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Todo: short bio thingy, add more todos |
- Squib/Drafts
- Squib/Drafts/10ed5
- Squib/Drafts/1830125∕1830101
- 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
pages to work on
miracle/mirage/extensions/prism
mirage dedicated page
miracle extensions (31-limit manna extension)
User:Squib/Miracle extensions and mirage
5.7.11.13
possible 5.7.11.13 comma pages to create
higher-limit commas
1830125/1830101 (!!)
other randomness
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)
104976/104975 (s324)
1403830272/1403737447 (equidistance 715/714, 833/832, 936/935)
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
No-twos commas
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...
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
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
Intervals with monzos containing only ones
Non-subgroup monzos
Superparticular intervals:
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 |