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We're used to 12 EDO or maybe 19.

But would it make sense to have a fractional value, say 12.5 EDO?

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    Maybe it's worth noting that if the divisions have a fractional value, then they are by definition not equal.
    – PiedPiper
    Oct 22 '21 at 21:55
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A fractional value would represent an equal division of multiple octaves rather than a single octave. 12.5 EDO, for example, would be equivalent to 25 ED2O (equal division of two octaves).

As an abstract concept it's certainly possible, but it's not clear what the musical motivation would be. Further, one could achieve 12.5 EDO by using 25 EDO and then composing using only every other pitch.

  • Would it make sense as an abstract idea? Yes.
  • Would it make sense as a system for composition? Matter of opinion.
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    Electronic composers have done this in the past. To me, such efforts come across more experimental than musical. There’s some work that has been done with dividing three octaves, even. Oct 21 '21 at 15:14
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    x EDO, if x is not a rational number, won't be even an equal division of multiple octaves. Would it make sense? As much or as little as other exotic systems. If we depart from 12 EDO, does it make much more difference if we depart from the octave as well? Oct 21 '21 at 22:10
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    The musical motivation IMO is to give access to more and/or juicier intervals that you can get with low-number EDOs, but without creating an overly-crammed spectrum like in e.g. 31-edo or 34-edo where it gets different to even distinguish adjacent scale steps. Oct 22 '21 at 13:00
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    @Aaron yes (I upvoted all the answers) but IMO the Carlos scales, like Bohlen-Pierce, shouldn't be labelled EDOs at all, because they eschew the very concept of octaves. An example of what I mean is 17.5-edo, which has good approximations to 9:8 (whole tone, allows for melodies that don't sound too alien), 5:2 (major tenth, can be used for third-ish counterpoint), 7:4 (harmonic seventh for that extra oomph) and of course 4:1 (double octave, which allows amongst other things for convincing final chords). Oct 22 '21 at 14:37
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    @leftaroundabout Although the Carlos scales and Bohlen-Pierce do not have an octave, they do divide the octave equally. Mathematically-speaking, if you divide one interval equally, you divide them all equally.
    – Theodore
    Oct 22 '21 at 15:06
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Wendy Carlos used her Alpha scale at about 15.385 steps per octave and Beta scale, approximately 18.8 steps per octave on "Beauty in the Beast". They give up octaves altogether in favor of closer approximations to other just intervals.

This makes sense when you consider that 12EDO is a compromise having just octaves and decent fifths, but poor thirds and really poor [harmonic] sevenths.

I have experimented a bit with more subtle non-octave equal scales that end up close to 12EDO: They have a pseudo-octave that's pretty close to a just octave, but with some other intervals tuned just, or closer to just than they would be in 12EDO:

  • 28 equal divisions of 5:1, with just-intoned 17ths (2 octaves + M3) and everything else compromised a bit. The pseudo-octave is a little flat at 1194¢.
  • 34 equal divisions of 7:1, with JI harmonic m21 (2 octaves + m7) and 1189¢ pseudo-octave.
  • 22 divisions of 7:2, with JI m14 (1 octave + m7) and 1183¢ pseudo-octave.
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  • Hope this gets voted up more...it's pretty rare, but obviously here the answer is yes, as Carlos' pieces are examples where it was done for musicality (rather than just as a random experiment.)
    – Beska
    Oct 22 '21 at 14:50
  • @Beska Unfortunately this piece (and most of Carlos' oeuvre) is hard to find for listening, since it is out of print and not available for streaming or download. Even used, an LP costs nearly US$100.
    – Theodore
    Oct 22 '21 at 14:59
  • Darn...I hadn't realized this...I'd known of it, and had checked a bit of it out a long time ago, and this post made me realize I wanted to explore it a bit more. Doesn't look like that's going to happen.
    – Beska
    Oct 22 '21 at 15:16
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The Bohlen-Pierce scale divides the perfect twelfth into 13 equal parts. This is equivalent to roughly 8.2 divisions of the octave.



Here's a video explaining the system:

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    This is true, but IMO it's a missing the point to label Bohlen-Pierce as 8.2-edo. The proper label is 13-edt. Oct 22 '21 at 12:24
  • Also I find it a bit silly to write monophonic music in a tuning like Bohlen-Pierce, because there's no way the ear can latch onto its key features this way. Oct 22 '21 at 13:03
  • One of the examples is a "12 bar blues"... pretty silly. Oct 22 '21 at 16:21
  • The Between Secrets example is quite neat. I also like Elaine Walker's Bohlen-Pierce music. (She arguably uses is as kind of as a fractional-EDO tuning, incidentally.) Oct 22 '21 at 19:44
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Expanding on my comments to Aaron's answer.

I consider fractional-EDOs to make mostly sense as divisions of multiple octaves. Technically speaking you can of course interpret any equal-steps scale as a real-valued EDO, but if the scale doesn't utilise octaves then I don't find it useful to define it in terms of octaves.

Now, why would you want to divide multiple octaves, instead of a single one? Well, first consider why we divide octaves in the first place. It's not something we think about a lot because the octave is so utterly central and taken-for-granted in music – not just Western but also traditions from around the globe.
The best way to realise the importance of octaves is probably to write some music that deliberately avoids them. To me, the most obvious issue is that it gets tricky to convey finality. In both common-practice music and rock etc., you tend to have both the melody and bass landing on the tonic... in different octaves. (Of course it doesn't always have to be done this way, but it's definitely something you'll find limiting if you don't even have that option, as in Bohlen-Pierce tuning.) But more often than not, there will actually be a spacing of at least one octave in between – so for this concern what seems to be most important is to have some multiple of an octave.
Another reason to have octaves is that it allows you to invert chords. TBH I think this is a bit overrated, because the inversions of a chord are in many ways actually quite different from its base position. Inversions definitely don't work as easy if you always need to jump at least two octaves at a time.

In addition to octaves, you'll want other, more interesting intervals – for purposes of both melody and harmony. What particular intervals – well, that depends on style. Different fractional EDOs will support different intervals.

One starting point would be to look for fifths and thirds, so we can have our common chords. Another option is to focus on melodic intervals, in particular whole steps. And then you can of course start with any 7-limit or 11-limit experiment your heart desires...

One concrete tuning I've been experimenting with lately is 17.5-edo, in other words 35-equal divisions of 2×octaves. That gives you amongst others the following intervals

  • 3 steps = 9.009 : 8, a very good Pythagorean whole-step.
  • 12 steps = 8.042 : 5, a minor sixth, inverted 23 steps = 4.974 : 2 major tenth. These 5-limit intervals aren't near-perfect like they are in 31-edo, but slightly better than the 12-edo approximations.
  • 14 steps = 6.964 : 4, a Barbershop-ey harmonic seventh.
  • 8 steps = 10.98 : 8, an 11-limit wide fourth.

Here's a short example of music written in this tuning (score using the nearest quarter-tone accidental for each pitch):

score for 17.5-edo example

https://soundcloud.com/leftaroundabout/175-edo-demo-0

...now, of course you could now argue that I could as well have written this piece in 35-edo, and just not used any odd-numbered scale degrees. Sure, that's a way to look at it.
But would I? Probably not. Having lots and lots of nearby notes at your disposal can make it only more difficult to find ones that actually work well musically.

And definitely, building actual instruments in 31-edo, 34-edo or 35-edo is a lot more challenging than in something like 17.5-edo.

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  • I think idea of finality is right-on. Most of the non-octave scales I've experimented with have had some kind of pseudo-octave. A notable exception was a just division of 7:5 which had a 1165¢ interval, but needed 3 octaves to come close enough to be considered a near octave: (7/5)^6×(21/20)≅7.91 or 3580¢. The intent of that composition was to avoid resolution and remain unsettled.
    – Theodore
    Oct 25 '21 at 13:19

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