a meme. 6edo contains 2 interlocked instances of 3edo, meaning it has the same echoey major third, and it also contains 3 interlocked instances of 2edo, meaning it has the same uninvertible tritone. The combination of these intervals really contributes to the infamous dreamy quality of this edo, better known as the whole-tone scale.

There are many pieces that make use of the whole-tone scale, but very few that exist strictly in 6edo. Claude Debussy is well known for his use of the scale, and it's a very interesting color to use while working in 6n-edos, but a very limiting scale to use in isolation. The best 6edo pieces I could find are the prelude and invention from Aaron Andrew Hunt's "The Equal Tempered Keyboard," and honestly they are pretty interesting for what they are. They don't really sound dreamy, which is the sound that I feel is most well suited for this edo, but they are actually able to guide the listener on a comprehensive harmonic journey which is certainly no easy task.

6edo is the first edo to contain a dominant 7th chord, which would be more interesting if there were a perfect fifth, but at least you can resolve it through the back door (up a whole step). In fact, that's really one of the only ways you can make satisfying chord progressions in this edo. The arguably more infamous Mario cadence (♭VI-♭VII-I) is ever-present here, as it is, in my opinion, the most satisfying progression that exists in this edo. So, if you really wanted to, you could create a Mario-style fanfare using just the whole-tone scale, but deliberately avoiding the perfect fifth in such a style would be uncharacteristic.

Anyway, it's just the whole-tone scale. I can't reasonably put it in F tier with the rest of the 12edo subsets, since it is an actual scale rather than just an interval or chord, but there's still only one sound I find this edo to be useful for. Not very versatile in the slightest, especially in isolation.

7edo

Prime, 1° = 171.43¢, Fifth = 685.71¢ (4\7)

Tier: A

7edo disco in B neutral

This is the first edo with a perfect fifth and a third. The fifth is pretty flat, coming in 16.25¢ flat of 3/2, and the third is a supraminor third of 342.86¢. In a way, 7edo is the polar opposite of 5edo, containing a fifth that is almost equally off in the opposite direction. This gives 7edo a noticeably less uplifting sound, and more of a "crying out in pain" sound as one of my subscribers put it. Quartal and quintal structures have a distinct "underwater" quality, as opposed to the skyward energy of such structures in 5edo.

The equiheptatonic scale that this edo sports is a bit reminiscent of the dorian scale of 12edo, but not really. The 2nd, 3rd, 6th, and 7th are all neutral, so this edo has a very distinct xen sound. These neutral intervals, along with the rather sour fourth and fifth, have a very intense sound on an acoustic piano, which is why I elected to use electric piano and synth pads in the demo. 7edo also works very well in tinky percussive timbres, like handpan or kalimba. These sorts of sounds are heard in Fuschiamarine by Sevish, a very high quality example of what this edo is capable of.

This edo may not be very versatile harmonically, but its sound is truly something to behold, and a sound that cannot be easily replicated in any other non-7n-edo. A very nice blend of xen, ease of use, and ethereality. It's also trivially easy to notate, which is always a bonus.

8edo

2³, 1° = 150¢, "Fifth" = 750¢ (5\8)

Tier: B

8edo march in E minor (?)

This is... an interesting one. Honestly, I really didn't used to like 8edo, but eventually I came to see how powerful it is when used in certain contexts. It doesn't have a perfect fifth or a whole tone, but it is extremely close to 12/11 equal step tuning. It also features a good approximation of 13/10, a rather ambiguous "third-fourth" that can either be used as an ultramajor third or a very flat fourth.

Cadences in this temperament are relatively satisfying, but not in a traditional sense. The 150¢ neutral second can function cadentially either as a wide semitone or a narrow whole tone, meaning cadences tend to feel a bit obtuse or acute, if you will. It's an intriguing motion that sets the harmony of this edo apart from pretty much any other edo, even other 8n-edos. It's a very foreboding motion that of course must be accentuated by a pipe organ and battle-ready percussion.

8edo contains 2 interlocked instances of 4edo, meaning it houses the familiar diminished tetrad. And, just like in 4edo, the diminished/minor sound is what tends to sound the most appealing here since the 450¢ Barbados third really isn't a stable point of rest. This is mainly why I used to think 8edo wasn't very good, since the room for harmonic expression without the major/minor third dichotomy just seemed bland. Plus, the only stable sounds being a minor chord with an omitted fifth and a diminished chord just didn't seem very appealing.

But, as you can hopefully hear in the demo, I believe to have found the perfect sound for this temperament. It was somewhat inspired by Aaron Andrew Hunt's Fantasia & Fugue a4 in 8ET, once again from The Equal Tempered Keyboard, another piece I find quite fascinating. Other neat environments for 8edo include, as always, the percussive style characteristic of Hideya's music, another one of my favorite xen composers. Like Ensor's paintings is a very cool example of what 8edo sounds like in a strictly melodic context. It's, as Hideya says, an unpleasant experience that really gets you thinking for some reason.

Overall, a bit of a niche edo, but the sounds it produces are ineffable. I found myself struggling to place 8edo on the tier list, because the utter absence of a perfect fifth isn't something to just gloss over, yet its inquisitive properties certainly warrant a placement above 6edo and the melodic nature of the neutral second puts it on par with 7edo. So, I met in the middle and placed it in B tier, above 5edo.

9edo

3², 1° = 133.33¢, Fifth = 666.67¢ (5\9)

Tier: A

9edo tango in A harmonic anti-minor

9edo is the first edo with 2 flavors of thirds. It is notable for its uncanny approximations of septimal intervals down to a fraction of a cent. These intervals have an almost ghostly quality to them, giving this edo an incredibly unique, resonant sound that other edos of this scale simply cannot match.

For example, the juxtaposition of the 266.67¢ subminor third (nearly exactly 7/6) and the 400¢ major third (the one from 3edo) is totally jarring. The incredibly acrid fifth also contributes to the general vibe of resonant septimal consonances battling against sour dissonances. It's an ineffable combination of opposing stimuli, giving this edo an unmistakable sound.

9edo is also the first edo to support mavila temperament, being one of 3 that support it natively in the patent val. It's certainly an extreme mavila temperament, but making the fifth any sharper would get rid of the septimal consonances, so this particular way of approaching mavila is truly unique.

As this edo is right on the border of what I consider to be the macrotonal edos, 1 degree of 9edo has a particular quality to it that makes it sound like a semitone in some contexts and a neutral second in other contexts. My piece Elegy of the Willow illustrates this by including many chromaticisms, sometimes in the context of a chromatic scale and sometimes in the context of cadential motion.

Other inspiring works in this temperament include Buttered Cat Paradox by Xolta and Caelum by miaxia. Caelum especially illustrates the nature that I've described, leading me to believe 9edo works best in such a style. The demo I wrote for this edo, however, is nowhere near the style of any of these pieces, yet I think it works quite well. The aforementioned "ghostly" quality of 9edo, particularly in the fifth, gives the tango an "abandoned" quality that would fit well in a old lost town, like the Hidden Village from The Legend of Zelda: Twilight Princess.

Overall, a very solid edo. You can choose to avoid the dissonant fifth or embrace it, with both approaches leading to an incredible sound. It's surprisingly versatile, more so than any edo smaller than it, putting it above 7edo.

Macrotonal EDO Recap

Tier list of edos 5-9

These edos are very cool and interesting. Just because they don't have very many notes doesn't necessarily mean they are limiting. Even 5edo, the most limiting of them all, has much room for expression through its interseptimal intervals. 9edo is especially intriguing to me, as it is the smallest edo with 2 flavors of thirds, making it the first edo to allow for any sort of complex chord movement. It is a beautiful edo that ends up being more versatile than you'd think, along with 7edo and even 8edo.

If you're in the market for edos that approximate simple ratios very well, you won't find much here. Sure, 5edo has a decent 7th harmonic and 9edo has spookily accurate approximations of 27/25 and 7/6 among other septimal intervals, but these edos shine in their simplicity. The main appeal here is how compact these edos are, and it's just really neat to think that you don't even need 12 notes to make awesome music.

Oh, and I just realized I have barely talked about notation! Fortunately, all five of these edos are relatively easy to notate. 5edo and 9edo are notable in this regard; 5edo can be notated pentatonic-wise on a 5-line staff by skipping B and F, and 9edo can be notated with either harmonic or melodic mavila accidentals. I prefer to preserve the harmonic integrity in mavila systems; I embrace the quirk of having reversed sharps and flats so I don't have to deal with F♭ being a perfect fifth above B. 6edo can be notated as a subset of 12edo, 7edo can be notated by simply using all seven naturals, and 8edo can be notated as a subset of 24edo with quarter-tone accidentals. As you will see with future edos, however, things will not be as simple.

Macro-Semitonal

10edo

2×5, 1° = 120¢, Fifth = 720¢ (3\5)

Tier: A+

10edo heist theme in ^⌄A bish (3|3 mode of 3L 4s)

Ah yes, the homeland of mosh. 10edo is actually my favorite of the edos less than 12, although 9edo does come pretty close. This edo is the largest one with a singular third, that being the 360¢ submajor third which is extremely close to 16/13. That means this temperament also features a stunning approximation of the 13th harmonic, coming in just 0.53¢ sharp. It also features all of the interseptimal intervals from 5edo, meaning it includes the same approximation of the 7th harmonic, as well as a relatively stable fifth (especially compared to 9edo).

10edo is a very xen tuning — more so than even 7edo in my opinion — because it doesn't even pretend to represent the diatonic scale (5L 2s). Instead, as mentioned previously, the heptatonic mos of 10edo is 3L 4s, or mosh. This is an incredibly intriguing mos that can be difficult to wrap your head around, but the sounds it produces are quite beautiful. Apartment in the Sky by HEHEHE I AM A SUPAHSTAR SAGA is an example of the brightest mode of mosh, dril.

The demo I wrote for this edo is in the middle mode, bish. It oscilates between "tonic" and "dominant," and it sounds quite reminiscent of such movement in the diatonic mos, until you realize it's totally different. This mode of mosh could be mapped to the double harmonic major scale in a diatonic system, yet doing that results in a very different sound. I was incredibly surprised when I retuned the demo to be in A♭ double harmonic major, only to be subjected to something the original could only dream of being as weird as. Of course, I probably only say that because I wasn't used to it.

Being the first edo with what I would consider a semitone, chromatic movement in this edo is relatively familiar, yet with a tang of xen. In fact, that's a fairly apt way of describing this temperament as a whole. The third feels like a major third sometimes, especially when combined with the 1080¢ major seventh, yet other times it feels totally neutral or even minor. It certainly doesn't feel minor in isolation, but it kinda does at some points in the demo. This supposed submajor third is even more ambiguous than the 240¢ second-third, which is literally ambiguous by definition.

Notating this edo is an interesting challenge. I choose to notate it with the seven naturals assigned to mosh, with A dril lying on the naturals, but this certainly isn't standard. The "standard" (if you can call it that) is to notate this edo pentatonic-wise, using ups and downs to reach the notes in between. I, however, am not a particular fan of ups and downs. I prefer sagittal, but even that looks a bit clunky in this edo. The problem is the fact that B and F are skipped when using this method, or that B=C and E=F. I just think that looks and feels weird to read because it's based on the diatonic mos which 10edo doesn't have.*

Instead, it makes much more sense to assign the naturals to mosh and use sharps and flats to reach the notes in between. Or, you could use Diamond-mos notation, which is what Apartment in the Sky is notated in. My problem with this system is how difficult it is to actually implement, plus the idea of transposing a score instead of using key signatures is not my style. I'd rather just write in the accidentals manually than transpose the entire score, unless it's an extreme case.

Anyway, notation aside, this edo has such an amazing sound. Every 5n-edo (especially those up to 30edo) has a particular 5n-edo sound, yet this edo stands out even among that group. Being the largest edo with a singular third, it really cant help but sound totally xen, even though its semitone step-size tends to make it sound almost like 12edo in certain cases. Mosh is a super cool landscape, but as Adam Neely shows in How to make Microtonal Lo-Fi Hip Hop, you can just as easily stick to the 2 interlocked equipentatonic scales and produce some great music.

Overall, a very unique and awesome edo that stands atop the macrotonal podium as the best edo smaller than 12edo (in my opinion).

*This problem creeps up in all 5n-edos up to 30edo, and while I say I prefer to assign the naturals to a different mos rather than stick to the diatonic mos which isn't even supported, I really only mean that for 10edo. 15edo is the other most common 5n-edo with the 720¢ fifth, and trying to notate it any other way than with accordance to the diatonic mos ends up looking weird. Even still, I do not like equating natural pitches, so I still choose to omit B and F. At the end of the day, some edos just aren't fit to be notated on a 5-line staff generated by fifths.

11edo

Prime, 1° = 109.09¢, "Fifth" = 654.55¢ (6\11)

Tier: C

11edo casino swing in A altered lydian (?)

This is a... fun one. It suffers from being almost 12edo, giving it a somewhat similar vibe to 9edo, but without the whole "battle between consonance and dissonance" thing since nearly everything here is dissonant. The fifth is more like 16/11 than anything close to 3/2, meaning it actually has the worst fifth since 6edo, although 8edo comes close. This edo is notable for its approximation of 9/7, being just 1.28¢ sharp, making it the first edo with a supermajor third. It also has a pretty good approximation of the 11th harmonic, being 5.87¢ flat.

While 10edo is the home of mosh, 11edo is the home of smitonic. I don't really like it as much as mosh, mostly because its fifth is never a good approximation of 3/2, but it has its uses. Anyway, I don't really dabble with smitonic in this edo. I prefer to write free of any scale, since this edo is already pretty limiting as it is. That's why the demo is full of jazz chords.

As always, The Equal Tempered Keyboard has some good examples of this edo. They're rather beautiful for what they are, but I find myself wishing they were in a different edo. Tostadosto by Francium and Longwayaway People by Sevish are other great examples of what this edo is capable of.

The best way to notate this edo is as a subset of 22edo, which also isn't doing it any favors. Overall, a niche edo with an alien sound that I just cannot find myself wanting to use very often. Its approximation of the 11th harmonic is somewhat appealing, and the "almost 12edo" quality can make for some interesting effects, but there are better edos for both of these things.

12edo

2²×3, 1° = 100¢, Fifth = 700¢ (7\12)

Tier: S+

12edo chorale in B major

I could say many things about 12edo, but I don't think you need to hear it. It's the first edo with the diatonic mos, the first edo with a decent approximation of 3/2 (1.96¢ flat), and the first edo that can claim to be a 5-limit system. It's a meantone temperament, but only a little. This means the thirds are closer to pure 5-limit ratios, but not enough to sacrifice the edo's ability to support Pythagorean melodies. It contains symmetrical augmented and diminished structures, as well as the whole-tone scale. It's essentially a perfect temperament that works well in basically any style, so I cannot honestly put it anywhere but the top of the tier list.

Yeah, that's all I'm going to say about 12edo. The purpose of this page is to enlighten you about the other 71 edos from 1-72, above stating my own opinions.