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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.
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 http://www.wendycarlos.com/resources/pitch.html]). [[Carlo Serafini]] has also made much use of the alpha, beta and gamma scales.
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 [http://www.youtube.com/watch?v=x_HSMND6RnA example] of it.
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 [http://www.youtube.com/watch?v=x_HSMND6RnA example] of it.
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*[[174edf]]
*[[174edf]]
*[[175edf]]
*[[175edf]]
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=EDF-EDO correspondence=
=EDF-EDO correspondence=
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| | [[16edf]]
| | [[16edf]]
| |  
| |  
| |  
| | 16edf falls exactly halfway between 27 and 28 edos. It entirely misses 2/1, and just barely does not miss the "double octave" 4/1.
|-
|-
| | [[17edf]]
| | [[17edf]]
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| | [[43edo]]
| | [[43edo]]
| | 25edf is 43edo with 7.4 cent stretched octaves, but a rough correspondence. Patent vals differ in the 5 limit.
| | 25edf is 43edo with 7.4 cent stretched octaves, but a rough correspondence. Patent vals differ in the 5 limit.
|-
|[[26edf]]
|
|Perhaps surprisingly, this is not very similar to [[44edo]] or [[45edo]]. Patent vals differ in the 5 limit.
|-
|-
| | [[27edf]]
| | [[27edf]]
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| | [[48edo]]
| | [[48edo]]
| | Same 3.4 cent octave stretch as 7edf~12edo. Patent vals match through the 5 limit, but not the 7 limit.
| | Same 3.4 cent octave stretch as 7edf~12edo. Patent vals match through the 5 limit, but not the 7 limit.
|-
|[[29edf]]
|[[50edo]]
|29edf is 50edo with 10.27 cent stretched octaves.
|-
|[[30edf]]
|51edo
|Same 6.6 cent octave compression as 10edf~17edo.
|-
|-
| | [[31edf]]
| | [[31edf]]
| | [[53edo]]
| | [[53edo]]
| | 31edf is 53edo with 0.12 cent stretched octaves. Patent vals match through the 61 limit.
| | 31edf is 53edo with 0.12 cent stretched octaves. Patent vals match through the 61 limit.
|-
|[[32edf]]
|55edo
|32edf is 55edo with 6.485 cent stretched octaves
|-
|-
| | [[34edf]]
| | [[34edf]]
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| | [[60edo]]
| | [[60edo]]
| | Same 3.4 cent octave stretch as 7edf~12edo. Patent vals match through the 7 limit.
| | Same 3.4 cent octave stretch as 7edf~12edo. Patent vals match through the 7 limit.
|-
|[[36edf]]
|
|Perhaps surprisingly, this is halfway between [[31edo|61edo]] and [[62edo]].
|-
|[[37edf]]
|[[63edo]]
|37edf is 63edo with 4.78 cent compressed octaves.
|-
|-
| | [[38edf]]
| | [[38edf]]
| | [[65edo]]
| | [[65edo]]
| | 38edf is 65edo with 0.71 cent stretched octaves. Patent vals match through the 11 limit.
| | 38edf is 65edo with 0.71 cent stretched octaves. Patent vals match through the 11 limit.
|-
|[[39edf]]
|[[67edo]]
|Surprisingly, 39edf is actually 67edo with 5.92 cent stretched octaves
|-
|[[40edf]]
|68edo
|Same 6.6 cent octave compression as 10edf~17edo.
|-
|-
| | [[41edf]]
| | [[41edf]]
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| | [[72edo]]
| | [[72edo]]
| | This is a rough correspondence, as the (7n)edf ~ (12n)edo sequence begins to break down. Patent vals match through the 7 limit.
| | This is a rough correspondence, as the (7n)edf ~ (12n)edo sequence begins to break down. Patent vals match through the 7 limit.
|-
|[[43edf]]
|[[74edo]]
|43edf is 74edo with 2.57 cent stretched octaves. In other words, it is an extended meantone with a just 3/2
|-
|[[44edf]]
|[[75edo]]
|44edf is 75edo with 3.49 cent compressed octaves.
|-
|-
| | [[45edf]]
| | [[45edf]]
| | [[77edo]]
| | [[77edo]]
| | 45edf is 77edo with 1.1 cent stretched octaves. Patent vals match through the 13 limit.
| | 45edf is 77edo with 1.1 cent stretched octaves. Patent vals match through the 13 limit.
|-
|[[46edf]]
|[[79edo]]
|46edf is 79edo with ~9.7 cent compressed octaves. Patent vals differ in the 7-limit.
|-
|[[47edf]]
|[[80edo]]
|47edf is 80edo with 5.18 cent compressed octaves.
|-
|-
| | [[48edf]]
| | [[48edf]]
| | [[82edo]]
| | [[82edo]]
| | Same 0.83 cent octave compression as 24edf~41edo. Patent vals match through the 11 limit, with the exception of 5.
| | Same 0.83 cent octave compression as 24edf~41edo. Patent vals match through the 11 limit, with the exception of 5.
|-
|[[49edf]]
|84edo
|This is a rough correspondence, as the (7n)edf ~ (12n)edo sequence continues to break down. Patent vals match through the 3 limit.
|-
|[[50edf]]
|
|The (10n)edf ~ (17n)edo sequence has broken down completely, 50edf falls exactly halfway between 85 and 86 edos. Technically, it may not entirely miss 2/1 (it falls within 7.4 cents on either side), but it nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\171 of an octave.
|-
|-
| | [[51edf]]
| | [[51edf]]
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| | [[89edo]]
| | [[89edo]]
| | 52edf is 89edo with 1.4 cent stretched octaves. Patent vals match through the 13 limit, with the exception of 5.
| | 52edf is 89edo with 1.4 cent stretched octaves. Patent vals match through the 13 limit, with the exception of 5.
|-
|[[53edf]]
|91edo
|53edf is 91edo with 5.24 cent stretched octaves.
|-
|[[54edf]]
|92edo
|Same 4.1 cent octave compression as 27edf~46edo. Patent vals also match through the same limit.
|-
|-
| | [[55edf]]
| | [[55edf]]
| | [[94edo]]
| | [[94edo]]
| | 55edf is 94edo with 0.3 cent compressed octaves. Patent vals match through the 47 limit.
| | 55edf is 94edo with 0.3 cent compressed octaves. Patent vals match through the 47 limit.
|-
|[[56edf]]
|[[96edo]]
|This is a rough correspondence, as the (7n)edf ~ (12n)edo sequence momentarily ceases to break down further. Patent vals match through the 3 limit.
|-
|[[57edf]]
|97edo
|57edo is 97edo with 4.455 cent copmressed octave.
|-
|-
| | [[58edf]]
| | [[58edf]]
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