User:BudjarnLambeth/12edo as a 2.3.5.17.19 tuning: Difference between revisions

← Older edit Newer edit →

VisualWikitext

@@ Line 23: / Line 23: @@
 === Aims ===
-The aim of these procedures is to make visibly available all of the simple [[consonant]] intervals an EDO has to offer, without falsely including ones it doesn’t have, and without allowing less-important intervals to create unnecessary clutter.
+The aim of these procedures is to make visibly available all of the simple [[consonant]] intervals an EDO has to offer, without falsely including too many ones it doesn’t have, and without allowing less-important intervals to create unnecessary clutter.
 The aim is also to make subgroups of similar-sized EDOs look fairly similar so that it’s easy to cross-compare between them at a glance.
@@ Line 36: / Line 36: @@
 Also, as EDOs get bigger, more notes per [[octave]] need to be labelled with a [[JI]] approximation, so more [[basis element]]s are needed to produce those labels. Whereas, as EDOs get smaller, too many basis elements just make it needlessly complicated to navigate them, and fewer basis elements are better. So this is another reason for differing approaches at different EDO sizes.
-=== EDOs with 1 to 6 tones/octave ===
+=== EDOs with 1 to 27 tones/octave ===
-# The subgroup should have 3 basis elements
+# The subgroup should have:
-# If the EDO approximates 3 or more primes 11 or lower within 15 [[cents]], then choose the best 3 and use those as its subgroup
+## 3 basis elements if the EDO has 1-6 tones
-# If it approximates less than 3 such primes, then include all the ones it does approximate, and fill the remaining spots with [[11-limit]] composite harmonics smaller than 60 that it approximates within 15 cents (giving preference to harmonics with lower prime factors first and excluding powers of two)
+## 5 basis elements if the EDO has 7-12 tones
-# If there are aren't enough of those to fill all 3 spots, fill the remaining spots with 11-limit [[taxicab distance|taxicab-2]] intervals the edo approximates within 15 cents, giving preference to intervals with lower primes
+## or 6 basis elements if the EDO has 13-27 tones
-# If there are aren't enough of those to fill all 3 spots, then allow primes, composites or taxicab intervals of any size which the edo approximates within 15 cents, giving preference to ones with lower prime factors and lower integer limits (and not discriminating based on whether they are prime, fractional or composite)
-=== EDOs with 7 to 27 tones/octave ===
-# The subgroup should have 5 basis elements if the EDO has 7-12 tones, or 6 basis elements if the EDO has 13-27 tones
 # Add prime 2 to the subgroup
-# If 3 is approximated within 15 cents, add 3 to the subgroup
+# Do whichever of the following things results in the most simple consonances being available (use your own discretion to decide):
-## If it is not, then add the smallest multiple of 3, 60 or lower, it approximates within 15 cents (if any)
+## Add the next 4-5 prime harmonics to be approximated within 15 cents
-## Optionally, add the second-smallest multiple of 3, 60 or lower, it approximates within 15 cents, which is not a multiple of the previous one (if any)
+## Add the next 4-5 prime harmonics to be approximated FLAT with <60% relative error (leave out sharp ones)
-## If 3 and all its multiples up to 60 are not approximated, then add the smallest 13-limit taxicab-2 interval containing 3 (if any)
+## Add the next 4-5 prime harmonics to be approximated SHARP with <60% relative error (leave out sharp ones)
-# If 5 is approximated within 15 cents, add 5 to the subgroup
+## Add the 4-5 smallest odd harmonics and/or [[taxicab distance|taxicab-2]] intervals to be approximated within 15 cents
-## If it is not, then add the smallest multiple of 5, 60 or lower, which it approximates within 15 cents (if any)
-## (''If this is the same as a number already added, just keep that one'')
-## If 5 and all its multiples up to 60 are not approximated, then add the smallest 13-limit taxicab-2 interval containing 5 (if any)
-## (''If this is the same as a fraction already added, just keep that one'')
-# If 7 is approximated within 15 cents, add 7 to the subgroup
-## If it is not, then add the smallest multiple of 7, 63 or lower, which it approximates within 15 cents (if any)
-## (''If this is the same as a number already added, just keep that one'')
-## If 7 and all its multiples up to 63 are not approximated, then add the smallest 13-limit taxicab-2 interval containing 7 (if any)
-## (''If this is the same as a fraction already added, just keep that one'')
-# If 11 is approximated within 15 cents, add 11 to the subgroup
-# If there are still spots free, and 13 is approximated within 7 cents, add 13 to the subgroup
-# If there are still spots free, and 11 is not in the subgroup:
-## Add the smallest multiple of 11, 66 or lower, approximated within 15 cents to the subgroup (if any)
-## (''If this is the same as a number already added, just keep that one'')
-## If 11 and all its multiples up to 66 are not approximated, then add the smallest 13-limit taxicab-2 interval containing 11 (if any)
-## (''If this is the same as a fraction already added, just keep that one'')
-# Check that each of the composite basis elements in the subgroup (if any) shares at least one common factor with at least one other thing in the subgroup; remove any composite basis elements that do not
-# If any primes 13, 17, 19 or 23 are approximated within 15 cents, include as many of those as there are basis element spots free (giving preference to harmonics with closer approximations first)
-# If there are aren't enough of those to fill all spots, fill the remaining spots with taxicab-2 intervals or composite harmonics which the edo approximates within 15 cents and which haven’t been added yet, giving preference to ones with small prime factors
-# ''Optionally'', replace any one basis element with any composite harmonic 60 or smaller, that shares factors in common with at least 2 other basis elements in the subgroup, is approximated within 15 cents, and has not yet been added (''do not replace pure 2, 3, 5, 7 or 11'')
 === EDOs with 28 to 52 tones/octave ===
@@ Line 118: / Line 94: @@
 * [[6edo]]: 2 • 9 • 5    (''comp'')
 ; 5 basis elements
-* [[7edo]]: 2 • 3 • <small><sup>11</sup>/<sub>3</sub></small> • <small><sup>11</sup>/<sub>5</sub></small> • 13    (''nth-b; 15th'')
+* [[7edo]]: 2 • 3 • 5 • 11 • 29    (''<dir'')
 * [[8edo]]: 2 • <small><sup>5</sup>/<sub>3</sub></small> • <small><sup>11</sup>/<sub>3</sub></small> • <small><sup>13</sup>/<sub>5</sub></small> • 19    (''nth-b; 15th'')
 * [[9edo]]: 2 • 5 • <small><sup>7</sup>/<sub>3</sub></small> • 11 • <small><sup>13</sup>/<sub>7</sub></small>    (''nth-b; 21st'')
@@ Line 126: / Line 102: @@
 ; 6 basis elements
 * [[13edo]]: 2 • 9 • 5 • 21 • 11 • 13    (''comp'')
-* [[14edo]]: 2 • 3 • <5 • 7 • 11 • 17    (''dir'')
+* [[14edo]]: 2 • 3 • <5 • 7 • 11 • 17    (''<dir'')
 * [[15edo]]: 2 • 3 • 5 • 7 • 11 • 23    (''no-n'')
-* [[16edo]]: 2 • 3 • 5 • 11 • 13 • 17    (''dir'')
+* [[16edo]]: 2 • 3 • 5 • 11 • 13 • 17    (''<dir'')
-* [[17edo]]: 2 • 3 • 35 • 7 • 11 • 13    (''comp'')
+* [[17edo]]: 2 • 3 • 5> • 7 • 11 • 13    (''dir>'')
-* [[18edo]]: 2 • 5 • 9 • 11 • 21 • 33   (''comp'')
+* [[18edo]]: 2 • 3 • 5 • 7 • 13 • 17   (''dir>'')
 * [[19edo]]: 2 • 3 • 5 • 7 • 11 • 13    (''lim'')