User:BudjarnLambeth/12edo as a 2.3.5.17.19 tuning

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.

This is a user page, not one of the main wiki pages.

This page is only opinion, not fact.

This user page details how I personally assign each EDO to a subgroup of just intonation.

Types of subgroups

• lim = Prime limit
• no-n = No-n subgroup
• dual = Dual-n subgroup
• EQ = Equalizer subgroup
• comp = Other composite subgroup
• nth-b Nth-basis subgroup
• frac = Other fractional subgroup

How to choose a type

1. If the edo has <40% relative error and <15-20c^ absolute error on all primes in a prime limit 7 or higher, use that prime limit.
2. If the edo has >40% relative error but <15-20c^ absolute error on any primes N where N is 11 or smaller, then use dual-N for all those primes; then also include every other prime as a single-prime up to the last prime before M, where M is the first prime above 11 with >40% relative error.
3. If the edo has >40% relative error and >15-20c^ absolute error on any primes N where N is 11 or smaller, then use no-N for all those primes; then include every other prime up to the last prime before M, where M is the first or second^ prime above 11 to have >15-20c^ absolute error.
4. An addition to the previous step: if the edo approximates any less-than-20 multiple of an excluded prime N, with <15-20c^ absolute error, then turn the edo's subgroup into a composite subgroup, and include that multiple as a basis element.
5. Another addition to the previous step: if the edo approximates any 30-integer-limit interval of an excluded prime N, with <~10c^ absolute error, then turn the edo's subgroup into a fractional subgroup, and include that multiple as a basis element, or include a fractional basis element which would make that interval accessible.
6. If none of the above cases are true, but the edo <15-20c^ absolute error on any primes N where N is 11 or smaller on all primes in a prime limit 7 or higher, use that prime limit. If that is the case for all but a small number^ of primes P, then just use the no-P version of the prime limit.
7. Avoid having more than one "no-n" where n is a prime 13 or higher. Just draw the cutoff there and leave out the second n and all primes higher than it.
8. Avoid having any "dual-n" where n is a prime 17 or higher. Just draw the cutoff there and leave out n and all primes higher than it.

^Use your own discretion when deciding or strict or lenient to set this value on a per-edo basis. For example if an edo has a great 17/1, don't leave it out just because a rule says to, or if it has a terrible 11/1 that just scrapes over the line, don't include it just because a rule says to. Each edo is its own unique and wild creature, use your own discretion and bend these rules to fit the edo, not the other way around.

Subgroups by EDO size (less dimensions)

Size categories taken from my human EDO size categorization (HUECAT).

Birthday EDOs (5-19)

• 5edo: 2.3.7 (no-n)
• 6edo: 2.9.5 (comp)
• 7edo: 2.3.11/3.11/5.13 (nth-b) (15th)
• 8edo: 2.11/3.13/5.19 (nth-b) (15th)
• 9edo: 2.5.11 (no-n)
• 10edo: 2.3.7.13 (no-n)
• 11edo: 2.9.15.7.11 (comp)
• 12edo: 2.3.5 (lim)
• 13edo: 2.9.5.11.13 (comp)
• 14edo: 2.3.7/5.9/5.11/5.13 (nth-b) (5th)
• 15edo: 2.3.5.7.11 (lim)
• 16edo: 2.5.7.13 (no-n)
• 17edo: 2.3.7.11.13 (no-n)
• 18edo: 2.9.5.7/3.11 (nth-b) (3rd)
• 19edo: 2.3.5.7.11.13 (lim)

Carousel EDOs (20-34)

• 20edo: 2.3.7.11.13 (no-n)
• 21edo: 2.3.5.7.13 (no-n)
• 22edo: 2.3.5.7.11 (lim)
• 23edo: 2.9.15.21.33.13 (comp)
• 24edo: 2.3.5.7.11.13 (lim)
• 25edo: 2.3.5.7.17 (no-n)
• 26edo: 2.3.5.7.11.13 (lim)
• 27edo: 2.3.5.7.11.13 (lim)
• 28edo: 2.3.5.7.11.13 (lim)
• 29edo: 2.3.5.7.11.13 (lim)
• 30edo: 2.3+.3-.5.7.11 (dual)
• 31edo: 2.3.5.7.11.13 (lim)
• 32edo: 2.3.5.7.11.13 (lim)
• 33edo: 2.3.5.7.11.13 (lim)
• 34edo: 2.3.5.7+.7-.11 (dual)

Schoolbus EDOs (35-54)

• 35edo: 2.3+.3-.5.7.11 (dual)
• 36edo: 2.3.5+.5-.7.11+.11- (dual)
• 37edo: 2.3.5.7.11.13 (lim)
• 38edo: 2.3.5.7.11+.11- (dual)
• 39edo: 2.3.5+.5-.7+.7-.11 (dual)
• 40edo: 2.3+.3-.5.7.11 (dual)
• 41edo: 2.3.5.7.11.13 (lim)
• 42edo: 2.3+.3-.5+.5-.7.11 (dual)
• 43edo: 2.3.5.7.11.13 (lim)
• 44edo: 2.3.5.7+.7-.11 (dual)
• 45edo: 2.3.5+.5-.7.11 (dual)
• 46edo: 2.3.5.7.11.13 (lim)
• 47edo: 2.3+.3-.5.7.11+.11- (dual)
• 48edo: 2.3.5+.5-.7.11 (dual)
• 49edo: 2.3.5.7+.7-.11+.11- (dual)
• 50edo: 2.3.5.7.11.13 (lim)
• 51edo: 2.3.5+.5-.7.11+.11- (dual)
• 52edo: 2.3+.3-.5.7.11 (dual)
• 53edo: 2.3.5.7.11.13 (lim)
• 54edo: 2.3+.3-.5+.5-.7+.7-.11 (dual)

Double-decker EDOs (55-74)

(May complete later.)

Subgroups by EDO size (more dimensions)

Birthday EDOs (5-19)

• 5edo: 2.3.7 (no-n)
• 6edo: 2.9.5 (comp)
• 7edo: 2.3.11/3.11/5.13 (nth-b) (15th)
• 8edo: 2.11/3.13/5.19 (nth-b) (15th)
• 9edo: 2.5.11 (no-n)
• 10edo: 2.3.7.13.17 (no-n)
• 11edo: 2.9.15.7.11.17 (comp)
• 12edo: 2.3.5.17.19 (no-n)
• 13edo: 2.9.5.11.13.17 (comp)
• 14edo: 2.3.7/5.9/5.11/5.13 (nth-b) (5th)
• 15edo: 2.3.5.7.11 (lim)
• 16edo: 2.5.7.13.19 (no-n)
• 17edo: 2.3.7.11.13 (no-n)
• 18edo: 2.9.5.7/3.11 (nth-b) (3rd)
• 19edo: full 43-limit (lim)

Carousel EDOs (20-34)

• 20edo: 2.3.7.11.13.17.19 (no-n)
• 21edo: 2.3.5.7.13.17.19.23.29.31 (no-n)
• 22edo: 2.3.5.7.11.17 (no-n)
• 23edo: 59-limit but with 3.5.7.11 removed and 9.15.21.33 added (comp)
• 24edo: 2.3.5.7.11.13.17.19 (lim)
• 25edo: 2.3.5.7.17.19.23 (no-n)
• 26edo: 2.3.5.7.11.13.17 (lim)
• 27edo: 2.3.5.7.11.13.17.19.23.29.31 (lim)
• 28edo: no-17 43-limit (no-n)
• 29edo: 2.3.5.7.11.13.19.23.29.31.37 (no-n)
• 30edo: 2.3+.3-.5.7.11.13.17 (dual)
• 31edo: 2.3.5.7.11.13.17.19.23 (lim)
• 32edo: 2.3.5.7.11.13.17.19.23 (lim)
• 33edo: 2.3.5.7.11.13.17.19.23.29 (lim)
• 34edo: 2.3.5.7+.7-.11.13.17 (dual)

Schoolbus EDOs (35-54)

• 35edo: 2.3+.3-.5.7.11.17 (dual)
• 36edo: dual-5 dual-11 29-limit (dual)
• 37edo: full 43-limit (lim)
• 38edo: 2.3.5.7.11+.11-.13.17 (dual)
• 39edo: 2.3.5+.5-.7+.7-.11.13 (dual)
• 40edo: 2.3+.3-.5.7.11.13 (dual)
• 41edo: 2.3.5.7.11.13.17.19 (lim)
• 42edo: 2.3+.3-.5+.5-.7.11.13+.13- (dual)
• 43edo: 2.3.5.7.11.13.17.19 (lim)
• 44edo: dual-7 43-limit (dual)
• 45edo: 2.3.5+.5-.7.11.13+.13-.17.19 (dual)
• 46edo: 2.3.5.7.11.13.17 (lim)
• 47edo: 2.3+.3-.5.7.11+.11-.13.17.19 (dual)
• 48edo: dual-5 41-limit (dual)
• 49edo: dual-7 dual-11 37-limit (dual)
• 50edo: 2.3.5.7.11.13.17.19.23.29.31 (lim)
• 51edo: 2.3.5+.5-.7.11+.11-.13 (dual)
• 52edo: 2.3+.3-.5.7.11.13+.13- (dual)
• 53edo: 2.3.5.7.11.13.17.19.23 (lim)
• 54edo: 2.3+.3-.5+.5-.7+.7-.11.13.17 (dual)

Double-decker EDOs (55-74)

(May complete later.)

Subgroups by subgroup type

(This list uses the complex high-dimension versions.)

Full prime limit

• 15edo: 2.3.5.7.11 (lim)
• 19edo: full 43-limit (lim)
• 24edo: 2.3.5.7.11.13.17.19 (lim)
• 26edo: 2.3.5.7.11.13.17 (lim)
• 27edo: 2.3.5.7.11.13.17.19.23.29.31 (lim)
• 31edo: 2.3.5.7.11.13.17.19.23 (lim)
• 32edo: 2.3.5.7.11.13.17.19.23 (lim)
• 33edo: 2.3.5.7.11.13.17.19.23.29 (lim)
• 37edo: full 43-limit (lim)
• 41edo: 2.3.5.7.11.13.17.19 (lim)
• 43edo: 2.3.5.7.11.13.17.19 (lim)
• 46edo: 2.3.5.7.11.13.17 (lim)
• 50edo: 2.3.5.7.11.13.17.19.23.29.31 (lim)
• 53edo: 2.3.5.7.11.13.17.19.23 (lim)

No-n

• 5edo: 2.3.7 (no-n)
• 9edo: 2.5.11 (no-n)
• 10edo: 2.3.7.13.17 (no-n)
• 12edo: 2.3.5.17.19 (no-n)
• 16edo: 2.5.7.13.19 (no-n)
• 17edo: 2.3.7.11.13 (no-n)
• 20edo: 2.3.7.11.13.17.19 (no-n)
• 21edo: 2.3.5.7.13.17.19.23.29.31 (no-n)
• 22edo: 2.3.5.7.11.17 (no-n)
• 25edo: 2.3.5.7.17.19.23 (no-n)
• 28edo: no-17 43-limit (no-n)
• 29edo: 2.3.5.7.11.13.19.23.29.31.37 (no-n)

Dual-n

• 30edo: 2.3+.3-.5.7.11.13.17 (dual)
• 34edo: 2.3.5.7+.7-.11.13.17 (dual)
• 35edo: 2.3+.3-.5.7.11.17 (dual)
• 36edo: dual-5 dual-11 29-limit (dual)
• 38edo: 2.3.5.7.11+.11-.13.17 (dual)
• 39edo: 2.3.5+.5-.7+.7-.11.13 (dual)
• 40edo: 2.3+.3-.5.7.11.13 (dual)
• 42edo: 2.3+.3-.5+.5-.7.11.13+.13- (dual)
• 44edo: dual-7 43-limit (dual)
• 45edo: 2.3.5+.5-.7.11.13+.13-.17.19 (dual)
• 47edo: 2.3+.3-.5.7.11+.11-.13.17.19 (dual)
• 48edo: dual-5 41-limit (dual)
• 49edo: dual-7 dual-11 37-limit (dual)
• 51edo: 2.3.5+.5-.7.11+.11-.13 (dual)
• 52edo: 2.3+.3-.5.7.11.13+.13- (dual)
• 54edo: 2.3+.3-.5+.5-.7+.7-.11.13.17 (dual)

Equalizer

No edos really fit this category.

Other composite

• 6edo: 2.9.5 (comp)
• 11edo: 2.9.15.7.11.17 (comp)
• 13edo: 2.9.5.11.13.17 (comp)
• 23edo: 59-limit but with 3.5.7.11 removed and 9.15.21.33 added (comp)

Nth-basis

Interestingly, all of these can be seen as 15th-basis. It might just be because in EDOs 2 is always pure, and 3 and 5 are the next simplest harmonics, so it just makes sense for them to show up as simple subgroup denominators.

• 7edo: 2.3.11/3.11/5.13 (nth-b)
• 8edo: 2.11/3.13/5.19 (nth-b)
• 14edo: 2.3.7/5.9/5.11/5.13 (nth-b)
• 18edo: 2.9.5.7/3.11 (nth-b)

Other fractional

No edos really fit this category.

(Technically any fractional subgroup can be said to be nth-basis, but if it were something absurdly big like 200th-basis, then it would belong in this category, not nth-basis, for the purpose of this list.

But, there aren't any edos where that kind of subgroup makes sense hence this category being empty.)