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| {{Idiosyncratic terms}} | | {{Idiosyncratic terms}} |
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| This is a user page, not one of the main wiki pages. | | This is a user page, '''not a real wiki page'''. |
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| This page is only opinion, not fact. | | '''This page is only opinion, not fact.''' |
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| This user page details how I personally assign each [[EDO]] to a [[subgroup]] of [[just intonation]].
| | Interpreting 12edo as a 2.3.5.17.19 system: |
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| == Types of subgroups ==
| | == Intervals == |
| * 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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| * EQ = [[Equalizer subgroup]]
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| * comp = [[Subgroup|Other composite subgroup]]
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| * nth-b [[Half-prime subgroup|Nth-basis subgroup]]
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| * frac = [[Subgroup|Other 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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| == How to choose a type ==
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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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| === 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 6 tones/octave ===
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| # The subgroup should have 3 basis elements
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| # 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
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| # 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)
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| # 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
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| # 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)
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| === EDOs with 7 to 27 tones/octave ===
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| # 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
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| # Add prime 2 to the subgroup
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| # If 3 is approximated within 15 cents, add 3 to the subgroup
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| ## If it is not, then add the smallest multiple of 3, 60 or lower, it approximates within 15 cents (if any)
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| ## 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)
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| ## 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)
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| # If 5 is approximated within 15 cents, add 5 to the subgroup
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| ## If it is not, then add the smallest multiple of 5, 60 or lower, which it approximates within 15 cents (if any)
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| ## (''If this is the same as a number already added, just keep that one'')
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| ## 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)
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| ## (''If this is the same as a fraction already added, just keep that one'')
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| # If 7 is approximated within 15 cents, add 7 to the subgroup
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| ## If it is not, then add the smallest multiple of 7, 56 or lower, which it approximates within 15 cents (if any)
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| ## (''If this is the same as a number already added, just keep that one'')
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| ## If 7 and all its multiples up to 56 are not approximated, then add the smallest 13-limit taxicab-2 interval containing 7 (if any)
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| ## (''If this is the same as a fraction already added, just keep that one'')
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| # If 11 is approximated within 15 cents, add 11 to the subgroup
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| # If there are still spots free, and 13 is approximated within 7 cents, add 13 to the subgroup
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| # If there are still spots free, and 11 is not in the subgroup:
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| ## Add the smallest multiple of 11, 77 or lower, approximated within 15 cents to the subgroup (if any)
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| ## (''If this is the same as a number already added, just keep that one'')
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| ## If 11 and all its multiples up to 77 are not approximated, then add the smallest 13-limit taxicab-2 interval containing 11 (if any)
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| ### (''If this is the same as a fraction already added, just keep that one'')
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| # If any composite basis elements now in the subgroup share zero common factors with anything else in the subgroup, remove them
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| # 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)
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| # If there are aren't enough of those to fill all spots, fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents which haven’t been added yet, giving preference to intervals with small primes
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| # 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'')
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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 ==
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| Size categories taken from my [[human EDO size categorization]] (HUECAT).
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| === Picnic EDOs (1-4) ===
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| ; 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> • <small><sup>7</sup>/<sub>5</sub></small> (''nth-b; 15th'')
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| === Birthday EDOs (5-19) ===
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| ; 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 • <small><sup>11</sup>/<sub>3</sub></small> • <small><sup>11</sup>/<sub>5</sub></small> • 13 (''nth-b; 15th'')
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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 • 15 • 7 • 13 (''comp'')
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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 • 25 • 35 • 45 • 55 (''comp'')
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| * [[15edo]]: 2 • 3 • 5 • 7 • 11 • 23 (''no-n'')
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| * [[16edo]]: 2 • 5 • 7 • 13 • 27 • 45 (''comp'')
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| * [[17edo]]: 2 • 3 • 35 • 7 • 11 • 13 (''comp'')
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| * [[18edo]]: 2 • 5 • 9 • 11 • 21 • 33 (''comp'')
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| * [[19edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'')
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| === Carousel EDOs (20-34) ===
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| ; 6 basis elements
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| * [[20edo]]: 2 • 3 • 15 • 7 • 11 • 13 (''comp'')
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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) ===
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| ; 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) ===
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| ; 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 • 5 • 7 • 11 • 13 • 17 • 19 (''lim'')
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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 ==
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| Most EDO notation systems, including the near-universal [[ups and downs notation]], are built upon [[chain-of-fifths notation]]. How then should an EDO be notated if it’s dual-fifth, i.e. it has two mappings of 3: 3+ and 3-?
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| The most straightforward solution is to just choose whichever 3 is closer to just 3/1, and pretend that’s the "real 3" for notation purposes. Treat the other 3 as just another prime, like 5 or 7. In most cases, I advise to do that.
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| If you happen to be mainly using an EDO as a tuning for one specific non-dual [[regular temperament]] like meantone, mavila, etc., then pretend that temperament’s mapping of 3 is the ‘real’ one for the purpose of notation, and pretend the other 3 is just like any other larger prime.
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| Of course, this results in multiple notation systems for the same EDO, since different people use different temperaments or none at all, but that’s already the case. All of those notation systems already exist, I’m not adding any new ones, I’m just saying that the ones we already have all have a valid place and it’s okay to use one some day and another some other day on a project-by-project basis.
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| As long as you name and briefly explain your notation system at the start of your score, use whatever system you want. Use whichever one works in practice for you and the musicians collaborating with you. Invent one, if the existing ones don’t work. It’s fine. Not everything has to be standardized and homogenized.
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| Because I’m personally a fan of mixing and matching multiple temperaments, and other things that aren’t temperaments like approximated [[JI]] scales, [[MOS scale]]s, [[MODMOS]] & [[inflected MOS]] scales and even randomly generated scales, I usually like to go with the first option: ups and downs notation, in particular using whichever 3 is closest to just for its chain of fifths, and the other 3 being treated as just another available prime like 5, 7 or 11.
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| == Interpreting 12edo as a 2.3.5.17.19 system==
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| === Intervals ===
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| === Chords ===
| | == Chords == |
| <br> | | <br> |
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| ==== Common chords ====
| | === Common chords === |
| My interpretation of what the just harmonies are, hiding behind common practice chords. | | My interpretation of what the just harmonies are, hiding behind common practice chords. |
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| ==== Parent chords ====
| | === Parent chords === |
| My list of in my opinion the most harmonious 'parent chords' in 12edo, which you can use as palettes to build novel and pretty smaller chords. Choose one of these chords, take any subset of 2 or more notes from it, and you will make another, also harmonious chord. | | My list of in my opinion the most harmonious 'parent chords' in 12edo, which you can use as palettes to build novel and pretty smaller chords. Choose one of these chords, take any subset of 2 or more notes from it, and you will make another, also harmonious chord. |
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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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