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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
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
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
Individual pages for EDFs
Individual pages for EDFs
9 (Carlos Alpha)
11 (Carlos Beta)
20 (Carlos Gamma)
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.
Very rough correspondence - patent vals disagree in the 5 limit.
Also very rough.
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.
Since 88cET/octacot is well known to approximate some intervals quite accurately,
it would be wrong to lump this in with 14edo.
The Carlos alpha scale is neither 15edo nor 16edo.
10edf is 17edo with 6.6 cent compressed octaves.
Patent vals match through the 13 limit, with the exception of 5 (as expected).
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.
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.
Perhaps surprisingly, this is not very similar to 22edo. Patent vals differ in the 5 limit.
14edf is 24edo with 3.4 cent stretched octaves. Patent vals agree through the 19 limit.
Fairly rough correspondence. 15edf is 26edo with ~17 cent stretched octaves.
Patent vals agree through the 5 limit, but not through the 7 limit.
17edf is 29edo with 2.5 cent compressed octaves. Patent vals disagree in the 7 limit.
Perhaps surprisingly, this is not very similar to 31edo. Patent vals differ in the 5 limit.
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.
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