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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

4edf5edf

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

Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents.

Patent vals match through the 5 limit. Only a rough correspondence.

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.

it would be wrong to lump this in with 14edo.

Patent vals match through the 13 limit, with the exception of 5 (as expected).

Patent vals match through the 7 limit.

If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence.

so it strongly resembles the scale with generator 2\41 of an octave.

Patent vals agree through the 5 limit, but not through the 7 limit.

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.