EDF

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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

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