Magrathean diacritics

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Dave Keenan
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Re: Magrathean diacritics

Post by Dave Keenan »

I was just reminded of another excellent resource regarding tinas. Joe Monzo's Tonalsoft Encyclopedia entry.
http://tonalsoft.com/enc/t/tina.aspx

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cmloegcmluin
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Re: Magrathean diacritics

Post by cmloegcmluin »

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.

															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.

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Dave Keenan
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Re: Magrathean diacritics

Post by Dave Keenan »

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.

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Re: Magrathean diacritics

Post by Dave Keenan »

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?
An excellent question. I'm glad you asked it before we submitted the Magratheans to Steinberg. :)

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.

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Re: Magrathean diacritics

Post by Ash9903b4 »

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

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Dave Keenan
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Re: Magrathean diacritics

Post by Dave Keenan »

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.

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Dave Keenan
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Re: Magrathean diacritics

Post by Dave Keenan »

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.

Ash9903b4
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Re: Magrathean diacritics

Post by Ash9903b4 »

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.

Ash9903b4
Posts: 16
Joined: Sun Mar 15, 2020 11:27 am

Re: Magrathean diacritics

Post by Ash9903b4 »

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
  3.86   5:1310309n    [ 18   0   1  -2  -2  -1  -1   0 >        1310720/1310309        0.542945     17      71
  3.87   221:931n      [ 10  -5   0  -2   0   1   1  -1 >         226304/226233         0.543238     19      63
  3.87   121:117649n   [ -2  -5   0   6  -2   0   0   0 >         117649/117612         0.544550     11      64
  3.90   119:4225n     [ 21 -10  -2   1   0  -2   1   0 >      249561088/249482025      0.548556     17      60
  3.93   49:1045n      [  6  -1  -1   2  -1   0   0  -1 >           3136/3135           0.552140     19      49
  3.98   442225n       [ 14   3  -2  -2   0   0   0  -2 >         442368/442225         0.559730     19      62
  4.00   17:6875n      [ 15  -4  -4   0  -1   0   1   0 >         557056/556875         0.562608     17      48
  4.02   8960875n      [-12  -7   3   3   1   0   0   1 >        8960875/8957952        0.564813     19      66
  4.05   91:11495n     [-26  12   1  -1   2  -1   0   1 >     6108914295/6106906624     0.569057     19      66
  4.07   7:3025n       [ -4  -3   2  -1   2   0   0   0 >           3025/3024           0.572403     11      39
  4.12   786695n       [-18  -1   1   2   0   2   0   1 >         786695/786432         0.578866     19      64
  4.14   221:18865n    [ -8   1   1   3   1  -1  -1   0 >          56595/56576          0.581305     17      67
  4.17   49:1859n      [-10   3   0  -2   1   2   0   0 >          50193/50176          0.586456     13      51
  4.21   325:1463n     [  1  -2  -2   1   1  -1   0   1 >           2926/2925           0.591774     19      60
  4.24   85:637n       [-14   7  -1   2   0   1  -1   0 >        1393119/1392640        0.595357     17      49

4.5 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  4.26   121:557375n   [  9   2  -3  -3   2  -1   0   0 >         557568/557375         0.599364     13      71
  4.31   247:625n      [ -5   4  -4   0   0   1   0   1 >          20007/20000          0.605826     19      52
  4.33   125:169n      [ 17 -11  -3   0   0   2   0   0 >       22151168/22143375       0.609172     13      41
  4.41   143:390625n   [-13   1   8   0  -1  -1   0   0 >        1171875/1171456        0.619108     13      64
  4.43   11:1421875n   [ 36 -12  -6  -1   1  -1   0   0 >   755914244096/755642671875   0.622081     13      61
  4.43   265625n       [ -1  12  -6   0   0   0  -1   0 >         531441/531250         0.622318     17      47
  4.53   13:19140625n   [ 30  -6  -8  -2   0   1   0   0 >    13958643712/13953515625    0.636133     13      67
  4.55   1770125n      [-16  -3   3   2   0   0   2   0 >        1770125/1769472        0.638771     17      63
  4.60   125:91091n    [  0   6   3  -2  -1  -2   0   0 >          91125/91091          0.646068     13      66
  4.62   93347n        [ -7  -6   0   0   0   0   3   1 >          93347/93312          0.649239     19      70
  4.64   1331:3325n    [-14   8   2   1  -3   0   0   1 >       21815325/21807104       0.652530     19      69
  4.66   247:4165n     [-12   5   1   2   0  -1   1  -1 >        1012095/1011712        0.655263     19      68
  4.67   1331:11375n   [  8  -7   3   1  -3   1   0   0 >        2912000/2910897        0.655877     13      68
  4.70   5:41503n      [ -6  12   1  -3  -2   0   0   0 >        2657205/2656192        0.660120     11      48
  4.74   289:325n      [ -3   2  -2   0   0  -1   2   0 >           2601/2600           0.665731     17      57

5 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  4.76   20449n        [ 27  -8   0   0  -2  -2   0   0 >      134217728/134165889      0.668785     13      48
  4.86   232925n       [ 21  -2  -2  -1  -3   0   0   0 >        2097152/2096325        0.682837     11      50
  4.92   55883275n     [-40   9   2   6   0   0   0   1 >  1099950501825/1099511627776  0.690890     19      71
  4.93   625:833n      [  2  -1   4  -2   0   0  -1   0 >           2500/2499           0.692632     17      51
  4.94   191179625n    [-18  -6   3   6   0   1   0   0 >      191179625/191102976      0.694237     13      70
  4.95   11:9025n      [  3  -8   2   0  -1   0   0   2 >          72200/72171          0.695511     19      59
  4.97   13:71995n     [  5 -11   1   1   2  -1   1   0 >        2303840/2302911        0.698243     17      64
  4.99   19:175175n    [-10  -2   2   2   1   1   0  -1 >         175175/175104         0.701827     19      67
  5.00   95:2401n      [ 11  -4   1  -4   0   0   0   1 >         194560/194481         0.703101     19      52
  5.04   25:3553n      [-23  10  -2   0   1   0   1   1 >      209801097/209715200      0.708949     19      57
  5.06   11:5764801n   [ 19   0   0  -8   1   0   0   0 >        5767168/5764801        0.710691     11      67
  5.07   19:2431n      [  7   0   0   0  -1  -1  -1   1 >           2432/2431           0.712002     19      60
  5.07   5:2431n       [ -1  -5  -1   0   1   1   1   0 >           2431/2430           0.712295     17      46
  5.09   19:15379n     [-16   4   0   1   0   3   0  -1 >        1245699/1245184        0.715879     19      65
  5.12   2653105n      [ 15   4  -1  -4   0  -1  -1   0 >        2654208/2653105        0.719592     17      63
  5.13   25:2401n      [ -5  -1  -2   4   0   0   0   0 >           2401/2400           0.721197      7      38
  5.14   775523n       [ 37 -11   0  -4   0   0  -1  -1 >   137438953472/137381572881   0.722939     19      64
  5.19   11:2375n      [  3   3  -3   0   1   0   0  -1 >           2376/2375           0.728787     19      45

5.5 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  5.28   885115n       [-15  -3   1   1   3   0   0   1 >         885115/884736         0.741461     19      64
  5.32   49:1015625n   [  8   4  -7   2   0  -1   0   0 >        1016064/1015625        0.748158     13      62
  5.33   343:14641n    [ -7   1   0  -3   4   0   0   0 >          43923/43904          0.749051     11      65
  5.35   49:296875n    [ 30 -11  -6   2   0   0   0  -1 >    52613349376/52590515625    0.751504     19      63
  5.40   289:1235n     [  9  -7   1   0   0   1  -2   1 >         632320/632043         0.758567     19      71
  5.40   265837n       [  1 -12   0   0   2   3   0   0 >         531674/531441         0.758860     13      61
  5.40   121:371875n   [-10  -1   5   1  -2   0   1   0 >         371875/371712         0.759000     17      71
  5.47   537109375n    [-29   0  11   0   1   0   0   0 >      537109375/536870912      0.768795     11      66
  5.49   1573:2125n    [-17  11  -3   0   2   1  -1   0 >      278652231/278528000      0.772005     17      67
  5.52   24565n        [ 13   1  -1   0   0   0  -3   0 >          24576/24565          0.775058     17      56
  5.59   17:11011n     [  3   4   0  -1  -2  -1   1   0 >          11016/11011          0.785960     17      59
  5.62   85:16093n     [ 25 -11   1  -1  -2   0   1  -1 >     2852126720/2850826671     0.789307     19      70
  5.66   637:171875n   [-16   5   6  -2   1  -1   0   0 >       41765625/41746432       0.795755     13      68

6 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  5.76   343:8125n     [-22  11   4  -3   0   1   0   0 >     1439319375/1438646272     0.809807     13      54
  5.80   169:475n      [-11   6   2   0   0  -2   0   1 >         346275/346112         0.815126     19      55
  5.80   174845n       [-19   1   1   0   2   0   2   0 >         524535/524288         0.815418     17      61
  5.82   13:125n       [ 11  -9   3   0   0  -1   0   0 >         256000/255879         0.818472     13      28
  5.85   55:405769n    [ -3  10   1  -4   1  -2   0   0 >        3247695/3246152        0.822715     13      70
  5.87   13:448987n    [-23   5   0   4   1  -1   1   0 >      109103841/109051904      0.824320     17      69
  5.90   323:637n      [ 20 -12   0  -2   0  -1   1   1 >      338690048/338527917      0.828941     19      63
  5.90   19:77n        [-17  12   0  -1  -1   0   0   1 >       10097379/10092544       0.829178     19      37
  5.92   11:199927n    [ 30 -10   0  -1   1  -4   0   0 >    11811160064/11805489423    0.831380     13      70
  5.92   65:77n        [  5  -3   1  -1  -1   1   0   0 >           2080/2079           0.832524     13      36
  5.94   5:54587n      [ 15  -1   1   0   0  -2  -1  -1 >         163840/163761         0.834964     19      67
  5.97   25:75803n     [-29  11  -2   3   0   1   1   0 >    13428274041/13421772800    0.838372     17      61
  5.98   13:319333n    [ 13   1   0  -5   0   1   0  -1 >         319488/319333         0.840114     19      67
  5.99   343:2057n     [  1   1   0   3  -2   0  -1   0 >           2058/2057           0.841426     17      60
  6.02   207025n       [ 24  -4  -2  -2   0  -2   0   0 >       16777216/16769025       0.845432     13      50
  6.04   124355n       [  9   5  -1  -1  -1   0  -1  -1 >         124416/124355         0.849016     19      59
  6.06   133:15125n    [ 10  -2  -3   1  -2   0   0   1 >         136192/136125         0.851895     19      63
  6.12   2358125n      [ 18   2  -4  -3  -1   0   0   0 >        2359296/2358125        0.859484     11      52
  6.13   17:26125n     [ -9  -1   3   0   1   0  -1   1 >          26125/26112          0.861690     19      62
  6.20   18883865n     [-21  -2   1   4   2   1   0   0 >       18883865/18874368       0.870885     13      68
  6.21   49:3971n      [  0  -4   0  -2   1   0   0   2 >           3971/3969           0.872158     19      63
  6.25   55:3757n      [ -5   7   1   0   1  -1  -2   0 >         120285/120224         0.878182     17      63

6.5 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  6.25   19:17303n     [-13   2   0   0   3   1   0  -1 >         155727/155648         0.878474     19      65
  6.28   66171875n     [ 45 -12  -7  -1  -2   0   0   0 > 35184372088832/35166447421875 0.882201     11      64
  6.35   11:4165n      [ 26 -11  -1  -2   1   0  -1   0 >      738197504/737817255      0.891997     17      47
  6.39   625:5929n     [ -8   3  -4   2   2   0   0   0 >         160083/160000         0.897845     11      56
  6.41   49:93925n     [ 22  -7  -2   2   0  -1  -2   0 >      205520896/205413975      0.900898     17      71
  6.58   19:109375n    [-22   6   6   1   0   0   0  -1 >       79734375/79691776       0.925179     19      56
  6.66   7:2125n       [-13   3   3  -1   0   0   1   0 >          57375/57344          0.935647     17      39
  6.72   143:8575n     [-17   7   2   3  -1  -1   0   0 >       18753525/18743296       0.944549     13      55
  6.75   209:42875n    [  5  -8   3   3  -1   0   0  -1 >        1372000/1371249        0.947896     19      66

7 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  6.78   5:1729n       [ -9  11   1  -1   0  -1   0  -1 >         885735/885248         0.952139     19      44
  6.80   25:2527n      [ 13  -4   2  -1   0   0   0  -2 >         204800/204687         0.955486     19      55
  6.81   49:142805n    [  2   6  -1   2   0  -4   0   0 >         142884/142805         0.957457     13      71
  6.82   11:119n       [ 14 -11   0   1  -1   0   1   0 >        1949696/1948617        0.958364     17      35
  6.85   17:111475n    [  0   8  -2  -3   0  -1   1   0 >         111537/111475         0.962608     17      61
  6.87   17:32585n     [ 22  -7  -1  -3   0   0   1  -1 >       71303168/71263395       0.965954     19      62
  6.89   1890625n      [ -5 -10   6   0   2   0   0   0 >        1890625/1889568        0.968159     11      52
  6.91   17:274625n    [ 33 -12  -3   0   0  -3   1   0 >   146028888064/145946984625   0.971272     17      71
  6.91   7:232375n     [ -4  12  -3   1  -1  -2   0   0 >        3720087/3718000        0.971509     13      59
  6.94   7:67925n      [ 18  -3  -2   1  -1  -1   0  -1 >        1835008/1833975        0.974856     19      60
  6.97   7:339625n     [-33  11   3  -1   1   1   0   1 >    60163549875/60129542144    0.978865     19      65
  6.99   7:1161875n    [-11  -4   4  -1   1   2   0   0 >        1161875/1161216        0.982211     13      64
  7.03   13:256025n    [  0  -9   2   2   1  -1   0   1 >         256025/255879         0.987530     19      67
  7.05   17:557375n    [-15   0   3   3   0   1  -1   0 >         557375/557056         0.991113     17      66
  7.08   605:637n      [  8  -5   1  -2   2  -1   0   0 >         154880/154791         0.995120     13      54
  7.10   5:2527n       [-28  12  -1   1   0   0   0   2 >     1342951407/1342177280     0.998235     19      50
  7.13   1729n         [ -6  -3   0   1   0   1   0   1 >           1729/1728           1.001582     19      39
  7.14   539:2873n     [  4  -1   0   2   1  -2  -1   0 >           8624/8619           1.004021     17      68
  7.18   143:1715n     [  2   1  -1  -3   1   1   0   0 >           1716/1715           1.009172     13      50
  7.20   5:1090397n    [ 32  -9   1  -3  -1   0  -2   0 >    21474836480/21462284151    1.012225     17      71
  7.22   209:21125n    [ 13  -4  -3   0   1  -2   0   1 >        1712128/1711125        1.014490     19      71
  7.24   7:425n        [ -2   5  -2   1   0   0  -1   0 >           1701/1700           1.018074     17      34

7.5 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  7.25   55:28561n     [ 10 -12  -1   0  -1   4   0   0 >       29246464/29229255       1.018980     13      68
  7.27   91:1615n      [ 20 -10  -1   1   0   1  -1  -1 >       95420416/95364135       1.021420     19      61
  7.32   19:21875n     [  7   2  -5  -1   0   0   0   1 >          21888/21875          1.028542     19      51
  7.51   3125:5929n    [ -7   5   5  -2  -2   0   0   0 >         759375/758912         1.055876     11      61
  7.53   669921875n    [ 42  -8  -9  -3   0   0   0   0 >  4398046511104/4395357421875  1.058849      7      66
  7.55   425:2299n     [ 15 -11  -2   0   2   0  -1   1 >       75333632/75287475       1.061054     19      68
  7.55   13:50575n     [ -4  -5   2   1   0  -1   2   0 >          50575/50544          1.061487     17      64
  7.65   95:17303n     [ -2   6   1   0  -3  -1   0   1 >          69255/69212          1.075246     19      70
  7.65   11:3757n      [-10   1   0   0  -1   1   2   0 >          11271/11264          1.075539     17      58
  7.67   119:3211n     [  0   3   0   1   0  -2   1  -1 >           3213/3211           1.077979     19      69
  7.68   25:1331n      [ 20  -9   2   0  -3   0   0   0 >       26214400/26198073       1.078593     11      43
  7.70   245:3971n     [-15  12   1   2  -1   0   0  -2 >      130203045/130121728      1.081563     19      68
  7.71   3776773n      [  6  10   0  -4  -2  -1   0   0 >        3779136/3776773        1.082836     13      63
  7.74   119:390625n   [  1  -8   8  -1   0   0  -1   0 >         781250/780759         1.088388     17      64

8 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  7.77   17:5225n      [ -6   9  -2   0  -1   0   1  -1 >         334611/334400         1.092031     19      57
  7.79   11790625n     [-33   6   5   3   1   0   0   0 >     8595365625/8589934592     1.094236     11      57
  7.82   343:59375n    [ 10 -11   5  -3   0   0   0   1 >       60800000/60761421       1.098857     19      65
  7.84   133:75625n    [-25  10   4  -1   2   0   0  -1 >     4465580625/4462739456     1.101826     19      68
  7.86   621775n       [-24   3   2   1   1   0   1   1 >       16787925/16777216       1.104705     19      64
  7.88   2127125n      [ -2 -12   3   1   1   1   1   0 >        2127125/2125764        1.108051     17      63
  7.89   1035125n      [-39  12   3   2   0   2   0   0 >   550107865125/549755813888   1.108288     13      55
  7.91   343:10285n    [-16   7   1  -3   2   0   1   0 >       22493295/22478848       1.112295     17      65
  7.94   125:75803n    [ 14  -3   3  -3   0  -1  -1   0 >        2048000/2046681        1.115348     17      66
  7.95   420175n       [ -6  -8   2   5   0   0   0   0 >         420175/419904         1.116953      7      45
  7.96   54587n        [-30   9   0   0   0   2   1   1 >     1074435921/1073741824     1.118757     19      62
  7.98   13:77n        [-20  11   0   1   1  -1   0   0 >       13640319/13631488       1.121197     13      31
  7.98   186745n       [ -8  -6   1   0   0   3   1   0 >         186745/186624         1.122103     17      61
  8.00   19:385n       [  2  -4   1   1   1   0   0  -1 >           1540/1539           1.124543     19      42
  8.01   19:218491n    [ 23  -6   0  -5   0  -1   0   1 >      159383552/159279939      1.125817     19      67
  8.02   5:110789n     [  3 -11  -1   3   0   0   1   1 >         886312/885735         1.127422     19      62
  8.05   11:405769n    [-12  -2   0   4  -1   2   0   0 >         405769/405504         1.131005     13      65
  8.08   175:187n      [ 11  -7  -2  -1   1   0   1   0 >         382976/382725         1.135012     17      45
  8.10   34969n        [  4   7   0   0  -2   0  -2   0 >          34992/34969          1.138302     17      56
  8.10   95:169n       [ -4   2  -1   0   0   2   0  -1 >           1521/1520           1.138595     19      50
  8.14   343:1625n     [  7  -3  -3   3   0  -1   0   0 >          43904/43875          1.143913     13      49
  8.19   11:1080625n   [ 15   1  -4  -1   1  -1   0  -1 >        1081344/1080625        1.151503     19      70
  8.24   637:34375n    [  1   3  -5   2  -1   1   0   0 >          34398/34375          1.157966     13      63

8.5 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  8.28   13:25289n     [ -3  -5   0   0   3  -1   0   1 >          25289/25272          1.164177     19      65
  8.31   17:55055n     [-18   4   1   1   2   1  -1   0 >        4459455/4456448        1.167761     17      64
  8.38   175:29887n    [ -9   1  -2  -1   2   1   0   1 >          89661/89600          1.178229     19      71
  8.39   122825n       [-28   7   2   0   0   0   3   0 >      268618275/268435456      1.178662     17      61
  8.41   19:2023n      [ 21  -9   0  -1   0   0  -2   1 >       39845888/39818709       1.181283     19      60
  8.43   107421875n    [ 14   8 -10   0  -1   0   0   0 >      107495424/107421875      1.184926     11      61
  8.50   121:74375n    [ -5   9  -4  -1   2   0  -1   0 >        2381643/2380000        1.194721     17      66
  8.57   1715:14641n   [ -8   7   1   3  -4   0   0   0 >        3750705/3748096        1.204670     11      70
  8.58   49:5078125n   [-23   4   8  -2   0   1   0   0 >      411328125/411041792      1.205563     13      67
  8.63   177023n       [  0  11   0  -1  -3   0   0  -1 >         177147/177023         1.212260     19      59
  8.66   77:1328125n   [ 22  -5  -7   1   1   0  -1   0 >      322961408/322734375      1.217438     17      70
  8.72   11:11875n     [-18   5   4   0  -1   0   0   1 >        2885625/2883584        1.224934     19      50
  8.74   11:40625n     [  4 -10   5   0  -1   1   0   0 >         650000/649539         1.228280     13      49

9 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  8.77   125:2401n     [-10   9   3  -4   0   0   0   0 >        2460375/2458624        1.232524      7      43
  8.78   13265525n     [-30   4   2   4   0   1   1   0 >     1074507525/1073741824     1.234128     17      68
  8.81   95:15379n     [  1   4   1  -1   0  -3   0   1 >          15390/15379          1.237842     19      70
  8.83   25:1183n      [ 23 -11   2  -1   0  -2   0   0 >      209715200/209564901      1.241188     13      43
  8.84   19:12155n     [-22   8   1   0   1   1   1  -1 >       79748955/79691776       1.241718     19      65
  8.86   125:3553n     [  8  -2   3   0  -1   0  -1  -1 >          32000/31977          1.244772     19      62
  8.88   121:4655n     [  9  -9   1   2  -2   0   0   1 >        2383360/2381643        1.247650     19      60
  8.90   19:2401n      [-26  12   0   4   0   0   0  -1 >     1275989841/1275068416     1.250620     19      47
  8.91   19:35035n     [ -5  10  -1  -2  -1  -1   0   1 >        1121931/1121120        1.251894     19      62
  8.93   539n          [ 17  -5   0  -2  -1   0   0   0 >         131072/130977         1.255240     11      25
  8.97   125:833n      [-17   9  -3   2   0   0   1   0 >       16395939/16384000       1.261089     17      46
  8.99   11176655n     [ 25  -1  -1  -6   0   0   0  -1 >       33554432/33529965       1.262830     19      66
  9.00   49:133705n    [ 13  -1  -1   2  -2  -1  -1   0 >         401408/401115         1.264142     17      71
  9.04   1164625n      [-36  10   3   1   3   0   0   0 >    68769941625/68719476736    1.270884     11      55
  9.17   289:1625n     [-12   6   3   0   0   1  -2   0 >        1184625/1183744        1.287990     17      62
  9.19   17:1484375n   [ 18  -1  -7   0   0   0   1  -1 >        4456448/4453125        1.291395     19      71
  9.21   41503n        [ -9  -4   0   3   2   0   0   0 >          41503/41472          1.293601     11      43
  9.24   247:833n      [ -3   3   0  -2   0   1  -1   1 >           6669/6664           1.298458     19      63

9.5 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  9.26   665:1331n     [ -1   0  -1  -1   3   0   0  -1 >           1331/1330           1.301191     19      64
  9.26   833:845n      [ 19 -12   1  -2   0   2  -1   0 >      443023360/442690353      1.301805     17      62
  9.31   25:91091n     [-15   2  -2   2   1   2   0   0 >         819819/819200         1.307653     13      61
  9.36   354025n       [  1  11  -2  -2   0   0  -2   0 >         354294/354025         1.314950     17      58
  9.40   847:40625n    [  4   1  -5   1   2  -1   0   0 >          40656/40625          1.320561     13      67
  9.47   1328125n      [-14  -4   7   0   0   0   1   0 >        1328125/1327104        1.331403     17      52
  9.50   143:78125n    [ -2   7  -7   0   1   1   0   0 >         312741/312500         1.334613     13      59
  9.52   9934375n      [ 28  -3  -5   0  -1   0  -2   0 >      268435456/268228125      1.337667     17      70
  9.59   247:3125n     [-10   4   5   0   0  -1   0  -1 >         253125/252928         1.347895     19      57
  9.62   361:15625n    [ 12 -11   6   0   0   0   0  -2 >       64000000/63950067       1.351241     19      68
  9.67   425:637n      [ -1   1   2  -2   0  -1   1   0 >           1275/1274           1.358363     17      54
  9.69   1463:1625n    [-16  10   3  -1  -1   1   0  -1 >       95954625/95879168       1.361947     19      65
  9.73   245:1859n     [ -5   5   1   2  -1  -2   0   0 >          59535/59488          1.367265     13      56

10 tinas
tinas    comma         monzo                                      interval              cents       limit   popularity
  9.75   1225:2717n    [ 17 -10   2   2  -1  -1   0  -1 >      160563200/160436133      1.370612     19      67
  9.77   221:3773n     [ -7   7   0  -3  -1   1   1   0 >         483327/482944         1.372416     17      62
  9.78   157339n       [  3   9   0  -2   0  -2   0  -1 >         157464/157339         1.374855     19      59
  9.81   5:229957n     [ 25  -6   1  -2   0  -1   0  -2 >      167772160/167638653      1.378202     19      70
  9.83   7:605n        [-11  11  -1   1  -2   0   0   0 >        1240029/1239040        1.381317     11      34
  9.85   91:2299n      [ 11  -4   0   1  -2   1   0  -1 >         186368/186219         1.384664     19      61
  9.90   91:378125n    [  7 -12   5  -1   2  -1   0   0 >       48400000/48361131       1.390875     13      67
  9.90   17:34375n     [-30  12   5   0   1   0  -1   0 >    18268284375/18253611008    1.391113     17      53
  9.92   2211125n      [-29   5   3   2   0   0   0   2 >      537303375/536870912      1.393991     19      67
  9.92   15104375n     [  8  10  -4   0  -1  -3   0   0 >       15116544/15104375       1.394226     13      70
  9.94   7564375n      [ -7 -10   4   2   0   1   0   1 >        7564375/7558272        1.397338     19      66
  9.94   4415125n      [ 30  -5  -3   0  -1  -2   0  -1 >     1073741824/1072875375     1.397572     19      71
  9.97   49:5225n      [-21   9   2  -2   1   0   0   1 >      102843675/102760448      1.401581     19      54
 10.00   49:17875n     [  1  -6   3  -2   1   1   0   0 >          35750/35721          1.404928     13      53
 10.04   169:7315n     [ 12 -11   1   1   1  -2   0   1 >       29962240/29937843       1.410246     19      68
 10.04   1310309n      [-33   8   0   2   2   1   1   0 >     8596937349/8589934592     1.410776     17      66
 10.06   17:1225n      [ -3  -2   2   2   0   0  -1   0 >           1225/1224           1.413829     17      41
 10.09   121:57967n    [ 20  -7   0  -3   2  -2   0   0 >      126877696/126773829      1.417836     13      69
 10.10   343:10985n    [ -5   0   1  -3   0   3   0   0 >          10985/10976          1.418980     13      65
 10.10   5:1294139n    [-37  12  -1   6   1   0   0   0 >   687758524299/687194767360   1.419678     11      58
 10.11   325:361n      [-16  10  -2   0   0  -1   0   2 >       21316689/21299200       1.420951     19      61
 10.11   55:15827n     [  5   2   1  -2   1   0  -1  -1 >          15840/15827          1.421419     19      66
 10.14   5:19n         [  6  -5  -1   0   0   0   0   1 >           1216/1215           1.424298     19      24
 10.16   187:1183n     [ -9   4   0   1  -1   2  -1   0 >          95823/95744          1.427882     17      61
 10.19   11:60025n     [ 14  -1  -2  -4   1   0   0   0 >         180224/180075         1.431888     11      49

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

Ash9903b4
Posts: 16
Joined: Sun Mar 15, 2020 11:27 am

Re: Magrathean diacritics

Post by Ash9903b4 »

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:
  • {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.
The missing values are all 5-schisma complements of intervals that require at most 3 elements to represent, so we can stop here. Additive representations are more efficient for intervals 7 and under, and subtractive ones are more efficient for 10 tinas and over, but the issue is that from 7.5 to 9.5 tinas, addition and subtraction are equally efficient.

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?

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