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| == Types of subgroups == | | == Types of subgroups == |
| '''WIP: I plan to heavily rewrite this section after carefully studying the new ''[[User:Dummy index/Heuristics for picking a nonstandard basis of JI subgroup]]''.''' | | '''WIP: I plan to heavily rewrite this section after carefully studying the new ''[[User:Dummy index/Heuristics for picking a nonstandard basis of JI subgroup]]''.''' |
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| ; Prime
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| * lim = [[Prime limit]]
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| * no-n = [[Subgroup|No-n subgroup]]
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| * dual = [[Dual-n|Dual-n subgroup]]
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| * cen = Directional subgroup
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| ** ''Example: -40cen+1 includes all primes where -40¢<absolute error<+1¢''
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| ** ''Example: -1cen+35 includes all primes where -1¢<absolute error<+35¢''
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| ** ''Example: -25cen+10 includes all primes where -25¢<relative error<+10¢''
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| ** Note that lim, no-n and most dual subgroups obey -15cen+15 or stricter
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| ; Composite
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| * comp = Other [[Subgroup|composite subgroup]]
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| ;Fractional
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| * nth-b = [[Half-prime subgroup|Nth-basis subgroup]]
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| * frac = Other [[Subgroup|fractional subgroup]]
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| <small>(''Technically any fractional subgroup can be said to be nth-basis, so an arbitrary cutoff must be drawn somewhere. This page considers 200th-basis or higher to not be nth-basis, while 199th or lower is accepted.'')</small>
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| == Procedure for choosing a subgroup == | | == Procedure for choosing a subgroup == |
| '''WIP: I plan to heavily rewrite this section after carefully studying the new ''[[User:Dummy index/Heuristics for picking a nonstandard basis of JI subgroup]]''.''' | | '''WIP: I plan to heavily rewrite this section after carefully studying the new ''[[User:Dummy index/Heuristics for picking a nonstandard basis of JI subgroup]]''.''' |
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| Remember: All of these rules are made to be broken. Bend the rules to fit the EDO. Don't bend the EDO to fit the rules.
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| === Aims ===
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| 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.
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| 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.
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| === Why different sized EDOs have different procedures ===
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| As EDOs get bigger and their step size gets smaller, their step size gets closer and closer to the [[just-noticeable difference]].
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| This means that if a smaller EDO has high [[relative error]] on a [[prime]], it will sound like the prime is not there at all (no-no), but if a larger EDO has high relative error on a prime, especially a small prime, it will sound like there are two versions of the prime (dual).
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| Different approaches are needed for different EDO sizes to reflect this.
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| 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.
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| === EDOs with 1 to 27 tones/octave ===
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| # The subgroup should have:
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| ## 3 basis elements if the EDO has 1-6 tones
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| ## 5 basis elements if the EDO has 7-12 tones
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| ## or 6 basis elements if the EDO has 13-27 tones
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| # Add prime 2 to the subgroup
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| # Do whichever of the following things results in the most simple consonances being available and the least out of tune consonances being listed (use your own discretion to decide):
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| ## Add the next 4-5 prime harmonics to be approximated within 15 cents
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| ## Add the next 4-5 prime harmonics with absolute error between -a¢ and +b¢ where (a+b) is no bigger than 50. Make a and b anything you like that fits the rule, try to make them as small as possible while still including most important intervals
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| ## Add the 4-5 smallest odd harmonics and/or [[taxicab distance|taxicab-2]] intervals to be approximated within 15 cents, avoiding any that include an already-added basis element as a factor
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| === EDOs with 28 to 52 tones/octave ===
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| # The subgroup should have 7 basis elements
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| # Primes 2, 3, 5, 7 and 11 must be added to the subgroup
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| # If any primes 3, 5, 7 or 11 have more than 40% relative error, then they should be made a dual prime
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| # If there are more than 2 dual-primes, then only the 2 lowest dual-primes should be kept dual, and the rest made single again
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| # If there are still spots left open, then they should be filled by every prime 13 and up which the EDO approximates with less than 35% relative error, preferencing the lowest primes first, until all spots are filled
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| === EDOs with 53 to 71 tones/octave ===
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| # The subgroup should have 8 basis elements
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| # Primes 2, 3, 5, 7 and 11 must be added to the subgroup
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| # If any primes 3, 5, 7 or 11 have more than 40% relative error, then they should be made a dual prime
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| # If there are more than 3 dual-primes, then only the 3 lowest dual-primes should be kept dual, and the rest made single again
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| # If there are still spots left open, then they should be filled by every prime 13 and up which the EDO approximates with less than 35% relative error, preferencing the lowest primes first, until all spots are filled
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| === EDOs with 72 to 98 tones/octave ===
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| # The subgroup should have 9 basis elements
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| # Primes 2, 3, 5, 7, 11 and 13 must be added to the subgroup
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| # If any primes 3, 5, 7, 11 or 13 have more than 40% relative error, then they should be made a dual prime
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| # If there are more than 4 dual-primes, then only the 4 lowest dual-primes should be kept dual, and the rest made single again
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| # If there are still spots left open, then they should be filled by every prime 17 and up which the EDO approximates with less than 35% relative error, preferencing the lowest primes first, until all spots are filled
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| === EDOs with 99 or more tones/octave ===
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| # The subgroup should have 11 basis elements
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| # Add primes 2, 3, 5, 7, 11, 13, 17, 19 and 23 to the subgroup
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| # Add the next two smallest primes with <35% relative error after 23 to the subgroup
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| # If any primes 23 or lower have >40% relative error, then they should be made a dual prime
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| # If there are now more than 11 basis elements, then the primes should be removed one by one starting with the highest and getting lower until there are 11 basis elements left
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| # If a dual-prime is the last one to be removed, and this causes there to be only 10 basis elements left, then add back the smallest non-dual prime that was removed (''if no non-dual primes were removed, add the next smallest prime with <35% relative error that's not already in the subgroup'')
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| == List of subgroups by EDO == | | == List of subgroups by EDO == |
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| === Picnic EDOs (1-4) === | | === Picnic EDOs (1-4) === |
| ; 3 basis elements
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| * [[1edo]]: 2 • 127 • 129 (''comp'')
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| * [[2edo]]: 2 • <small><sup>7</sup>/<sub>5</sub></small> • <small><sup>17</sup>/<sub>3</sub></small> (''nth-b; 15th'')
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| * [[3edo]]: 2 • 5 • <small><sup>19</sup>/<sub>3</sub></small> (''nth-b; 3rd'')
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| * [[4edo]]: 2 • <small><sup>5</sup>/<sub>3</sub></small> • 19 (''nth-b; 3rd'')
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| === Birthday EDOs (5-19) === | | === Birthday EDOs (5-19) === |
| ; 3 basis elements
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| * [[5edo]]: 2 • 3 • 7 (''no-n'')
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| * [[6edo]]: 2 • 9 • 5 (''comp'')
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| ; 5 basis elements
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| * [[7edo]]: 2 • 3 • 5 • 11 • 29 (''-44cen+1'')
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| * [[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'')
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| * [[9edo]]: 2 • 5 • <small><sup>7</sup>/<sub>3</sub></small> • 11 • <small><sup>13</sup>/<sub>7</sub></small> (''nth-b; 21st'')
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| * [[10edo]]: 2 • 3 • 7 • 13 • 17 (''no-n'')
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| * [[11edo]]: 2 • 9 • 15 • 7 • 11 (''comp'')
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| * [[12edo]]: 2 • 3 • 5 • 17 • 19 (''no-n'')
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| ; 6 basis elements
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| * [[13edo]]: 2 • 9 • 5 • 21 • 11 • 13 (''comp'')
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| * [[14edo]]: 2 • 3 • <5 • 7 • 11 • 17 (''-44cen+1'')
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| * [[15edo]]: 2 • 3 • 5 • 7 • 11 • 23 (''no-n'')
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| * [[16edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''-28cen+7'')
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| * [[17edo]]: 2 • 3 • 5> • 7 • 11 • 13 (''-1cen+38'')
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| * [[18edo]]: 2 • 3 • 5 • 7 • 13 • 17 (''-1cen+32'')
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| * [[19edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'')
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| === Carousel EDOs (20-34) === | | === Carousel EDOs (20-34) === |
| ; 6 basis elements
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| * [[20edo]]: 2 • 3 • 5> • 7 • 11 • 13 (''-12cen+34'')
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| * [[21edo]]: 2 • 3 • 5 • 7 • 17 • 19 (''no-n'')
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| * [[22edo]]: 2 • 3 • 5 • 7 • 11 • 17 (''no-n'')
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| * [[23edo]]: 2 • 9 • 15 • 21 • 33 • 13 (''comp'')
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| * [[24edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'')
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| * [[25edo]]: 2 • 3 • 5 • 7 • 17 • 19 (''no-n'')
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| * [[26edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'')
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| * [[27edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'')
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| ; 7 basis elements
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| * [[28edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 19 (''no-n'')
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| * [[29edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 19 (''no-n'')
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| * [[30edo]]: 2 • ''3+'' • ''3-'' • 5 • 7 • 11 • 13 (''dual'')
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| * [[31edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 (''lim'')
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| * [[32edo]]: 2 • 3 • 5 • 7 • 11 • 17 • 19 (''no-n'')
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| * [[33edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 (''lim'')
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| * [[34edo]]: 2 • 3 • 5 • ''7+'' • ''7-'' • 11 • 13 (''dual'')
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| === Schoolbus EDOs (35-54) === | | === Schoolbus EDOs (35-54) === |
| ; 7 basis elements
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| * [[35edo]]: 2 • ''3+'' • ''3-'' • 5 • 7 • 11 • 17 (''dual'')
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| * [[36edo]]: 2 • 3 • ''5+'' • ''5-'' • 7 • ''11+'' • ''11-'' (''dual'')
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| * [[37edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 (''lim'')
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| * [[38edo]]: 2 • 3 • 5 • 7 • ''11+'' • ''11-'' • 13 (''dual'')
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| * [[39edo]]: 2 • 3 • ''5+'' • ''5-'' • ''7+'' • ''7-'' • 11 (''dual'')
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| * [[40edo]]: 2 • ''3+'' • ''3-'' • 5 • 7 • 11 • 13 (''dual'')
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| * [[41edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 (''lim'')
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| * [[42edo]]: 2 • ''3+'' • ''3-'' • ''5+'' • ''5-'' • 7 • 11 (''dual'')
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| * [[43edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 (''lim'')
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| * [[44edo]]: 2 • 3 • 5 • ''7+'' • ''7-'' • 11 • 13 (''dual'')
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| * [[45edo]]: 2 • 3 • ''5+'' • ''5-'' • 7 • 11 • 17 (''dual'')
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| * [[46edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 (''lim'')
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| * [[47edo]]: 2 • ''3+'' • ''3-'' • 5 • 7 • ''11+'' • ''11-'' (''dual'')
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| * [[48edo]]: 2 • 3 • ''5+'' • ''5-'' • 7 • 11 • 13 (''dual'')
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| * [[49edo]]: 2 • 3 • 5 • ''7+'' • ''7-'' • ''11+'' • ''11-'' (''dual'')
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| * [[50edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 (''lim'')
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| * [[51edo]]: 2 • 3 • ''5+'' • ''5-'' • 7 • ''11+'' • ''11-'' (''dual'')
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| * [[52edo]]: 2 • ''3+'' • ''3-'' • 5 • 7 • 11 • 19 (''dual'')
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| ; 8 basis elements
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| * [[53edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 (''lim'')
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| * [[54edo]]: 2 • ''3+'' • ''3-'' • ''5+'' • ''5-'' • ''7+'' • ''7-'' • 11 (''dual'')
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| === Double-decker EDOs (55-74) === | | === Double-decker EDOs (55-74) === |
| ; 8 basis elements
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| * [[55edo]]: 2 • 3 • 5 • ''7+'' • ''7-'' • 11 • 17 • 23 (''dual'')
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| * [[56edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 (''lim'')
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| * [[57edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 (''lim'')
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| * [[58edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 (''lim'')
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| * [[59edo]]: 2 • ''3+'' • ''3-'' • 5 • 7 • 11 • 13 • 17 (''dual'')
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| * [[60edo]]: 2 • 3 • 5 • ''7+'' • ''7-'' • ''11+'' • ''11-'' • 13 (''dual'')
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| * [[61edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 (''lim'')
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| * [[62edo]]: 2 • 3 • 5 • 7 • ''11+'' • ''11-'' • 29 • 31 (''dual'')
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| * [[63edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 23 • 29 (''no-n'')
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| * [[64edo]]: 2 • ''3+'' • ''3-'' • ''5+'' • ''5-'' • 7 • ''11+'' • ''11-'' (''dual'')
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| * [[65edo]]: 2 • 3 • 5 • ''7+'' • ''7-'' • 11 • 19 • 23 (''dual'')
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| * [[66edo]]: 2 • ''3+'' • ''3-'' • 5 • 7 • 11 • 13 • 17 (''dual'')
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| * [[67edo]]: 2 • 3 • ''5+'' • ''5-'' • 7 • 11 • 13 • 17 (''dual'')
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| * [[68edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 (''lim'')
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| * [[69edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 (''lim'')
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| * [[70edo]]: 2 • 3 • ''5+'' • ''5-'' • ''7+'' • ''7-'' • 11 • 13 (''dual'')
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| * [[71edo]]: 2 • ''3+'' • ''3-'' • 5 • 7 • 11 • 13 • 17 (''dual'')
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| ; 9 basis elements
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| * [[72edo]]: 2 • 3 • 5 • 7 • 11 • ''13+'' • ''13-'' • 17 • 19 (''dual'')
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| * [[73edo]]: 2 • 3 • ''5+'' • ''5-'' • 7 • ''11+'' • ''11-'' • 13 • 19 (''dual'')
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| * [[74edo]]: 2 • 3 • 5 • 7 • 11 • 13 • 19 • 23 • 31 (''no-n'')
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| == Notation of dual-3 EDOs == | | == Notation of dual-3 EDOs == |
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| Note names: C, D, E, F, F#/Gb, G, G#/Ab, A | | Note names: C, D, E, F, F#/Gb, G, G#/Ab, A |
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| [[Category:Impression]]
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