Magrathean diacritics
- Dave Keenan
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Re: Magrathean diacritics
I was just reminded of another excellent resource regarding tinas. Joe Monzo's Tonalsoft Encyclopedia entry.
http://tonalsoft.com/enc/t/tina.aspx
http://tonalsoft.com/enc/t/tina.aspx
- cmloegcmluin
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Re: Magrathean diacritics
I have been working with @Dave Keenan to design the glyphs for the new Olympian accents. While I was at it, I designed glyphs for the Magrathean accents too. I will be sure to share those out once designs are finalized. I bring up this work now because in doing it Dave had the idea to reduce the count of code points we'd take up in the SMuFL standard by — instead of including a separate glyph for each n.5, n>0 tina — creating n.5, n>0 tina marks by combining glyphs for the 0.5 tina and the n tina. So, as Dave points out, we only really need to define primary commas for the whole tinas and then the 0.5 tina. And we've already got the 3 tina and 6 tina covered via the 1 mina and 2 mina.
@Ash9903b4 made some helpful suggestions, not all of which agree with George's. I thought I would go back and get the full information for George's suggestions, to help us make our decisions.
The 539n, by the way, is exactly the 455n + the 65:77n (the 3 tina + 6 tina)
He also gave another approximation for 5 tinas:
Ash9903b4, would you be able to get information for your suggestions in this form? You offer several suggestions for each of these tina values. Yours include George's in every case except the 0.5 tina and 1 tina.
Some observations about George's commas: they are quite low limit, and many of them are superparticular.
George's 0.5 tina value is rather nice, being only 7-limit, and only 0.00004862097¢ off from the half-step of 8539-EDO; it's only concern really is having a large apotome slope, greater than the maximum current (correcting my earlier statements, 11.558 is the max absolute value apotome slope, and 4.439 is the average).
I have a script that is able to find many more possibilities. I can give it a limit, an error threshold, a maximum count of primes in the monzo (beyond 3). Many of the commas I find coincide with Ash9903b4's too. Let me know if you have specific criteria you want me to meet and I can run the numbers.
@Ash9903b4 made some helpful suggestions, not all of which agree with George's. I thought I would go back and get the full information for George's suggestions, to help us make our decisions.
apotome popularity tina comma monzo cents 8539edo cents limit ratio slope "rank" 0.5 49:9765625n [ 3 -13 10 -2 ⟩ 0.07031446006 0.07026583909 7 78125000/78121827 -13.0 64 1 121:1225n [ -3 4 -2 -2 2 ⟩ 0.17664752314 0.14053167818 11 9801/9800 3.99 46 2 13:6655n [ 9 0 -1 0 -3 1 ⟩ 0.2601208102 0.28106335636 13 6656/6655 -0.0160 51 4 7:3025n [ -4 -3 2 -1 2 ⟩ 0.57240339389 0.56212671273 11 3025/3024 -3.04 39 5 25:2401n [ -5 -1 -2 4 ⟩ 0.72119728144 0.70265839091 7 2401/2400 -1.04 38 7 143:1715n [ 2 1 -1 -3 1 1 ⟩ 1.00917172725 0.98372174727 13 1716/1715 0.938 50 8 420175n [ -6 -8 2 5 ⟩ 1.1169531522 1.12425342546 7 420175/419904 -8.07 45 9 539n [ 17 -5 0 -2 -1 ⟩ 1.25524037006 1.26478510364 11 131072/130977 -5.08 25
The 539n, by the way, is exactly the 455n + the 65:77n (the 3 tina + 6 tina)
He also gave another approximation for 5 tinas:
apotome popularity tina comma monzo cents 8539eda cents limit ratio slope "rank" 5 343:14641n [ -7 1 0 -3 4 ⟩ 0.74905091704 0.70265839091 11 43923/43904 0.954 65
Ash9903b4, would you be able to get information for your suggestions in this form? You offer several suggestions for each of these tina values. Yours include George's in every case except the 0.5 tina and 1 tina.
Some observations about George's commas: they are quite low limit, and many of them are superparticular.
George's 0.5 tina value is rather nice, being only 7-limit, and only 0.00004862097¢ off from the half-step of 8539-EDO; it's only concern really is having a large apotome slope, greater than the maximum current (correcting my earlier statements, 11.558 is the max absolute value apotome slope, and 4.439 is the average).
I have a script that is able to find many more possibilities. I can give it a limit, an error threshold, a maximum count of primes in the monzo (beyond 3). Many of the commas I find coincide with Ash9903b4's too. Let me know if you have specific criteria you want me to meet and I can run the numbers.
- Dave Keenan
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Re: Magrathean diacritics
It seems Ash may be busy with other things. So you might have to put his 0.5 tina and whole tina suggestions in that form yourself. I note that, because these commas are so small, there is no significant difference between the slope and the 3-exponent, so bolding the 3-exponent in the monzo could save a column.
It would also be more useful to see the error expressed in tinas (with only 2 decimal places), in place of those two columns of cents. And the tina you use should be based on 809eda, not 8539edo. I note that you have a hybrid heading "8539eda".
It would be good to include the (settled) 3-tina and 6-tina commas in the list for comparison. And it would be good to have all candidates for a given number of tinas grouped together in the one list.
Criteria for the 0.5 tina would be "no worse than the one we've got" in either absolute 3-exponent or in the popularity "rank" (sum of prime factors above 3, SoPF>3), while having error no more than a quarter tina. I suppose we should allow primes up to 37.
For the whole tinas, perhaps "rank" ≤ 51, abs 3-exponent ≤ 12, error < 0.25 tinas.
It would also be more useful to see the error expressed in tinas (with only 2 decimal places), in place of those two columns of cents. And the tina you use should be based on 809eda, not 8539edo. I note that you have a hybrid heading "8539eda".
It would be good to include the (settled) 3-tina and 6-tina commas in the list for comparison. And it would be good to have all candidates for a given number of tinas grouped together in the one list.
Criteria for the 0.5 tina would be "no worse than the one we've got" in either absolute 3-exponent or in the popularity "rank" (sum of prime factors above 3, SoPF>3), while having error no more than a quarter tina. I suppose we should allow primes up to 37.
For the whole tinas, perhaps "rank" ≤ 51, abs 3-exponent ≤ 12, error < 0.25 tinas.
- Dave Keenan
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Re: Magrathean diacritics
An excellent question. I'm glad you asked it before we submitted the Magratheans to Steinberg.cmloegcmluin wrote: ↑Wed May 20, 2020 4:23 am Somehow this thought never surfaced until just now -- but do we really need up to 9.5 tinas? We don't even go past 6 tinas (2 minas) in Olympian. Do we have reason to suspect we'll need them when defining symbol+comma+boundaries for the Magrathean precision?
Of course you get the last 0.5 for free since you already have the dot. But I don't remember why we went to 9 in whole tinas. I'd have to dig back through emails. It may be simply that we needed to go beyond 6 tinas and so we were combining up to 3 elements graphically, so we thought: Why not include every whole number of tinas that can be done with 3 elements.
But when we think about what we might actually need, we should look at how many tinas there are in the smallest non-tina element, which is the schisma accent. (I'm considering the mina to be a tina element here because it uses the same element as 3 tinas). Whether we take the schisma accent to be 5s (32805/32768) or 1 degree of 58eda, the answer is 13.9, call it 14 tinas.
So you might say we only need symbols for up to 7 tinas, however we should consider the principle that results in us having single symbols up to about 60% of the apotome in the evo notation.
That principle is that, up to some point, it's easier to deal with fractions greater than half than to deal with fractions subtracted from the whole. Or putting it another way: At some point it's better to stop making symbols for larger fractions and instead symbolise it as subtracting a small fraction from the whole. So where is that point?
In the case of symbols within the apotome, we settled on something close to 60% for reasons of convenience having to do with square roots of 2,3-commas. In the case of minas within the schisma, in Olympian, the figure is about 62.5% since we go from for 50% to instead of for 75%, with the capture zone boundary roughly halfway between.
60% of 14 is 8.4. 62.5% of 14 is 8.75 which rounds up to 9.
The fact that we don't use as a 3 mina symbol in Olympian, could be considered an artifact of the fact that in 233eda there are only 4 tempered 455n in the tempered 5s. Had there been 5 (as per a recent mistake of mine) would we have preferred or for 3 minas? What if someone in future wants to notate a temperament in which there are 15 tinas to the schisma. Then would be only 60% of the schisma.
Re: Magrathean diacritics
Code: Select all
tina comma monzo cents 8539cents limit ratio popularity
0.0 11:228095n [ 8 4 -1 -4 1 0 0 -1 > 0.007590 0.000000 19 228096/228095 63
0.0 323:6125n [ 9 -3 -3 -2 0 0 1 1 > 0.010469 0.000000 19 165376/165375 65
0.0 169:1925n [ -6 6 -2 -1 -1 2 0 0 > 0.014052 0.000000 13 123201/123200 54
0.0 1:104975n [ 4 8 -2 0 0 -1 -1 -1 > 0.016492 0.000000 19 104976/104975 59
0.5 17:9295n [ 2 -7 1 0 1 2 -1 0 > 0.046564 0.070266 17 37180/37179 59
0.5 289:1463n [ -4 4 0 -1 -1 0 2 -1 > 0.073957 0.070266 19 23409/23408 71
1.0 19:23375n [ 4 -9 3 0 1 0 1 -1 > 0.106470 0.140532 19 374000/373977 62
1.0 247:60025n [ 0 -5 2 4 0 -1 0 -1 > 0.115371 0.140532 19 60025/60021 70
1.0 25:14399n [ 6 2 2 -1 -2 0 -1 0 > 0.120229 0.140532 17 14400/14399 56
1.0 55:4693n [ 8 -1 1 0 1 -1 0 -2 > 0.122961 0.140532 19 14080/14079 67
1.0 65:833n [ 9 -8 -1 2 0 -1 1 0 > 0.125840 0.140532 17 426496/426465 49
1.0 209:4459n [ -6 1 0 3 -1 1 0 -1 > 0.129424 0.140532 19 13377/13376 64
1.0 19:207515n [ 15 -1 -1 -3 -2 0 0 1 > 0.130697 0.140532 19 622592/622545 67
1.0 13:12635n [ 2 5 -1 -1 0 1 0 -2 > 0.137013 0.140532 19 12636/12635 63
1.0 1375:1547n [ 3 -2 -3 1 -1 1 1 0 > 0.139892 0.140532 17 12376/12375 63
1.0 1:196625n [-16 -1 3 0 2 1 0 0 > 0.149687 0.140532 13 196625/196608 50
1.0 7:653125n [ 7 6 -5 1 -1 0 0 -1 > 0.156384 0.140532 19 653184/653125 62
1.0 1183:1331n [ 3 -2 0 -1 3 -2 0 0 > 0.162595 0.140532 13 10648/10647 66
1.0 5:10241n [-11 0 -1 2 1 0 0 1 > 0.169058 0.140532 19 10241/10240 49
1.5 121:1225n [ -3 4 -2 -2 2 0 0 0 > 0.176648 0.210798 11 9801/9800 46
1.5 25:15379n [ 5 -9 -2 1 0 3 0 0 > 0.186456 0.210798 13 492128/492075 56
1.5 2197:15625n [ -6 2 6 0 0 -3 0 0 > 0.209300 0.210798 13 140625/140608 69
1.5 5:2023n [-15 4 -1 1 0 0 2 0 > 0.243015 0.210798 17 163863/163840 46
2.0 13:6655n [ 9 0 -1 0 -3 1 0 0 > 0.260121 0.281063 13 6656/6655 51
2.0 119:40625n [-10 1 5 -1 0 1 -1 0 > 0.269916 0.281063 17 121875/121856 62
2.0 343:6175n [ -1 -2 2 -3 0 1 0 1 > 0.280384 0.281063 19 6175/6174 63
2.0 169:361n [ 10 -7 0 0 0 -2 0 2 > 0.285703 0.281063 19 369664/369603 64
2.0 187:665n [ -5 2 1 1 -1 0 -1 1 > 0.289286 0.281063 19 5985/5984 59
2.0 1:221221n [-13 -3 0 1 1 2 1 0 > 0.289579 0.281063 17 221221/221184 61
2.0 247:5929n [ -3 -1 0 2 2 -1 0 -1 > 0.292019 0.281063 19 5929/5928 68
2.0 1:5831n [ 3 6 0 -3 0 0 -1 0 > 0.296876 0.281063 17 5832/5831 38
2.0 5:218491n [-17 1 -1 5 0 1 0 0 > 0.298481 0.281063 13 655473/655360 53
2.0 361:1925n [ 4 -1 -2 -1 -1 0 0 2 > 0.299755 0.281063 19 5776/5775 66
2.0 1625:2057n [ 6 -4 -3 0 2 -1 1 0 > 0.302487 0.281063 17 131648/131625 67
2.0 7:206635n [ -1 10 -1 1 -1 -1 -2 0 > 0.305778 0.281063 17 413343/413270 70
2.0 475:1001n [ -9 5 -2 1 1 1 0 -1 > 0.306071 0.281063 19 243243/243200 60
2.5 343:15625n [ -4 6 -6 3 0 0 0 0 > 0.325441 0.351329 7 250047/250000 51
2.5 85:121n [ -9 6 1 0 -2 0 1 0 > 0.363244 0.351329 17 61965/61952 44
2.5 1:43681n [ 17 -1 0 0 -2 0 0 -2 > 0.383082 0.351329 19 131072/131043 60
3.0 1:4375n [ -1 -7 4 1 0 0 0 0 > 0.395756 0.421595 7 4375/4374 27
3.0 1375:6517n [ 7 -3 3 -3 1 0 0 -1 > 0.403346 0.421595 19 176000/175959 66
3.0 7:1615n [ 8 -10 1 -1 0 0 1 1 > 0.406224 0.421595 19 413440/413343 48
3.0 11:4225n [ -7 -1 2 0 -1 2 0 0 > 0.409808 0.421595 13 4225/4224 47
3.0 175:4199n [ 3 1 2 1 0 -1 -1 -1 > 0.412248 0.421595 19 4200/4199 66
3.0 143:6517n [ 4 -6 0 3 -1 -1 0 1 > 0.415126 0.421595 19 104272/104247 64
3.0 1:455n [ 12 -2 -1 -1 0 -1 0 0 > 0.422716 0.421595 13 4096/4095 25
3.0 13:3553n [ -3 7 0 0 -1 1 -1 -1 > 0.426300 0.421595 19 28431/28424 60
3.0 13:67375n [ 6 4 -3 -2 -1 1 0 0 > 0.436768 0.421595 13 67392/67375 53
3.0 1:248897n [-10 -5 0 0 4 0 1 0 > 0.452175 0.421595 17 248897/248832 61
3.5 209:595n [ -8 6 -1 -1 1 0 -1 1 > 0.465934 0.491861 19 152361/152320 59
3.5 343:1445n [ 10 -5 -1 3 0 0 -2 0 > 0.478182 0.491861 17 351232/351135 60
3.5 2057:15625n [ 5 -5 6 0 -2 0 -1 0 > 0.515985 0.491861 17 500000/499851 69
3.5 935:1183n [ -6 4 1 -1 1 -2 1 0 > 0.525839 0.491861 17 75735/75712 66
4.0 361:1625n [ 1 -2 3 0 0 1 0 -2 > 0.532769 0.562127 19 3250/3249 66
4.0 11:54145n [ 2 -9 1 2 -1 1 1 0 > 0.535648 0.562127 17 216580/216513 60
4.0 221:931n [ 10 -5 0 -2 0 1 1 -1 > 0.543238 0.562127 19 226304/226233 63
4.0 121:117649n [ -2 -5 0 6 -2 0 0 0 > 0.544550 0.562127 11 117649/117612 64
4.0 49:1045n [ 6 -1 -1 2 -1 0 0 -1 > 0.552140 0.562127 19 3136/3135 49
4.0 1:442225n [ 14 3 -2 -2 0 0 0 -2 > 0.559730 0.562127 19 442368/442225 62
4.0 17:6875n [ 15 -4 -4 0 -1 0 1 0 > 0.562608 0.562127 17 557056/556875 48
4.0 7:3025n [ -4 -3 2 -1 2 0 0 0 > 0.572403 0.562127 11 3025/3024 39
4.0 221:18865n [ -8 1 1 3 1 -1 -1 0 > 0.581305 0.562127 17 56595/56576 67
4.0 49:1859n [-10 3 0 -2 1 2 0 0 > 0.586456 0.562127 13 50193/50176 51
4.0 325:1463n [ 1 -2 -2 1 1 -1 0 1 > 0.591774 0.562127 19 2926/2925 60
4.5 121:557375n [ 9 2 -3 -3 2 -1 0 0 > 0.599364 0.632393 13 557568/557375 71
4.5 247:625n [ -5 4 -4 0 0 1 0 1 > 0.605826 0.632393 19 20007/20000 52
4.5 1:265625n [ -1 12 -6 0 0 0 -1 0 > 0.622318 0.632393 17 531441/531250 47
4.5 125:91091n [ 0 6 3 -2 -1 -2 0 0 > 0.646068 0.632393 13 91125/91091 66
4.5 1:93347n [ -7 -6 0 0 0 0 3 1 > 0.649239 0.632393 19 93347/93312 70
4.5 289:325n [ -3 2 -2 0 0 -1 2 0 > 0.665731 0.632393 17 2601/2600 57
5.0 625:833n [ 2 -1 4 -2 0 0 -1 0 > 0.692632 0.702658 17 2500/2499 51
5.0 11:9025n [ 3 -8 2 0 -1 0 0 2 > 0.695511 0.702658 19 72200/72171 59
5.0 19:175175n [-10 -2 2 2 1 1 0 -1 > 0.701827 0.702658 19 175175/175104 67
5.0 95:2401n [ 11 -4 1 -4 0 0 0 1 > 0.703101 0.702658 19 194560/194481 52
5.0 19:2431n [ 7 0 0 0 -1 -1 -1 1 > 0.712002 0.702658 19 2432/2431 60
5.0 5:2431n [ -1 -5 -1 0 1 1 1 0 > 0.712295 0.702658 17 2431/2430 46
5.0 25:2401n [ -5 -1 -2 4 0 0 0 0 > 0.721197 0.702658 7 2401/2400 38
5.0 11:2375n [ 3 3 -3 0 1 0 0 -1 > 0.728787 0.702658 19 2376/2375 45
5.5 343:14641n [ -7 1 0 -3 4 0 0 0 > 0.749051 0.772924 11 43923/43904 65
5.5 289:1235n [ 9 -7 1 0 0 1 -2 1 > 0.758567 0.772924 19 632320/632043 71
5.5 1:265837n [ 1 -12 0 0 2 3 0 0 > 0.758860 0.772924 13 531674/531441 61
5.5 121:371875n [-10 -1 5 1 -2 0 1 0 > 0.759000 0.772924 17 371875/371712 71
5.5 1:24565n [ 13 1 -1 0 0 0 -3 0 > 0.775058 0.772924 17 24576/24565 56
5.5 17:11011n [ 3 4 0 -1 -2 -1 1 0 > 0.785960 0.772924 17 11016/11011 59
6.0 169:475n [-11 6 2 0 0 -2 0 1 > 0.815126 0.843190 19 346275/346112 55
6.0 1:174845n [-19 1 1 0 2 0 2 0 > 0.815418 0.843190 17 524535/524288 61
6.0 13:125n [ 11 -9 3 0 0 -1 0 0 > 0.818472 0.843190 13 256000/255879 28
6.0 65:77n [ 5 -3 1 -1 -1 1 0 0 > 0.832524 0.843190 13 2080/2079 36
6.0 5:54587n [ 15 -1 1 0 0 -2 -1 -1 > 0.834964 0.843190 19 163840/163761 67
6.0 13:319333n [ 13 1 0 -5 0 1 0 -1 > 0.840114 0.843190 19 319488/319333 67
6.0 343:2057n [ 1 1 0 3 -2 0 -1 0 > 0.841426 0.843190 17 2058/2057 60
6.0 1:124355n [ 9 5 -1 -1 -1 0 -1 -1 > 0.849016 0.843190 19 124416/124355 59
6.0 133:15125n [ 10 -2 -3 1 -2 0 0 1 > 0.851895 0.843190 19 136192/136125 63
6.0 17:26125n [ -9 -1 3 0 1 0 -1 1 > 0.861690 0.843190 19 26125/26112 62
6.0 49:3971n [ 0 -4 0 -2 1 0 0 2 > 0.872158 0.843190 19 3971/3969 63
6.0 55:3757n [ -5 7 1 0 1 -1 -2 0 > 0.878182 0.843190 17 120285/120224 63
6.5 19:17303n [-13 2 0 0 3 1 0 -1 > 0.878474 0.913456 19 155727/155648 65
6.5 625:5929n [ -8 3 -4 2 2 0 0 0 > 0.897845 0.913456 11 160083/160000 56
6.5 7:2125n [-13 3 3 -1 0 0 1 0 > 0.935647 0.913456 17 57375/57344 39
7.0 25:2527n [ 13 -4 2 -1 0 0 0 -2 > 0.955486 0.983722 19 204800/204687 55
7.0 17:111475n [ 0 8 -2 -3 0 -1 1 0 > 0.962608 0.983722 17 111537/111475 61
7.0 13:256025n [ 0 -9 2 2 1 -1 0 1 > 0.987530 0.983722 19 256025/255879 67
7.0 17:557375n [-15 0 3 3 0 1 -1 0 > 0.991113 0.983722 17 557375/557056 66
7.0 605:637n [ 8 -5 1 -2 2 -1 0 0 > 0.995120 0.983722 13 154880/154791 54
7.0 1:1729n [ -6 -3 0 1 0 1 0 1 > 1.001582 0.983722 19 1729/1728 39
7.0 539:2873n [ 4 -1 0 2 1 -2 -1 0 > 1.004021 0.983722 17 8624/8619 68
7.0 143:1715n [ 2 1 -1 -3 1 1 0 0 > 1.009172 0.983722 13 1716/1715 50
7.0 7:425n [ -2 5 -2 1 0 0 -1 0 > 1.018074 0.983722 17 1701/1700 34
7.5 19:21875n [ 7 2 -5 -1 0 0 0 1 > 1.028542 1.053988 19 21888/21875 51
7.5 13:50575n [ -4 -5 2 1 0 -1 2 0 > 1.061487 1.053988 17 50575/50544 64
7.5 95:17303n [ -2 6 1 0 -3 -1 0 1 > 1.075246 1.053988 19 69255/69212 70
7.5 11:3757n [-10 1 0 0 -1 1 2 0 > 1.075539 1.053988 17 11271/11264 58
7.5 119:3211n [ 0 3 0 1 0 -2 1 -1 > 1.077979 1.053988 19 3213/3211 69
8.0 17:5225n [ -6 9 -2 0 -1 0 1 -1 > 1.092031 1.124253 19 334611/334400 57
8.0 1:420175n [ -6 -8 2 5 0 0 0 0 > 1.116953 1.124253 7 420175/419904 45
8.0 1:186745n [ -8 -6 1 0 0 3 1 0 > 1.122103 1.124253 17 186745/186624 61
8.0 19:385n [ 2 -4 1 1 1 0 0 -1 > 1.124543 1.124253 19 1540/1539 42
8.0 11:405769n [-12 -2 0 4 -1 2 0 0 > 1.131005 1.124253 13 405769/405504 65
8.0 175:187n [ 11 -7 -2 -1 1 0 1 0 > 1.135012 1.124253 17 382976/382725 45
8.0 1:34969n [ 4 7 0 0 -2 0 -2 0 > 1.138302 1.124253 17 34992/34969 56
8.0 95:169n [ -4 2 -1 0 0 2 0 -1 > 1.138595 1.124253 19 1521/1520 50
8.0 343:1625n [ 7 -3 -3 3 0 -1 0 0 > 1.143913 1.124253 13 43904/43875 49
8.0 637:34375n [ 1 3 -5 2 -1 1 0 0 > 1.157966 1.124253 13 34398/34375 63
8.5 13:25289n [ -3 -5 0 0 3 -1 0 1 > 1.164177 1.194519 19 25289/25272 65
8.5 175:29887n [ -9 1 -2 -1 2 1 0 1 > 1.178229 1.194519 19 89661/89600 71
8.5 1:177023n [ 0 11 0 -1 -3 0 0 -1 > 1.212260 1.194519 19 177147/177023 59
8.5 11:40625n [ 4 -10 5 0 -1 1 0 0 > 1.228280 1.194519 13 650000/649539 49
9.0 95:15379n [ 1 4 1 -1 0 -3 0 1 > 1.237842 1.264785 19 15390/15379 70
9.0 125:3553n [ 8 -2 3 0 -1 0 -1 -1 > 1.244772 1.264785 19 32000/31977 62
9.0 1:539n [ 17 -5 0 -2 -1 0 0 0 > 1.255240 1.264785 11 131072/130977 25
9.0 49:133705n [ 13 -1 -1 2 -2 -1 -1 0 > 1.264142 1.264785 17 401408/401115 71
9.0 1:41503n [ -9 -4 0 3 2 0 0 0 > 1.293601 1.264785 11 41503/41472 43
9.0 247:833n [ -3 3 0 -2 0 1 -1 1 > 1.298458 1.264785 19 6669/6664 63
9.5 665:1331n [ -1 0 -1 -1 3 0 0 -1 > 1.301191 1.335051 19 1331/1330 64
9.5 1:354025n [ 1 11 -2 -2 0 0 -2 0 > 1.314950 1.335051 17 354294/354025 58
9.5 847:40625n [ 4 1 -5 1 2 -1 0 0 > 1.320561 1.335051 13 40656/40625 67
9.5 143:78125n [ -2 7 -7 0 1 1 0 0 > 1.334613 1.335051 13 312741/312500 59
9.5 247:3125n [-10 4 5 0 0 -1 0 -1 > 1.347895 1.335051 19 253125/252928 57
9.5 425:637n [ -1 1 2 -2 0 -1 1 0 > 1.358363 1.335051 17 1275/1274 54
9.5 245:1859n [ -5 5 1 2 -1 -2 0 0 > 1.367265 1.335051 13 59535/59488 56
10.0 221:3773n [ -7 7 0 -3 -1 1 1 0 > 1.372416 1.405317 17 483327/482944 62
10.0 1:157339n [ 3 9 0 -2 0 -2 0 -1 > 1.374855 1.405317 19 157464/157339 59
10.0 91:2299n [ 11 -4 0 1 -2 1 0 -1 > 1.384664 1.405317 19 186368/186219 61
10.0 49:17875n [ 1 -6 3 -2 1 1 0 0 > 1.404928 1.405317 13 35750/35721 53
10.0 17:1225n [ -3 -2 2 2 0 0 -1 0 > 1.413829 1.405317 17 1225/1224 41
10.0 343:10985n [ -5 0 1 -3 0 3 0 0 > 1.418980 1.405317 13 10985/10976 65
10.0 55:15827n [ 5 2 1 -2 1 0 -1 -1 > 1.421419 1.405317 19 15840/15827 66
10.0 5:19n [ 6 -5 -1 0 0 0 0 1 > 1.424298 1.405317 19 1216/1215 24
10.0 187:1183n [ -9 4 0 1 -1 2 -1 0 > 1.427882 1.405317 17 95823/95744 61
10.0 11:60025n [ 14 -1 -2 -4 1 0 0 0 > 1.431888 1.405317 11 180224/180075 49
10.5 1:27625n [ 10 3 -3 0 0 -1 -1 0 > 1.440790 1.475583 17 27648/27625 45
10.5 25:102487n [-12 0 -2 1 4 0 0 0 > 1.470248 1.475583 11 102487/102400 61
10.5 169:1445n [ 8 -7 1 0 0 -2 2 0 > 1.484203 1.475583 17 369920/369603 65
10.5 143:7225n [ 12 -4 -2 0 1 1 -2 0 > 1.487354 1.475583 17 585728/585225 68
10.5 209:2975n [ -7 2 2 1 -1 0 1 -1 > 1.487787 1.475583 19 26775/26752 64
10.5 289:385n [ 2 -1 -1 -1 -1 0 2 0 > 1.498255 1.475583 17 1156/1155 57
10.5 7:51623n [ -3 10 0 1 -1 -1 0 -2 > 1.504279 1.475583 19 413343/412984 69
11.0 13:17765n [-12 1 1 0 1 -1 1 1 > 1.527421 1.545848 19 53295/53248 65
11.0 19:147875n [ -5 -5 3 1 0 2 0 -1 > 1.534351 1.545848 19 147875/147744 67
11.0 11:845n [ -8 9 -1 0 1 -2 0 0 > 1.543913 1.545848 13 216513/216320 42
11.0 95:14399n [ 12 -3 1 -1 -2 0 -1 1 > 1.544527 1.545848 19 389120/388773 70
11.0 7:2873n [ 4 -8 0 -1 0 2 1 0 > 1.544820 1.545848 17 45968/45927 50
11.0 11:247n [ 14 -6 0 0 1 -1 0 -1 > 1.547259 1.545848 19 180224/180063 43
11.0 55:4459n [ 0 -4 -1 3 -1 1 0 0 > 1.553721 1.545848 13 4459/4455 50
11.0 13:3325n [ 8 0 -2 -1 0 1 0 -1 > 1.561311 1.545848 19 3328/3325 49
11.0 7:171875n [ 13 1 -6 1 -1 0 0 0 > 1.580682 1.545848 11 172032/171875 48
11.5 17:121n [ -6 2 0 0 2 0 -1 0 > 1.590477 1.616114 17 1089/1088 39
11.5 2299:6125n [ 3 -1 -3 -2 2 0 0 1 > 1.600945 1.616114 19 18392/18375 70
11.5 85:65219n [ 8 1 1 -2 -3 0 1 0 > 1.618484 1.616114 17 65280/65219 69
11.5 343:78125n [-11 2 7 -3 0 0 0 0 > 1.628279 1.616114 7 703125/702464 56
11.5 1001:2375n [ -6 3 3 -1 -1 -1 0 1 > 1.647650 1.616114 19 64125/64064 65
12.0 25:218491n [ 2 7 2 -5 0 -1 0 0 > 1.655240 1.686380 13 218700/218491 58
12.0 1:29155n [-18 2 1 3 0 0 1 0 > 1.656845 1.686380 17 262395/262144 43
12.0 133:187n [-10 6 0 -1 1 0 1 -1 > 1.664434 1.686380 19 136323/136192 54
12.0 25:38437n [ -9 -1 -2 1 0 0 2 1 > 1.667313 1.686380 19 38437/38400 70
12.0 343:361n [ 8 -5 0 3 0 0 0 -2 > 1.676683 1.686380 19 87808/87723 59
12.0 119:8125n [ -5 7 -4 1 0 -1 1 0 > 1.683805 1.686380 17 260253/260000 57
12.0 19:166375n [ -2 -7 3 0 3 0 0 -1 > 1.696946 1.686380 19 166375/166212 67
12.0 25:2023n [ 0 4 2 -1 0 0 -2 0 > 1.710706 1.686380 17 2025/2023 51
12.0 65:5929n [ 3 -6 -1 2 2 -1 0 0 > 1.716317 1.686380 13 47432/47385 54
12.0 19:29155n [ 9 1 -1 -3 0 0 -1 1 > 1.721174 1.686380 19 29184/29155 62
12.5 125:1001n [ -3 0 -3 1 1 1 0 0 > 1.730369 1.756646 13 1001/1000 46
12.5 91:265625n [ -2 -6 6 -1 0 -1 1 0 > 1.754119 1.756646 17 265625/265356 67
12.5 121:6125n [-12 4 3 2 -2 0 0 0 > 1.777073 1.756646 11 496125/495616 51
12.5 85:57967n [ 11 -1 1 -3 0 -2 1 0 > 1.781080 1.756646 17 174080/173901 69
12.5 25:10241n [ -4 8 2 -2 -1 0 0 -1 > 1.784663 1.756646 19 164025/163856 54
12.5 121:323n [ -3 1 0 0 -2 0 1 1 > 1.787542 1.756646 19 969/968 58
12.5 7:24167n [ 7 3 0 1 -1 -3 0 0 > 1.789981 1.756646 13 24192/24167 57
13.0 17:132055n [ 5 5 -1 -4 -1 0 1 0 > 1.795132 1.826912 17 132192/132055 61
13.0 1:39325n [ 1 9 -2 0 -2 -1 0 0 > 1.804034 1.826912 13 39366/39325 45
13.0 1:16625n [ 5 -12 3 1 0 0 0 1 > 1.820054 1.826912 19 532000/531441 41
13.0 11:16055n [ -1 -6 1 0 -1 2 0 1 > 1.834106 1.826912 19 16055/16038 61
13.0 35:221n [ 9 -4 1 1 0 -1 -1 0 > 1.836546 1.826912 17 17920/17901 42
13.0 169:2401n [ 7 -2 0 -4 0 2 0 0 > 1.841696 1.826912 13 21632/21609 54
13.0 95:184877n [ -3 -5 -1 5 1 0 0 -1 > 1.845740 1.826912 19 184877/184680 70
13.0 13:935n [ 3 2 -1 0 -1 1 -1 0 > 1.850598 1.826912 17 936/935 46
13.5 11:36125n [ 1 -8 3 0 -1 0 2 0 > 1.894011 1.897178 17 72250/72171 60
13.5 85:2717n [ 5 0 1 0 -1 -1 1 -1 > 1.910503 1.897178 19 2720/2717 65
14.0 1:524875n [-19 0 3 0 0 1 1 1 > 1.937229 1.967443 19 524875/524288 64
14.0 169:9625n [ -9 2 3 1 1 -2 0 0 > 1.939669 1.967443 13 86625/86528 59
14.0 1:5n [-15 8 1 0 0 0 0 0 > 1.953721 1.967443 5 32805/32768 5
- Dave Keenan
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Re: Magrathean diacritics
Thanks for that, Ash. It's good that you went all the way up to 14 tinas (the 5-schisma), because it made me realise that in deciding the default commas for whole tinas less than 6, we need to consider the possible commas for the combinations where they are subtracted from the 5-schisma.
Because the 5-schisma has a 3-exponent of +8, that suggests we should be biased towards low-numbered tina commas having positive 3-exponents.
It also suggests choosing n-tina commas that stay as approximately the same fraction of a 5-schisma when the 5-schisma is tempered. i.e. it suggests calculating the 5-schisma-slope of the commas. However the 5-schisma can be tempered in two dimensions, 3-prime and 5-prime. But I think the 5-prime slope is less important.
It also suggests that the commas for n>7-tinas should be the 5-schisma complements of the n<7-tina commas.
Because the 5-schisma has a 3-exponent of +8, that suggests we should be biased towards low-numbered tina commas having positive 3-exponents.
It also suggests choosing n-tina commas that stay as approximately the same fraction of a 5-schisma when the 5-schisma is tempered. i.e. it suggests calculating the 5-schisma-slope of the commas. However the 5-schisma can be tempered in two dimensions, 3-prime and 5-prime. But I think the 5-prime slope is less important.
It also suggests that the commas for n>7-tinas should be the 5-schisma complements of the n<7-tina commas.
- Dave Keenan
- Site Admin
- Posts: 2180
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- Contact:
Re: Magrathean diacritics
Ash, I know you never claimed this was an exhaustive list, even of those limited to 3-exponents from -12 thru 12, 19-prime-limit and sum-of-prime-factors 51 or less. And I can see that it is not, because I can generate others meeting those criteria by subtracting some of yours from the 5-schisma.
But this list will be very useful as a check, if cmloegcmluin cranks up some code that does claim to be exhaustive. Thanks.
But this list will be very useful as a check, if cmloegcmluin cranks up some code that does claim to be exhaustive. Thanks.
Re: Magrathean diacritics
I started by generating a list of intervals with odd limit 720720 or less. This is admittedly suboptimal, because there are intervals with odd limits in the millions that meet those criteria, for example, the 1890625-schismina that I had listed previously, but I didn't save the original script and didn't have much time to work on algorithm design.
I'll edit my post as soon as I can produce an exhaustive list.
I'll edit my post as soon as I can produce an exhaustive list.
Re: Magrathean diacritics
I'm fairly sure this is an exhaustive list of all 19-limit commas smaller than the 5-schisma with a 3-exponent between -12 and 12, and with a popularity rank below 72.
Code: Select all
0 tinas
tinas comma monzo interval cents limit popularity
0.00 1n [ 0 0 0 0 0 0 0 0 > 1 0.000000 1 0
0.05 11:228095n [ 8 4 -1 -4 1 0 0 -1 > 228096/228095 0.007590 19 63
0.07 323:6125n [ 9 -3 -3 -2 0 0 1 1 > 165376/165375 0.010469 19 65
0.10 169:1925n [ -6 6 -2 -1 -1 2 0 0 > 123201/123200 0.014052 13 54
0.12 104975n [ 4 8 -2 0 0 -1 -1 -1 > 104976/104975 0.016492 19 59
0.14 30685n [ 26 -7 -1 0 0 0 -1 -2 > 67108864/67108095 0.019838 19 60
0.19 1990625n [ 13 5 -5 -2 0 -1 0 0 > 1990656/1990625 0.026960 13 52
0.22 581875n [ 35 -10 -4 -2 0 0 0 -1 > 34359738368/34359136875 0.030307 19 53
0.5 tinas
tinas comma monzo interval cents limit popularity
0.27 22370117n [-26 1 0 5 3 0 0 0 > 67110351/67108864 0.038360 11 68
0.31 17:2717n [-20 8 0 0 1 1 -1 1 > 17826237/17825792 0.043218 19 60
0.33 17:9295n [ 2 -7 1 0 1 2 -1 0 > 37180/37179 0.046564 17 59
0.35 55859375n [ 40 -9 -8 0 -1 -1 0 0 > 1099511627776/1099480078125 0.049677 13 64
0.42 121:38675n [ 21 -8 -2 -1 2 -1 -1 0 > 253755392/253746675 0.059472 17 69
0.52 5:1574573n [ 4 9 1 -1 -3 -2 0 0 > 1574640/1574573 0.073665 13 71
0.53 289:1463n [ -4 4 0 -1 -1 0 2 -1 > 23409/23408 0.073957 19 71
0.63 5:1771561n [ -1 -11 -1 0 6 0 0 0 > 1771561/1771470 0.088931 11 71
0.69 3125:31213n [ 16 -8 5 -4 0 -1 0 0 > 204800000/204788493 0.097275 13 66
0.73 13:4675n [-18 6 2 0 1 -1 1 0 > 3408075/3407872 0.103123 17 51
1 tinas
tinas comma monzo interval cents limit popularity
0.76 19:23375n [ 4 -9 3 0 1 0 1 -1 > 374000/373977 0.106470 19 62
0.80 169:12005n [-22 10 1 4 0 -2 0 0 > 708883245/708837376 0.112025 13 59
0.82 247:60025n [ 0 -5 2 4 0 -1 0 -1 > 60025/60021 0.115371 19 70
0.83 49:3179n [ 13 -12 0 -2 1 0 2 0 > 26042368/26040609 0.116938 17 59
0.83 7:221n [-24 12 0 -1 0 1 1 0 > 117448461/117440512 0.117175 17 37
0.86 25:14399n [ 6 2 2 -1 -2 0 -1 0 > 14400/14399 0.120229 17 56
0.87 55:4693n [ 8 -1 1 0 1 -1 0 -2 > 14080/14079 0.122961 19 67
0.88 19882681n [ 29 -3 0 -6 0 -2 0 0 > 536870912/536832387 0.124235 13 68
0.90 65:833n [ 9 -8 -1 2 0 -1 1 0 > 426496/426465 0.125840 17 49
0.92 209:4459n [ -6 1 0 3 -1 1 0 -1 > 13377/13376 0.129424 19 64
0.93 19:207515n [ 15 -1 -1 -3 -2 0 0 1 > 622592/622545 0.130697 19 67
0.97 13:12635n [ 2 5 -1 -1 0 1 0 -2 > 12636/12635 0.137013 19 63
0.98 226474325n [ 23 3 -2 -7 -1 0 0 0 > 226492416/226474325 0.138287 11 70
1.00 1375:1547n [ 3 -2 -3 1 -1 1 1 0 > 12376/12375 0.139892 17 63
1.02 7:113135n [ 33 -12 -1 1 -3 0 -1 0 > 60129542144/60124577535 0.142946 17 62
1.07 196625n [-16 -1 3 0 2 1 0 0 > 196625/196608 0.149687 13 50
1.09 289:3125n [ 14 -11 5 0 0 0 -2 0 > 51200000/51195483 0.152741 17 59
1.11 7:653125n [ 7 6 -5 1 -1 0 0 -1 > 653184/653125 0.156384 19 62
1.13 17:660275n [-20 3 2 4 1 0 -1 0 > 17827425/17825792 0.158589 17 66
1.16 1183:1331n [ 3 -2 0 -1 3 -2 0 0 > 10648/10647 0.162595 13 66
1.17 7:120835n [-22 5 1 -1 1 3 0 0 > 29362905/29360128 0.163739 13 62
1.18 323:605n [-12 7 1 0 2 0 -1 -1 > 1323135/1323008 0.166179 19 63
1.20 5:10241n [-11 0 -1 2 1 0 0 1 > 10241/10240 0.169058 19 49
1.23 17:57967n [-26 9 0 3 0 2 -1 0 > 1140964461/1140850688 0.172641 17 64
1.5 tinas
tinas comma monzo interval cents limit popularity
1.26 121:1225n [ -3 4 -2 -2 2 0 0 0 > 9801/9800 0.176648 11 46
1.28 184093n [ 27 -6 0 -2 0 -1 -2 0 > 134217728/134203797 0.179701 17 61
1.29 85:117649n [ 7 -11 -1 6 0 0 -1 0 > 15059072/15057495 0.181306 17 64
1.30 125:22477n [-17 6 -3 1 0 2 0 1 > 16385733/16384000 0.183110 19 67
1.32 539:27625n [ -7 8 -3 2 1 -1 -1 0 > 3536379/3536000 0.185549 17 70
1.33 25:15379n [ 5 -9 -2 1 0 3 0 0 > 492128/492075 0.186456 13 56
1.34 539:8075n [ 15 -7 -2 2 1 0 -1 -1 > 17661952/17660025 0.188896 19 71
1.40 11:13671875n [-25 3 9 1 -1 0 0 0 > 369140625/369098752 0.196391 11 63
1.42 11:3125n [ 24 -10 -5 0 1 0 0 0 > 184549376/184528125 0.199364 11 36
1.49 2197:15625n [ -6 2 6 0 0 -3 0 0 > 140625/140608 0.209300 13 69
1.52 169:546875n [ 18 -4 -7 -1 0 2 0 0 > 44302336/44296875 0.213417 13 68
1.59 625:1001n [-12 8 4 -1 -1 -1 0 0 > 4100625/4100096 0.223352 13 51
1.61 1463:3125n [ 10 -7 5 -1 -1 0 0 -1 > 3200000/3199581 0.226698 19 62
1.65 19:145775n [-24 7 2 3 0 0 1 -1 > 318809925/318767104 0.232547 19 67
1.68 5:1399489n [ 7 7 1 -2 0 -4 0 0 > 1399680/1399489 0.236260 13 71
1.69 425:3773n [ 19 -10 2 -3 -1 0 1 0 > 222822400/222791877 0.237167 17 59
1.73 5:2023n [-15 4 -1 1 0 0 2 0 > 163863/163840 0.243015 17 46
2 tinas
tinas comma monzo interval cents limit popularity
1.75 35:1573n [ 15 -6 1 1 -2 -1 0 0 > 1146880/1146717 0.246069 13 47
1.80 931931n [ 23 -2 0 -3 -1 -1 0 -1 > 8388608/8387379 0.253659 19 64
1.85 119:30875n [-11 12 -3 1 0 -1 1 -1 > 63241479/63232000 0.259507 19 71
1.85 13:6655n [ 9 0 -1 0 -3 1 0 0 > 6656/6655 0.260121 13 51
1.92 119:40625n [-10 1 5 -1 0 1 -1 0 > 121875/121856 0.269916 17 62
1.96 2519825n [ -7 -9 2 2 2 0 1 0 > 2519825/2519424 0.275527 17 63
2.00 343:6175n [ -1 -2 2 -3 0 1 0 1 > 6175/6174 0.280384 19 63
2.02 13:6470695n [-11 -5 1 6 1 -1 0 0 > 6470695/6469632 0.284429 13 71
2.03 169:361n [ 10 -7 0 0 0 -2 0 2 > 369664/369603 0.285703 19 64
2.06 187:665n [ -5 2 1 1 -1 0 -1 1 > 5985/5984 0.289286 19 59
2.06 221221n [-13 -3 0 1 1 2 1 0 > 221221/221184 0.289579 17 61
2.08 247:5929n [ -3 -1 0 2 2 -1 0 -1 > 5929/5928 0.292019 19 68
2.11 5831n [ 3 6 0 -3 0 0 -1 0 > 5832/5831 0.296876 17 38
2.12 5:218491n [-17 1 -1 5 0 1 0 0 > 655473/655360 0.298481 13 53
2.13 361:1925n [ 4 -1 -2 -1 -1 0 0 2 > 5776/5775 0.299755 19 66
2.15 1625:2057n [ 6 -4 -3 0 2 -1 1 0 > 131648/131625 0.302487 17 67
2.18 7:206635n [ -1 10 -1 1 -1 -1 -2 0 > 413343/413270 0.305778 17 70
2.18 475:1001n [ -9 5 -2 1 1 1 0 -1 > 243243/243200 0.306071 19 60
2.19 19:2100875n [ 12 3 -3 -5 0 0 0 1 > 2101248/2100875 0.307345 19 69
2.20 7:60401n [ 21 -5 0 1 -1 0 -2 -1 > 14680064/14677443 0.309125 19 71
2.21 13:420175n [ 34 -12 -2 -5 0 1 0 0 > 223338299392/223298222175 0.310691 13 58
2.5 tinas
tinas comma monzo interval cents limit popularity
2.27 163625n [ 30 -8 -3 -1 -1 0 -1 0 > 1073741824/1073543625 0.319593 17 50
2.32 343:15625n [ -4 6 -6 3 0 0 0 0 > 250047/250000 0.325441 7 51
2.39 17:65219n [-23 7 0 2 3 0 -1 0 > 142633953/142606336 0.335237 17 64
2.46 125:25289n [-14 4 -3 0 3 0 0 1 > 2048409/2048000 0.345705 19 67
2.48 25:17303n [ 8 -11 -2 0 3 1 0 0 > 4429568/4428675 0.349052 13 56
2.51 91:4913n [ 1 3 0 1 0 1 -3 0 > 4914/4913 0.352342 17 71
2.58 85:121n [ -9 6 1 0 -2 0 1 0 > 61965/61952 0.363244 17 44
2.65 7:859375n [-28 7 7 -1 1 0 0 0 > 1879453125/1879048192 0.373039 11 53
2.73 43681n [ 17 -1 0 0 -2 0 0 -2 > 131072/131043 0.383082 19 60
3 tinas
tinas comma monzo interval cents limit popularity
2.79 13:16625n [-23 8 3 1 0 -1 0 1 > 109076625/109051904 0.392409 19 54
2.82 4375n [ -1 -7 4 1 0 0 0 0 > 4375/4374 0.395756 7 27
2.85 275:4459n [-15 12 2 -3 1 -1 0 0 > 146146275/146112512 0.399999 13 55
2.86 15714545n [-35 7 1 5 1 0 1 0 > 34367709915/34359738368 0.401604 17 68
2.87 1375:6517n [ 7 -3 3 -3 1 0 0 -1 > 176000/175959 0.403346 19 66
2.89 7:1615n [ 8 -10 1 -1 0 0 1 1 > 413440/413343 0.406224 19 48
2.91 55:2197n [ 18 -8 1 0 1 -3 0 0 > 14417920/14414517 0.408664 13 55
2.91 19:14399n [-27 11 0 1 2 0 1 -1 > 2550739653/2550136832 0.409194 19 65
2.92 11:4225n [ -7 -1 2 0 -1 2 0 0 > 4225/4224 0.409808 13 47
2.93 175:4199n [ 3 1 2 1 0 -1 -1 -1 > 4200/4199 0.412248 19 66
2.94 187:16807n [ 16 -6 0 -5 1 0 1 0 > 12255232/12252303 0.413814 17 63
2.95 143:6517n [ 4 -6 0 3 -1 -1 0 1 > 104272/104247 0.415126 19 64
3.01 455n [ 12 -2 -1 -1 0 -1 0 0 > 4096/4095 0.422716 13 25
3.03 13:3553n [ -3 7 0 0 -1 1 -1 -1 > 28431/28424 0.426300 19 60
3.11 13:67375n [ 6 4 -3 -2 -1 1 0 0 > 67392/67375 0.436768 13 53
3.22 248897n [-10 -5 0 0 4 0 1 0 > 248897/248832 0.452175 17 61
3.24 289:1925n [-17 9 2 1 1 0 -2 0 > 37889775/37879808 0.455465 17 62
3.5 tinas
tinas comma monzo interval cents limit popularity
3.27 13:1890625n [ 33 -10 -6 0 -2 1 0 0 > 111669149696/111639515625 0.459485 13 65
3.32 209:595n [ -8 6 -1 -1 1 0 -1 1 > 152361/152320 0.465934 19 59
3.34 119:143n [ 14 -9 0 -1 1 1 -1 0 > 2342912/2342277 0.469280 17 48
3.38 125:539539n [-20 5 -3 3 2 1 0 0 > 131107977/131072000 0.475129 13 71
3.40 343:1445n [ 10 -5 -1 3 0 0 -2 0 > 351232/351135 0.478182 17 60
3.44 125:102487n [ -3 8 3 -1 -4 0 0 0 > 820125/819896 0.483473 11 66
3.48 133:10625n [ 19 -8 -4 1 0 0 -1 1 > 69730304/69710625 0.488651 19 63
3.57 847:390625n [ -7 10 -8 1 2 0 0 0 > 50014503/50000000 0.502089 11 69
3.60 7:161051n [ 24 -6 0 1 -5 0 0 0 > 117440512/117406179 0.506189 11 62
3.65 138125n [-25 5 4 0 0 1 1 0 > 33564375/33554432 0.512931 17 50
3.67 2057:15625n [ 5 -5 6 0 -2 0 -1 0 > 500000/499851 0.515985 17 69
3.71 11:300125n [-29 9 3 4 -1 0 0 0 > 5907360375/5905580032 0.521833 11 54
3.74 935:1183n [ -6 4 1 -1 1 -2 1 0 > 75735/75712 0.525839 17 66
3.75 2375:5929n [ 14 -8 3 -2 -2 0 0 1 > 38912000/38900169 0.526453 19 70
4 tinas
tinas comma monzo interval cents limit popularity
3.79 11:15827n [-20 6 0 2 -1 0 1 1 > 11537883/11534336 0.532301 19 61
3.79 361:1625n [ 1 -2 3 0 0 1 0 -2 > 3250/3249 0.532769 19 66
3.80 25:1294139n [ 22 -4 2 -6 -1 0 0 0 > 104857600/104825259 0.534043 11 63
3.81 343:2197n [-10 8 0 3 0 -3 0 0 > 2250423/2249728 0.534741 13 60
3.81 11:54145n [ 2 -9 1 2 -1 1 1 0 > 216580/216513 0.535648 17 60
3.83 1715:3211n [ 12 -7 1 3 0 -2 0 -1 > 7024640/7022457 0.538088 19 71
3.84 17:245n [-12 10 -1 -2 0 0 1 0 > 1003833/1003520 0.539891 17 36
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tinas comma monzo interval cents limit popularity
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7 tinas
tinas comma monzo interval cents limit popularity
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8 tinas
tinas comma monzo interval cents limit popularity
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9 tinas
tinas comma monzo interval cents limit popularity
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9.5 tinas
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10 tinas
tinas comma monzo interval cents limit popularity
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10.5 tinas
tinas comma monzo interval cents limit popularity
10.25 27625n [ 10 3 -3 0 0 -1 -1 0 > 27648/27625 1.440790 17 45
10.28 8075n [ 32 -12 -2 0 0 0 -1 -1 > 4294967296/4291386075 1.444136 19 46
10.46 25:102487n [-12 0 -2 1 4 0 0 0 > 102487/102400 1.470248 11 61
10.56 169:1445n [ 8 -7 1 0 0 -2 2 0 > 369920/369603 1.484203 17 65
10.58 143:7225n [ 12 -4 -2 0 1 1 -2 0 > 585728/585225 1.487354 17 68
10.59 209:2975n [ -7 2 2 1 -1 0 1 -1 > 26775/26752 1.487787 19 64
10.66 289:385n [ 2 -1 -1 -1 -1 0 2 0 > 1156/1155 1.498255 17 57
10.68 5:121121n [ 32 -11 1 -1 -3 -1 0 0 > 21474836480/21456221787 1.501309 13 58
10.70 7:51623n [ -3 10 0 1 -1 -1 0 -2 > 413343/412984 1.504279 19 69
11 tinas
tinas comma monzo interval cents limit popularity
10.79 13:336875n [-21 4 4 2 1 -1 0 0 > 27286875/27262976 1.516952 13 58
10.82 19:1684375n [ 1 -11 5 2 1 0 0 -1 > 3368750/3365793 1.520299 19 69
10.87 13:17765n [-12 1 1 0 1 -1 1 1 > 53295/53248 1.527421 19 65
10.89 2275n [-27 10 2 1 0 1 0 0 > 134336475/134217728 1.531005 13 30
10.92 19:147875n [ -5 -5 3 1 0 2 0 -1 > 147875/147744 1.534351 19 67
10.96 13:12005n [ 6 -10 1 4 0 -1 0 0 > 768320/767637 1.539669 13 46
10.97 35:4199n [-18 7 -1 -1 0 1 1 1 > 9183213/9175040 1.541473 19 61
10.99 11:845n [ -8 9 -1 0 1 -2 0 0 > 216513/216320 1.543913 13 42
10.99 95:14399n [ 12 -3 1 -1 -2 0 -1 1 > 389120/388773 1.544527 19 70
10.99 7:2873n [ 4 -8 0 -1 0 2 1 0 > 45968/45927 1.544820 17 50
11.01 11:247n [ 14 -6 0 0 1 -1 0 -1 > 180224/180063 1.547259 19 43
11.03 275:6517n [-22 11 -2 3 -1 0 0 1 > 1154466999/1153433600 1.550375 19 61
11.04 3142909n [ 20 1 0 -5 -1 0 -1 0 > 3145728/3142909 1.552117 17 63
11.06 55:4459n [ 0 -4 -1 3 -1 1 0 0 > 4459/4455 1.553721 13 50
11.08 187:79625n [ 23 -9 -3 -2 1 -1 1 0 > 1568669696/1567258875 1.557728 17 70
11.11 13:3325n [ 8 0 -2 -1 0 1 0 -1 > 3328/3325 1.561311 19 49
11.15 49:105625n [ 19 -5 -4 2 0 -2 0 0 > 25690112/25666875 1.566630 13 60
11.18 218405n [-32 9 1 0 2 0 0 2 > 4298865615/4294967296 1.570639 19 65
11.20 747175n [-10 -6 2 0 2 1 0 1 > 747175/746496 1.573985 19 64
11.25 7:171875n [ 13 1 -6 1 -1 0 0 0 > 172032/171875 1.580682 11 48
11.5 tinas
tinas comma monzo interval cents limit popularity
11.32 17:121n [ -6 2 0 0 2 0 -1 0 > 1089/1088 1.590477 17 39
11.39 2299:6125n [ 3 -1 -3 -2 2 0 0 1 > 18392/18375 1.600945 19 70
11.42 1859:2975n [-12 8 -2 -1 1 2 -1 0 > 12196899/12185600 1.604529 17 71
11.48 539:2125n [ 21 -12 -3 2 1 0 -1 0 > 1130364928/1129312125 1.613194 17 57
11.52 85:65219n [ 8 1 1 -2 -3 0 1 0 > 65280/65219 1.618484 17 69
11.58 49:190333n [ 4 5 0 2 -4 -1 0 0 > 190512/190333 1.627386 13 71
11.59 343:78125n [-11 2 7 -3 0 0 0 0 > 703125/702464 1.628279 7 56
11.72 1001:2375n [ -6 3 3 -1 -1 -1 0 1 > 64125/64064 1.647650 19 65
11.75 77:625n [ 16 -12 4 -1 -1 0 0 0 > 40960000/40920957 1.650996 11 38
12 tinas
tinas comma monzo interval cents limit popularity
11.77 361:9625n [-19 9 3 1 1 0 0 -2 > 189448875/189267968 1.653966 19 71
11.78 25:218491n [ 2 7 2 -5 0 -1 0 0 > 218700/218491 1.655240 13 58
11.79 29155n [-18 2 1 3 0 0 1 0 > 262395/262144 1.656845 17 43
11.80 125:319333n [ 24 -8 3 -5 0 0 0 -1 > 2097152000/2095143813 1.658586 19 69
11.84 5:221221n [ -2 11 1 -1 -1 -2 -1 0 > 885735/884884 1.664142 17 66
11.84 133:187n [-10 6 0 -1 1 0 1 -1 > 136323/136192 1.664434 19 54
11.86 25:38437n [ -9 -1 -2 1 0 0 2 1 > 38437/38400 1.667313 19 70
11.88 17:823543n [ 33 -11 0 -7 0 0 1 0 > 146028888064/145888171821 1.669055 17 66
11.89 133:1573n [ 21 -11 0 1 -2 -1 0 1 > 278921216/278652231 1.670367 19 61
11.91 343:1235n [-14 10 -1 3 0 -1 0 -1 > 20253807/20234240 1.673336 19 58
11.93 343:361n [ 8 -5 0 3 0 0 0 -2 > 87808/87723 1.676683 19 59
11.94 245245n [ 29 -7 -1 -3 -1 -1 0 0 > 536870912/536350815 1.677957 13 50
11.98 119:8125n [ -5 7 -4 1 0 -1 1 0 > 260253/260000 1.683805 17 57
12.01 119:2375n [ 17 -8 -3 1 0 0 1 -1 > 15597568/15582375 1.687151 19 58
12.02 181575625n [-10 -11 4 4 2 0 0 0 > 181575625/181398528 1.689357 11 70
12.05 13:33275n [-24 8 2 0 3 -1 0 0 > 218317275/218103808 1.693600 13 56
12.08 19:166375n [ -2 -7 3 0 3 0 0 -1 > 166375/166212 1.696946 19 67
12.08 289:109375n [ 26 -11 -6 -1 0 0 2 0 > 19394461696/19375453125 1.697620 17 71
12.10 4659655n [-38 10 1 3 1 1 0 1 > 275147968095/274877906944 1.700062 19 69
12.10 14141875n [ 19 3 -4 0 -3 0 -1 0 > 14155776/14141875 1.700911 17 70
12.12 15940925n [-16 -5 2 3 1 2 0 0 > 15940925/15925248 1.703409 13 68
12.15 49:22627n [ 7 -10 0 -2 3 0 1 0 > 2896256/2893401 1.707415 17 64
12.17 25:2023n [ 0 4 2 -1 0 0 -2 0 > 2025/2023 1.710706 17 51
12.21 65:5929n [ 3 -6 -1 2 2 -1 0 0 > 47432/47385 1.716317 13 54
12.22 1399489n [-22 1 0 2 0 4 0 0 > 4198467/4194304 1.717461 13 66
12.25 19:29155n [ 9 1 -1 -3 0 0 -1 1 > 29184/29155 1.721174 19 62
12.5 tinas
tinas comma monzo interval cents limit popularity
12.31 125:1001n [ -3 0 -3 1 1 1 0 0 > 1001/1000 1.730369 13 46
12.33 7:15895n [ 27 -10 -1 1 -1 0 -2 0 > 939524096/938583855 1.733423 17 57
12.41 2197:3125n [ -9 6 -5 0 0 3 0 0 > 1601613/1600000 1.744421 13 64
12.48 91:265625n [ -2 -6 6 -1 0 -1 1 0 > 265625/265356 1.754119 17 67
12.50 11:2734375n [ 10 5 -8 -1 1 0 0 0 > 2737152/2734375 1.757329 11 58
12.65 121:6125n [-12 4 3 2 -2 0 0 0 > 496125/495616 1.777073 11 51
12.67 7:48828125n [ 37 -9 -11 1 0 0 0 0 > 962072674304/961083984375 1.780046 7 62
12.67 85:57967n [ 11 -1 1 -3 0 -2 1 0 > 174080/173901 1.781080 17 69
12.70 25:10241n [ -4 8 2 -2 -1 0 0 -1 > 164025/163856 1.784663 19 54
12.72 121:323n [ -3 1 0 0 -2 0 1 1 > 969/968 1.787542 19 58
12.74 7:24167n [ 7 3 0 1 -1 -3 0 0 > 24192/24167 1.789981 13 57
12.75 1331:5915n [-18 10 1 1 -3 2 0 0 > 349274835/348913664 1.791125 13 71
13 tinas
tinas comma monzo interval cents limit popularity
12.76 35:35321n [ 29 -12 1 1 -1 -2 0 -1 > 18790481920/18771027561 1.793328 19 68
12.77 17:132055n [ 5 5 -1 -4 -1 0 1 0 > 132192/132055 1.795132 17 61
12.79 7:3265625n [-22 2 6 -1 1 0 0 1 > 29390625/29360128 1.797337 19 67
12.81 1331:2401n [ 15 -10 0 4 -3 0 0 0 > 78675968/78594219 1.799790 11 61
12.84 39325n [ 1 9 -2 0 -2 -1 0 0 > 39366/39325 1.804034 13 45
12.86 11495n [ 23 -6 -1 0 -2 0 0 -1 > 8388608/8379855 1.807380 19 46
12.88 17:5359375s [ -4 -9 6 3 0 0 -1 0 > 5359375/5353776 1.809585 17 68
12.91 1547:6875s [-18 10 4 -1 1 -1 -1 0 > 405961875/405536768 1.813829 17 68
12.93 13:63175s [-17 3 2 1 0 -1 0 2 > 1705725/1703936 1.816707 19 68
12.95 16625s [ 5 -12 3 1 0 0 0 1 > 532000/531441 1.820054 19 41
12.98 1045:4459s [ -9 7 1 -3 1 -1 0 1 > 2285415/2283008 1.824297 19 69
13.01 275:343s [ 13 -8 2 -3 1 0 0 0 > 2252800/2250423 1.827644 11 42
13.03 11:4693s [-23 9 0 0 -1 1 0 2 > 92372319/92274688 1.830759 19 62
13.04 133:2873s [-13 11 0 1 0 -2 -1 1 > 23560551/23535616 1.833199 19 69
13.05 5:14399s [-21 6 -1 1 2 0 1 0 > 10496871/10485760 1.833492 17 51
13.05 11:16055s [ -1 -6 1 0 -1 2 0 1 > 16055/16038 1.834106 19 61
13.07 35:221s [ 9 -4 1 1 0 -1 -1 0 > 17920/17901 1.836546 17 42
13.08 40353607s [-11 -9 0 9 0 0 0 0 > 40353607/40310784 1.838150 7 63
13.11 169:2401s [ 7 -2 0 -4 0 2 0 0 > 21632/21609 1.841696 13 54
13.11 325:184877s [-25 10 -2 5 1 -1 0 0 > 10916801973/10905190400 1.842394 13 69
13.13 95:184877s [ -3 -5 -1 5 1 0 0 -1 > 184877/184680 1.845740 19 70
13.14 19:2275s [ 18 -7 -2 -1 0 -1 0 1 > 4980736/4975425 1.847014 19 49
13.15 125:31603s [-27 12 -3 0 1 2 1 0 > 16795129923/16777216000 1.847544 17 69
13.17 13:935s [ 3 2 -1 0 -1 1 -1 0 > 936/935 1.850598 17 46
13.21 625:64141s [ 6 -8 -4 3 1 0 1 0 > 4105024/4100625 1.856209 17 69
13.5 tinas
tinas comma monzo interval cents limit popularity
13.26 12569375s [ 22 1 -4 -1 0 -2 -1 0 > 12582912/12569375 1.863506 17 70
13.28 17935225s [-13 -7 2 2 4 0 0 0 > 17935225/17915904 1.866004 11 68
13.38 1574573s [-19 -1 0 1 3 2 0 0 > 1574573/1572864 1.880056 13 66
13.48 11:36125s [ 1 -8 3 0 -1 0 2 0 > 72250/72171 1.894011 17 60
13.59 11:252875s [ 24 -6 -3 -1 1 0 -2 0 > 184549376/184345875 1.910070 17 67
13.59 85:2717s [ 5 0 1 0 -1 -1 1 -1 > 2720/2717 1.910503 19 65
13.71 9953125s [-28 3 6 2 0 1 0 0 > 268734375/268435456 1.926760 13 57
14 tinas
tinas comma monzo interval cents limit popularity
13.78 524875s [-19 0 3 0 0 1 1 1 > 524875/524288 1.937229 19 64
13.80 169:9625s [ -9 2 3 1 1 -2 0 0 > 86625/86528 1.939669 13 59
13.83 323:30625s [-24 11 4 2 0 0 -1 -1 > 5425126875/5419040768 1.943252 19 70
13.85 11:1140475s [-23 4 2 4 -1 0 0 1 > 92378475/92274688 1.946131 19 68
13.87 11:3901625s [ -1 -11 3 4 -1 1 0 0 > 3901625/3897234 1.949477 13 67
13.90 5s [-15 8 1 0 0 0 0 0 > 32805/32768 1.953721 5 5
Re: Magrathean diacritics
I think the question of when to switch over into subtracting from the 5-schisma can be reduced to a sort of coin-counting problem. The tina accents are built from elements that stand for a half tina, 1, 2 and 3 tinas, and the 5-schisma, so the question is: Given denominations of 0.5t, 1t, 2t, 3t and 14t, at which point does it become more efficient to give the cashier a 14-tina note instead of exact change?
If we just consider addition, then:
This doesn't really help resolve anything, though, so let's look at some specific temperaments.
In 2460/Olympian, the 5-schisma is equivalent to 4 minas: (4375n~455n), (13:125n), (539n~5:19n), (5s).
The next reasonable equal temperament in the 19-limit is 3395, but there, the 5-schisma is equivalent to 6 minas, making the 3-mina diacritic necessary to notate it: (4375n~455n), (13:125n), (539n), (5:19n), (2275n), (5s).
In 4349edo, there isn't a straightforward mina mapping: the ragisma (4375n) maps to 1 step, but the tridecimal schismina (455n) maps to 2 steps. A possible notation would use every second tina accent: 2t (11:3125n), 4t (17:245n), (13:125n~65:77n), 8t or (13:77n), -4t (5:19n), -2t (77:625n), (5s).
I'm not sure what's going on with 4501. The best mapping for 5 is almost half a step flat, so a lot of commas aren't the ideal size. In any case, though they're significantly larger, both the 4375n and the 455n map to 1 step, so that has to be the value of the mina accent. Unfortunately, the 5-schisma maps to 8 steps; does this mean that we need a 4-mina diacritic?
If we just consider addition, then:
- {0.5, 1, 2, 3} are unitary symbols,
- {1.5, 2.5, 3.5, 4, 5, 6} need 2 elements, and
- {4.5, 5.5, 6.5, 7, 8, 9} need 3 elements.
This doesn't really help resolve anything, though, so let's look at some specific temperaments.
In 2460/Olympian, the 5-schisma is equivalent to 4 minas: (4375n~455n), (13:125n), (539n~5:19n), (5s).
The next reasonable equal temperament in the 19-limit is 3395, but there, the 5-schisma is equivalent to 6 minas, making the 3-mina diacritic necessary to notate it: (4375n~455n), (13:125n), (539n), (5:19n), (2275n), (5s).
In 4349edo, there isn't a straightforward mina mapping: the ragisma (4375n) maps to 1 step, but the tridecimal schismina (455n) maps to 2 steps. A possible notation would use every second tina accent: 2t (11:3125n), 4t (17:245n), (13:125n~65:77n), 8t or (13:77n), -4t (5:19n), -2t (77:625n), (5s).
I'm not sure what's going on with 4501. The best mapping for 5 is almost half a step flat, so a lot of commas aren't the ideal size. In any case, though they're significantly larger, both the 4375n and the 455n map to 1 step, so that has to be the value of the mina accent. Unfortunately, the 5-schisma maps to 8 steps; does this mean that we need a 4-mina diacritic?