User:Eboone/EDO Impressions: Difference between revisions

Revision as of 16:30, 22 May 2024

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.

Of course, the tier list above is subject to change in the future, as are all of the impressions on this page.

All demos are created by me in MuseScore 3. They are all short loops at 120 bpm and are meant to be in a style I think works well in that edo.

EDOs I haven't gotten to yet are marked with "placeholder."

Categorization

I have split these edos into categories based on step size:

1-9edo — Macrotonal (133.33-1200¢)
10-19edo — Semitonal (63.16-120¢)
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.

Macrotonal

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.

By the way, you won't be seeing F tier for a while after this. 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. Now, on to the edos people actually care about.

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 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 the fifth, gives the tango an "abandoned" quality that would fit well in an abandoned ghost 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. More people should use 9edo.

Semitonal

10edo

Tier: A+

10edo heist theme in G bish

placeholder

11edo

Tier: B

11edo casino swing in A altered lydian (?)

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 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 xenharmony.
+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.
 By the way, you won't be seeing F tier for a while after this. 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. Now, on to the edos people actually care about.

User:Eboone/EDO Impressions: Difference between revisions

Revision as of 16:30, 22 May 2024

Categorization

Macrotonal

Semitonal

Macro-diesitonal

Micro-diesitonal

Super-syntonic

Sub-syntonic

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