EDF: Difference between revisions
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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, and conversely one way to treat secundal chords (relative to scales where the large step is no larger than 253¢) as the one true type of triad is the use of 3/2 as the (formal) equivalence. 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, and conversely one way to treat secundal chords (relative to scales where the large step is no larger than 253¢) as the one true type of triad is the use of 3/2 as the (formal) equivalence. 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. | ||
== Individual pages for EDFs == | == Individual pages for EDFs == | ||
*[[2edf]] | *[[2edf]] | ||
*[[3edf]] | *[[3edf]] | ||
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== EDF-EDO correspondence == | == EDF-EDO correspondence == | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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| | [[4edf]] | | | [[4edf]] | ||
| | [[7edo]] | | | [[7edo]] | ||
| | 4edf is 7edo with 28.5 cent stretched octaves.<br>Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents.<br>Patent vals match through the 5 limit. Only a rough correspondence. | | | 4edf is 7edo with 28.5 cent stretched octaves. <br>Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents. <br>Patent vals match through the 5 limit. Only a rough correspondence. | ||
|- | |- | ||
| | [[5edf]] | | | [[5edf]] | ||
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| | [[11edf]] | | | [[11edf]] | ||
| | [[19edo]] | | | [[19edo]] | ||
| | 11edf is 19edo with 12.5 cent stretched octaves. Patent vals match through the 7 limit.<br>If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence. | | | 11edf is 19edo with 12.5 cent stretched octaves. Patent vals match through the 7 limit. <br>If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence. | ||
|- | |- | ||
| | [[12edf]] | | | [[12edf]] | ||
| | | | | | ||
| | 12edf falls exactly halfway between 20 and 21 | | | 12edf falls exactly halfway between 20 and 21 EDOs. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\41 of an octave. | ||
|- | |- | ||
| | [[13edf]] | | | [[13edf]] | ||
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| | [[16edf]] | | | [[16edf]] | ||
| | | | | | ||
| | 16edf falls exactly halfway between 27 and 28 | | | 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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| | [[19edf]] | | | [[19edf]] | ||
| | | | | | ||
| | 19edf falls exactly halfway between 32 and 33 | | | 19edf falls exactly halfway between 32 and 33 EDOs. | ||
|- | |- | ||
| | [[20edf]] | | | [[20edf]] | ||
| | [[34edo]] | | | [[34edo]] | ||
| | Same 6.6 cent octave compression as 10edf~17edo. Patent vals match through the 5 limit, but not the 7 limit.<br>If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence. | | | Same 6.6 cent octave compression as 10edf~17edo. Patent vals match through the 5 limit, but not the 7 limit. <br>If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence. | ||
|- | |- | ||
| | [[21edf]] | | | [[21edf]] | ||
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|[[32edf]] | |[[32edf]] | ||
|[[55edo]] | |[[55edo]] | ||
|32edf is 55edo with 6.485 cent stretched octaves | |32edf is 55edo with 6.485 cent stretched octaves. | ||
|- | |- | ||
|[[33edf]] | |[[33edf]] | ||
|[[56edo]] | |[[56edo]] | ||
|33edf is 56edo with 8.8 cent compressed octaves | |33edf is 56edo with 8.8 cent compressed octaves. | ||
|- | |- | ||
| | [[34edf]] | | | [[34edf]] | ||
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== See also == | == See also == | ||
* [[Relative | * [[Relative errors of small EDFs]] | ||
[[Category:Equal-step tuning]] | [[Category:Equal-step tuning]] | ||
[[Category:Edf| ]] <!-- main article --> | [[Category:Edf| ]] <!-- main article --> | ||