User:Squib

hi im squib :)

i think miracle temperament is the one that's worth investing a large portion of my time in, although i might dabble in other things sometimes

i am currently working on figuring out how to use software to turn my ideas into actual music.

my user subpages are listed here for my own reference

• Squib/Drafts
• Squib/Drafts/10ed5
• Squib/Drafts/1830125∕1830101
• Squib/Drafts/3ed11∕9
• Squib/Drafts/5.7.11.13 subgroup
• Squib/Drafts/75803∕75625
• Squib/Drafts/847∕845
• Squib/Drafts/Miracle extensions and mirage
• Squib/List of edos sorted by accuracy of the perfect fifth
• Squib/List of tunings
• Squib/Simple rank-2 temperaments by subgroup
• Squib/Theory
• Squib/Todo
• Squib/Unnamed music theory
• Squib/sandbox

things i want to do on this wiki

• create pages about my ideas
  • User:Squib/Equivalence is a construct
  • User:Squib/Efficiency (comma metric)
  • User:Squib/2d MOS
  • User:Squib/monzos with only ones (with a better title) (not exploring this anymore, mostly just want to document it)
• miracle extensions
  • add extension info to pages about miracle
    • rastmic rank-3 clan #mirage (including #prism)
    • miracle extensions (including the 31-limit manna extension)
    • gamelismic clan #miracle
  • User:Squib/Miracle extensions and mirage
  • dedicated page for mirage
• 5.7.11.13 (no-2 no-3 13-limit)
  • 5.7.11.13 subgroup
  • 847/845
  • 10ed5
  • 125/121
  • 175/169
• misc
  • superpyth (tritave-equivalent)

some things i do not like about the wiki

this list is here because listing the things i do like would take too long.

• things on here can be very hard to understand. this is not controversial.
• It's hard to find a page you're looking for even if you know what it's about, but especially if you don't know whether such a page exists in the first place. Important pages for starters should be accessible by following links from the main page. In particular, I'd like a "bird's eye view of bird's eye view pages" page to be linked on the main page.
• Period equivalence is assumed everywhere. 5/2 and 5/4 are the same as much as 9/8 and 10/9 are; treating them identically can be useful in certain contexts, but they are not fundamentally the same thing. In a space dedicated to exploring new tuning and music, it is frankly ridiculous to make period equivalence one of the fundamental assumptions you build your theory and terminology on. To be clear, I don't have an issue with the concepts of periods or equivalence, nor am I denying their usefulness. I have an issue with the Xenharmonic Wiki (of all places) assuming that these things must exist in every musical context.
  • Cuthbert chords is a page about the chords enabled by tempering out 847/845, a comma in the 5.7.11.13 subgroup. Why are we talking about the 2.5.7.11.13 subgroup? What is prime 2 doing here??

Random stuff that i don't have the heart to delete yet

No-twos commas

245/243

here's a family of them

S(4n-1)/S(4n+1)

27/25, 245/243, 847/845, 2025/2023, 3971/3969, 6877/6875, 10935/10933, 16337/16335, 23275/23273, 31941/31939, 42527/42525, 55225/55223, 70227/70225, 87725/87723, 107911/107909, 130977/130975, 157115/157113, 186517/186515, 219375/219373, 255881/255879... 26578125/26578123...

No-threes commas

176/175 245/242 1001/1000 6656/6655 170/169 221/220 2200/2197 833/832

19-limit

209/208 476/475 1331/1330 1445/1444 2432/2431 6860/6859 10241/10240

here's a family of them

S(9n-5)/S(9n-4)

128/125, 10985/10976, 85184/85169, 327701/327680, 896000/895973...

random commas to make pages for maybe

13013/13005

104976/104975 (s324)

364/361

1403830272/1403737447 (equidistance 715/714, 833/832, 936/935)

21736/21735

117/115

=5.7.11.13 subgroup

343/325

637/625

15625/15488

17303/16807

78125/77077

831875/823543

2941225/2924207

27217619/26796875

49098049/48828125

236513641/236328125

higher-limit no-2 no-3

121/119

325/323 (no-2 no-3, tempered out by 19-limit mirage) (210/209 * 715/714) (273/272 * 400/399) (286/285 * 375/374) (325/324 * 324/323)

1830125/1830101 (!!)

structurally important edos

edo	subgroup	notes
10	13-limit	higher primes?
12	2.3.5.13/11.19
17	2.3.7.11	?
19	2.3.5.7.13
22	2.3.5.7.11.17
24	2.3.5.11.13
31	2.3.5.7.11.17/13.19/13
34	2.3.5.11.13.17.23
41
46
53
58
72
87
99	2.3.5.7.13/11	higher primes?
159
171
205

list of detemperaments

7-limit edos

12: septimal meantone, garibaldi, septimal compton, misty, term, (12 & 270), 12 & 612

19: septimal meantone, sensi, kleismic, parakleismic, enneadecal, (19 & 270), 19 & 2859bcddd (splits 140/1 in 135 parts)

22: 22 & 118, 22 & 171

rank-twos

miracle: portent, canopus, freya, 31 & 41 & 278cd, ..., 31 & 41 & 994bbbccccddee

orwell: 22 & 31 & 311, 22 & 31 & 494

squares: jove, parimo + breedsma

23-limit 24 & 34: 24 & 34 & 41(g), 24 & 34 & 53, 24 & 34 & 94, 24 & 34 & 217

Intervals with monzos containing only ones

Non-subgroup monzos

Superparticular intervals:

• 2/1
• 3/2
• 6/5
• 15/14
• 715/714

No other such superparticular intervals exist (at least in the first 100,000 prime limits).

Smallest for each prime limit:

2: 2/1

3: 3/2

5: 6/5

7: 15/14

11: 55/42

13: 182/165

17: 715/714

19: 3135/3094

23: 15015/14858

29: 81345/79534

31: 448630/447051

37: 2733549/2714690

41: 17490603/17395070

Subgroup monzos

A superparticular interval of this type exists if and only if the square root of 4n+1 is an integer, where n is the product of all primes in the subgroup. The result is the sum of the numerator and denominator of the superparticular interval.

(This method also works for intervals containing any number of the same prime. For example, with factors 2, 2, 2, 2, 3, and 5, n is 240 and (4n+1)^0.5 is 31, which is an integer. So these factors can form a superparticular interval whose numerator and denominator add to 31: 16/15.)

(For subgroups with rational or non-prime elements, split them into prime factors and multiply all together to get n, then determine if the final result is in the subgroup. For example, for the 11/2.13.15.19 subgroup, n is 81510 and (4n+1)^0.5 is 571, so the resulting superparticular interval is 286/285, but this is not in the subgroup because 11 and 2 are on the same side of the fraction. So no superparticular interval exists in the subgroup.)

(note about intervals like 35/33)

(this should probably get its own page lol)

All superparticular intervals with no duplicate primes, by prime limit

Found by applying this method to every possible subgroup in the prime limit, using this desmos graph.

	1 (superparticular)	2 (odd-particular)	3 (throdd-particular)
2-limit	2/1	-	-
3-limit	3/2	3/1	-
5-limit	6/5	5/3	5/2
7-limit	7/6, 15/14	7/5	10/7
11-limit	11/10, 22/21	35/33	14/11
13-limit	14/13, 66/65, 78/77	13/11, 15/13	13/10
17-limit	34/33, 35/34, 715/714	17/15	17/14
19-limit	39/38, 210/209, 286/285	19/17, 21/19, 57/55, 665/663	22/19, 38/35, 133/130, 190/187
23-limit	23/22, 70/69, 115/114, 231/230, 323/322, 391/390	23/21, 255/253, 1311/1309	26/23, 598/595, 2093/2090
29-limit	30/29, 58/57, 494/493, 2002/2001, 2262/2261	87/85, 145/143, 437/435, 667/665	29/26, 58/55, 322/319, 377/374, 1105/1102
31-limit	31/30, 155/154, 187/186, 435/434, 714/713, 806/805, 12122/12121	31/29, 33/31, 93/91, 95/93, 715/713, 899/897, 7163/7161	34/31, 65/62, 406/403, 437/434, 10013/10010

strong temperaments by rank

temperaments that are strong extensions of all of their restrictions

rank-1

every prime is mapped to 1 step (or -1 step)

rank-2

max 3 primes, 1 comma. equates one prime with the product of the other two (or tempers the product of all three). examples: 14/13, 23/21, 165/1

rank-3

max 4 primes 1 comma, although i'm not confident about that. examples: 31/30, 145/143

rank-4

5 primes 1 comma: 406/403, 494/493, 667/665

6 primes 2 commas: uh oh i think it might just be 1 comma max for all the ranks