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Division of the perfect fifth (3/2) into n equal parts

Division of the 3:2 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of equivalence is still in its infancy. The utility of 3:2 as a base though, is apparent by being one of the strongest consonances after the octave. Many, if not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.

Perhaps the first to divide the perfect fifth was Wendy Carlos ( http://www.wendycarlos.com/resources/pitch.html). Carlo Serafini has also made much use of the alpha, beta and gamma scales.

Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes six 5/4 to get to 9/8 (tempering out the comma 15625/15552. So, doing this yields 9, 11, and 20 note MOS which the Carlos scales temper equally. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it if it hasn't been named yet, but in any case here is an example of it.

Individual pages for EDFs | EDO-EDF correspondence

Individual pages for EDFs

4edf
5edf
6edf
7edf
8edf (88cET)
9 (Carlos Alpha)
10edf
11 (Carlos Beta)
12edf
13edf
14edf
15edf
16edf
17edf
18edf
19edf
20 (Carlos Gamma)
21edf
22edf
23edf
24edf
25edf

EDO-EDF correspondence

EDO
EDF
Comments
7edo
4edf
4edf is 7edo with 28.5 cent stretched octaves.
Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents.
Patent vals match through the 5 limit. Only a rough correspondence.
8edo


9edo
5edf
Very rough correspondence - patent vals disagree in the 5 limit.
10edo
6edf
Also very rough.
11edo


12edo
7edf
7edf is 12edo with 3.4 cent stretched octaves.
Equivalently, 12edo is 7edf with 2.0 cent compressed 3/2s.
With the exception of 11 (which falls almost exactly halfway between steps in both cases),
the patent vals match through the 31 limit, so the agreement is excellent.
13edo



8edf
Since 88cET/octacot is well known to approximate some intervals quite accurately,
it would be wrong to lump this in with 14edo.
14edo


15edo



9edf
The Carlos alpha scale is neither 15edo nor 16edo.
16edo


17edo
10edf
10edf is 17edo with 6.6 cent compressed octaves.
Patent vals match through the 13 limit, with the exception of 5 (as expected).
18edo


19edo
11edf
11edf is 19edo with 12.5 cent stretched octaves.
Patent vals match through the 7 limit.
If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence.
20edo



12edf
12edf entirely misses 2/1, but nails the "double octave" 4/1,
so it strongly resembles the scale with generator 2\41 of an octave.
21edo


22edo



13edf
Perhaps surprisingly, this is not very similar to 22edo. Patent vals differ in the 5 limit.
23edo


24edo
14edf
14edf is 24edo with 3.4 cent stretched octaves. Patent vals agree through the 19 limit.
25edo


26edo
15edf
Fairly rough correspondence. 15edf is 26edo with ~17 cent stretched octaves.
Patent vals agree through the 5 limit, but not through the 7 limit.
27edo



16edf

28edo


29edo
17edf
17edf is 29edo with 2.5 cent compressed octaves. Patent vals disagree in the 7 limit.
30edo



18edf
Perhaps surprisingly, this is not very similar to 31edo. Patent vals differ in the 5 limit.
31edo


32edo



19edf

33edo


34edo
20edf
20edf is 34edo with 6.6 cent compressed octaves.
Patent vals match through the 5 limit, but not the 7 limit.
If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence.
35edo


36edo
21edf

37edo


38edo
22edf

39edo
23edf

40edo


41edo
24edf

42edo


43edo
25edf