## Magrathean diacritics

Dave Keenan
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### Re: Magrathean diacritics

To be clear: it would disallow as 5s because the 1/5C (Pythagorean comma, 81/80) has a better apotome slope than it (2.676 beats 7.880).
Correct.
And that 17/1 bit is related to Dave Ryan's preference covered here. It's the choice between the 17k w/ apotome slope 6.462 and the 17C w/ apotome slope 4.093. Sagittal uses the 17C to represent prime 17 in its Prime Factor notation, eschewing the 17k (which Dave Ryan prefers). Sagittal's (precision level) JI notation, on the other hand, allows either one. A prime-factor-combination notation would have the same limitation as the Prime Factor notation, though.
Agreed. But others have preferred 17k before Dave Ryan.
But this is all not terribly relevant to the current work at hand. It seems we agree that we should not actually pursue release of such a "prime-factor-combination" or "meta-comma" notation. However, we can use it to help make the best decisions about the tina commas.
Agreed.
In other words, such a notation can be worked into the JI precision level notation and become part of it at this precision level (and perhaps use this strategy when reassigning those couple of bad apple commas in the Extreme notation which we recently found w/ N2D3P9).
I don't know what you mean by "worked into the JI precision level notation" or "this precision level".
Let me know if you find [apportioning votes between commas for the same 2,3-free ratio]. I just reviewed the thread and couldn't find it.
Different thread. Sorry. viewtopic.php?p=1636#p1636
Are you saying that had apotome slope / abs3exp not knocked some commas out of the running, it would have been even more skewed toward the most popular ratios getting all the symbols?
Yes. But as you say, not relevant to this thread.
I still owe us some tina smileys for the forum, too, apparently!
Not important.

I thought I should resurface a thought we were working with earlier in the thread: that we should choose tina commas with 3-exponents close to +8 since when subtracted from the 5-schisma they would lead to a low 3-exponent. That was only for tinas 1 through 7, though, and it was because we were planning to set tinas 8 though 14 as the difference between the 5-schisma and the corresponding tina.
As mentioned before: I think abs3exp is irrelevant for meta-commas.
... Or should I only look for ones within 9 tinas, which is the range we directly need commas for? Okay, yes, I see that you specifically said 9 tinas earlier. Got it.
Yeah. 9 tinas max.
It looks like we have not already 100% locked down a half-tina, so I should group possibilities by half-tina.
No, as addressed in my previous post.
And the reason I brought up this 5-schisma-difference bit was to confirm that we no longer need the 7.5-, 8-, 8.5-, and 9- tina symbols to be half-5-schisma-mirrors of the 6.5-, 6-, 5.5-, and 5- tina symbols. That actually it might be a feature for them to not be complements of each other, if that allows us to exactly notate more as-of-yet not exactly notated 2,3-free classes.
Commas not needed for n.5 tinas. It might still be a good idea to make n>7 tinas 5s-complements, but lets see what turns up when we allow independent n>7 tinas.

Dave Keenan
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### Re: Magrathean diacritics

cmloegcmluin wrote: Wed Sep 02, 2020 3:07 pm Just this morning I added tests for half-apotome mirroring, and I soon plan to add tests for protected secondary commas, etc.
Cool. Thanks.
More realistically, the dot might be used for either 1/2-tina or 1/3-tina. We don't need to specify.
Elsewhere I had suggested that for purposes of playback, it would alter pitch by nothing (AKA its primary comma would be unison). But I suppose it could be a 1/3-tina.
I see that you really still want a default size for the dot in your software, even if we don't specify it in the SMuFL documentation.

I remind us of @Ash9903b4's post here:
viewtopic.php?p=1583#p1583
Ash's suggestions are more like 1/2-tina or 1/4-tina, not 1/3-tina, so I say default to 1/2-tina. There's the precedent of dotted notes (duration) that might lead people to expect it to correspond to 1/2 a tina, and that was the original definition.
But more importantly, yes, I do think I'm settling on the half-tina dot as this "full-stop... end the insanity here". We may lose the support of some sentient extraterrestrials who are physiologically capable of distinguishing pitches that are hundredths of cents apart, but they've all been fairly quiet on the forum recently...
Ha!
Sorry, I'm just laughing/crying for a moment in remembering that we're dealing with an EDO for which the tiny intervals we're concerning ourselves with map to multiple degrees...
Yeah. This whole thread is pretty much indulgent puzzle-solving fun on our part.
No. That just reintroduces the need to rank commas according to some combination of their abs3exp and their N2D3P9, in order to decide which is the most deserving meta-comma for a given number of tinas. When you only use the comma with the lowest absSlope (or alternatively the lowest abs3exp) for each 2,3-equivalence-class, then you don't have that problem.
Alright, I trust you on this one. My assumption was that including all these notating commas was that we'd find some new (tina-sized) cents-differences. It sounds like you have already convinced yourself that the additional notating commas per 2,3-class would be redundant, i.e. introduce no additional cents-differences. I could probably convince myself of this with a little bit of ... what do you call it... "reasoning"... but I'm afraid I'm spent for the day already.
I think it's likely to be redundant. But if it isn't, I really don't care. I only want to consider meta-commas between lowest-absSlope commas (and on a separate pass, lowest-abs3exp commas, to see if it makes any difference).
I'll get that soon. But tomorrow I've allocated to another project so you shouldn't plan to wake up to it. Sorry for the delay.
Thanks for the warning.

cmloegcmluin
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### Re: Magrathean diacritics

Just dropping a quick note to say that I did have an unexpected pocket of time last night so I thought I'd take a stab at getting you the chart. Well, the script ran for an hour and then just crashed with no explanation. I think it's the too-big-JavaScript-object problem we saw when developing N2D3P9. I don't have time to look into fixing the issue, but max N2D3P9 at the 136 we used previously works within seconds, so I'll at least get you that. I also haven't had time to check the results; I almost, for instance, forgot to actually take the abs of the apotome slopes (I did remember to filter out those > than half apotome; it looks like the "best" notating comma for 31/5 comes perilously close at 56.767¢). I chose "best" just as an expedient and concise title, because it was shorter than "least absolute apotome slope"; I guess I could have used LAAS; I don't actually mean they're "best" in some encompassing way. Let me know what if anything you'd like added/removed/changed here before I get at the 307-one. I decided to include the monzos to make it easier on us in calculating the differences, and also because we'd find the 3-exp in it. But then stuff was getting scrolled off the edge, so I decided to get rid of the Scala occurrences and rank and N2D3P9, keeping only the N2D3P9 rank. I figure I'll want to write a script that'll automatically enumerate all the <9 tina differences for us, though, and in particular the ones which exactly notate as-of-yet not exactly notated 2,3-free classes.

2,3-equivalent
best
best
best
pitch
notating
notating
notating
ratio
N2D3P9
comma
comma
comma
class
rank
cents
monzo
symbol
1/1
1
 23.460
[ -19  12 ⟩
5/1
2
 21.506
[  -4   4  -1 ⟩
7/1
3
 27.264
[   6  -2   0  -1 ⟩
25/1
4
 19.553
[  11  -4  -2 ⟩
7/5
5
 29.218
[  -9   6   1  -1 ⟩
11/1
6
 53.273
[  -5   1   0   0   1 ⟩
35/1
7
 48.770
[   2   2  -1  -1 ⟩
125/1
8
 41.059
[   7   0  -3 ⟩
13/1
9
 48.348
[ -10   4   0   0   0   1 ⟩
49/1
10
 35.697
[  -4  -1   0   2 ⟩
11/5
11
 38.906
[  -2   2   1   0  -1 ⟩
25/7
12
  7.712
[  -5   2   2  -1 ⟩
13/5
13
 43.831
[   3  -1   1   0   0  -1 ⟩
11/7
14
  9.688
[   7  -4   0   1  -1 ⟩
49/5
15
 56.482
[  -3   4   1  -2 ⟩
17/1
16
 14.730
[ -12   5   0   0   0   0   1 ⟩
55/1
17
 31.767
[  -1  -3   1   0   1 ⟩
175/1
18
 43.408
[  -9   1   2   1 ⟩
19/1
19
  3.378
[  -9   3   0   0   0   0   0   1 ⟩
625/1
20
 51.120
[ -14   3   4 ⟩
13/7
21
 38.073
[  -7   5   0   1   0  -1 ⟩
65/1
22
 26.841
[  -6   0   1   0   0   1 ⟩
77/1
23
 26.009
[ -11   3   0   1   1 ⟩
245/1
24
 37.651
[ -19   7   1   2 ⟩
49/25
25
 34.976
[   1   0   2  -2 ⟩
17/5
26
 12.777
[   3  -3  -1   0   0   0   1 ⟩
25/11
27
 17.399
[   2  -2   2   0  -1 ⟩
125/7
28
 13.795
[   1   2  -3   1 ⟩
23/1
29
 40.004
[ -14   6   0   0   0   0   0   0   1 ⟩
91/1
30
 21.084
[ -16   6   0   1   0   1 ⟩
343/1
31
  8.433
[ -10   1   0   3 ⟩
19/5
32
  1.424
[   6  -5  -1   0   0   0   0   1 ⟩
13/11
33
  4.925
[   5  -3   0   0   1  -1 ⟩
121/1
34
  7.139
[  -1   5   0   0  -2 ⟩
17/7
35
 41.995
[  -6   3   0  -1   0   0   1 ⟩
25/13
36
 45.785
[ -12   7   2   0   0  -1 ⟩
35/11
38
 11.642
[  -8   4   1   1  -1 ⟩
55/7
38
 31.194
[   3   0  -1   1  -1 ⟩
77/5
38
 47.515
[ -15   7  -1   1   1 ⟩
85/1
40
  6.776
[   8  -1  -1   0   0   0  -1 ⟩
275/1
41
 33.720
[ -16   5   2   0   1 ⟩
875/1
42
 21.902
[  -5  -3   3   1 ⟩
29/1
43
 33.487
[  -8   2   0   0   0   0   0   0   0   1 ⟩
19/7
44
 30.642
[  -3   1   0  -1   0   0   0   1 ⟩
23/5
45
 38.051
[   1  -2  -1   0   0   0   0   0   1 ⟩
95/1
46
 18.128
[   5   1  -1   0   0   0   0  -1 ⟩
143/1
47
 12.064
[   4   2   0   0  -1  -1 ⟩
31/1
48
 54.964
[   5   0   0   0   0   0   0   0   0   0  -1 ⟩
3125/1
49
 29.614
[ -10  -1   5 ⟩
35/13
51
 16.567
[  -3   1   1   1   0  -1 ⟩
65/7
51
 36.120
[   8  -3  -1   1   0  -1 ⟩
91/5
51
 19.130
[  -1  -2  -1   1   0   1 ⟩
49/11
53
 17.576
[  -1   2   0  -2   1 ⟩
343/5
54
 29.939
[ -14   5  -1   3 ⟩
119/1
55
 35.994
[  -1   5   0  -1   0   0  -1 ⟩
325/1
56
  5.335
[  -2  -4   2   0   0   1 ⟩
385/1
57
  4.503
[  -7  -1   1   1   1 ⟩
17/11
58
 51.682
[   1  -1   0   0  -1   0   1 ⟩
1225/1
59
 16.144
[ -15   3   2   2 ⟩
169/1
60
 16.990
[   9  -1   0   0   0  -2 ⟩
121/5
61
 14.367
[  -3  -1  -1   0   2 ⟩
77/25
62
 45.561
[   0  -1  -2   1   1 ⟩
125/49
63
 13.469
[   5  -4   3  -2 ⟩
25/17
64
 34.283
[  -1   1  -2   0   0   0   1 ⟩
23/7
65
 46.417
[  -3   3   0   1   0   0   0   0  -1 ⟩
17/13
66
 33.617
[   2  -1   0   0   0   1  -1 ⟩
125/11
67
 19.353
[ -13   6   3   0  -1 ⟩
133/1
68
 47.346
[  -4   7   0  -1   0   0   0  -1 ⟩
625/7
69
 35.301
[  -3   6  -4   1 ⟩
115/1
70
 18.498
[ -10   2   1   0   0   0   0   0   1 ⟩
19/11
71
 49.895
[   4  -2   0   0   1   0   0  -1 ⟩
37/1
72
 47.434
[  -2  -2   0   0   0   0   0   0   0   0   0   1 ⟩
49/13
73
 12.651
[  -6   5   0  -2   0   1 ⟩
29/5
74
 54.993
[ -12   6  -1   0   0   0   0   0   0   1 ⟩
455/1
75
  0.423
[  12  -2  -1  -1   0  -1 ⟩
539/1
76
  1.255
[  17  -5   0  -2  -1 ⟩
1715/1
77
 13.074
[   6   3  -1  -3 ⟩
25/19
78
 22.931
[   2  -1  -2   0   0   0   0   1 ⟩
55/13
80
  6.879
[ -10   5   1   0   1  -1 ⟩
65/11
80
 26.432
[   1   1  -1   0   1  -1 ⟩
143/5
80
  9.442
[  -8   2  -1   0   1   1 ⟩
121/7
82
 43.585
[ -12   5   0  -1   2 ⟩
19/13
83
 44.970
[  -1   1   0   0   0   1   0  -1 ⟩
187/1
84
 45.682
[   6   1   0   0  -1   0  -1 ⟩
31/5
85
 56.767
[  -1  -1  -1   0   0   0   0   0   0   0   1 ⟩
91/25
86
 40.636
[  -5   2  -2   1   0   1 ⟩
55/49
87
  3.930
[  -3   2  -1   2  -1 ⟩
605/1
88
  5.185
[  14  -3  -1   0  -2 ⟩
343/25
89
 27.985
[   1  -3  -2   3 ⟩
41/1
90
 44.702
[ -18   8   0   0   0   0   0   0   0   0   0   0   1 ⟩
35/17
92
 50.184
[  -1   0   1   1   0   0  -1 ⟩
85/7
92
 20.488
[  -2  -1   1  -1   0   0   1 ⟩
119/5
92
 14.487
[   3   1   1  -1   0   0  -1 ⟩
125/13
94
 24.278
[  -8   3   3   0   0  -1 ⟩
175/11
95.5
  9.865
[   4   0  -2  -1   1 ⟩
275/7
95.5
 52.701
[  -1   4  -2   1  -1 ⟩
425/1
97
 28.282
[   4   3  -2   0   0   0  -1 ⟩
169/5
98
  4.517
[ -13   5  -1   0   0   2 ⟩
121/25
99
 35.873
[  -7   3  -2   0   2 ⟩
43/1
100
 13.473
[  -7   1   0   0   0   0   0   0   0   0   0   0   0   1 ⟩
161/1
101
 10.720
[   1   4   0  -1   0   0   0   0  -1 ⟩
221/1
102
 50.607
[  11  -2   0   0   0  -1  -1 ⟩
1375/1
103
 12.214
[ -12   1   3   0   1 ⟩
4375/1
104
 23.856
[ -20   5   4   1 ⟩
23/11
105
 36.729
[ -10   7   0   0   1   0   0   0  -1 ⟩
29/7
106
 52.934
[  -9   7   0   1   0   0   0   0   0  -1 ⟩
209/1
107
 56.651
[ -14   4   0   0   1   0   0   1 ⟩
19/17
108
 11.352
[  -3   2   0   0   0   0   1  -1 ⟩
637/1
109
 29.641
[   3   4   0  -2   0  -1 ⟩
2401/1
110
 18.831
[  16  -3   0  -4 ⟩
145/1
111
 11.981
[  -4  -2   1   0   0   0   0   0   0   1 ⟩
35/19
113
 52.148
[  -7   5  -1  -1   0   0   0   1 ⟩
95/7
113
  9.136
[   1  -3   1  -1   0   0   0   1 ⟩
133/5
113
 25.840
[   0   3   1  -1   0   0   0  -1 ⟩
77/13
116
 22.339
[   1   1   0  -1  -1   1 ⟩
91/11
116
 55.649
[  -8   7   0  -1   1  -1 ⟩
143/7
116
 15.200
[   2  -4   0  -1   1   1 ⟩
25/23
118
 54.128
[  -8   5   2   0   0   0   0   0  -1 ⟩
47/1
119
 36.448
[   4   1   0   0   0   0   0   0   0   0   0   0   0   0  -1 ⟩
31/7
120
 27.700
[  -1   2   0   1   0   0   0   0   0   0  -1 ⟩
475/1
121
 39.635
[   1   5  -2   0   0   0   0  -1 ⟩
37/5
122
 44.745
[  -5   5   1   0   0   0   0   0   0   0   0  -1 ⟩
23/13
123
  8.343
[   4  -2   0   0   0   1   0   0  -1 ⟩
65/49
124
  8.855
[   2  -1  -1   2   0  -1 ⟩
715/1
125
 33.571
[   0   6  -1   0  -1  -1 ⟩
847/1
126
 34.403
[   5   3   0  -1  -2 ⟩
29/25
127
 53.040
[   3  -2  -2   0   0   0   0   0   0   1 ⟩
247/1
128
 51.726
[ -19   7   0   0   0   1   0   1 ⟩
49/17
129
 44.426
[ -11   6   0   2   0   0  -1 ⟩
155/1
130
 37.214
[ -12   3   1   0   0   0   0   0   0   0   1 ⟩
15625/1
131
 31.567
[ -25   7   6 ⟩

Dave Keenan
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Contact:

### Re: Magrathean diacritics

That's great. Thanks! I've put it into a spreadsheet and can give some results.

Of course the first meta-comma less than 9.5 tinas to appear is 3 tinas (1 mina) as rank7 minus rank9, which is 1/35M - 13M.
The next is 3 tinas as rank13-rank18 = 5/13M-175M which is the same meta-comma. I won't mention repeated numbers of tinas from here on, unless they are close in max-rank and have different metacommas.
3 tinas as rank7-rank9
6 tinas as rank11-rank21
9 tinas as rank3-rank23
5 tinas as rank10-rank25
7 tinas as rank16-rank28
7 tinas as rank28-rank26
7.5 tinas as rank8-rank29 (This could give us a ratio for the dot, as a meta-meta-comma)
8 tinas as rank29-rank11
4 tinas as rank12-rank34 *
2 tinas as rank41-rank43 *
1 tina as rank53-rank27 *

* Includes a not-yet-exactly-notated comma.

So we don't need to go any further in N2D3P9, but I'd be glad if you would write a script to check these results. And repeat the exercise with "best" redefined as lowest abs3exp, with smallest size in cents as a tie-breaker.

cmloegcmluin
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### Re: Magrathean diacritics

Dave Keenan wrote: Wed Sep 02, 2020 4:18 pm
More realistically, the dot might be used for either 1/2-tina or 1/3-tina. We don't need to specify.
Elsewhere I had suggested that for purposes of playback, it would alter pitch by nothing (AKA its primary comma would be unison). But I suppose it could be a 1/3-tina.
I see that you really still want a default size for the dot in your software, even if we don't specify it in the SMuFL documentation.

I remind us of @Ash9903b4's post here:
viewtopic.php?p=1583#p1583
Ash's suggestions are more like 1/2-tina or 1/4-tina, not 1/3-tina, so I say default to 1/2-tina. There's the precedent of dotted notes (duration) that might lead people to expect it to correspond to 1/2 a tina, and that was the original definition.
The web calculator for Sagittal should have some playback at some point, but I was really just thinking about this doc and wanting to having an answer for developers of music notation software.

^ Thanks for demonstrating that [hr] for me! I'll stop using my ugly ol' ------ trick.
This whole thread is pretty much indulgent puzzle-solving fun on our part.
Welp. I was supposed to get started on this other project today, but tried as I might to stop indulging in this puzzle, I couldn't help myself...

To the delight of many Node developers across the internet, this particular fail condition gives one very little to work off of. However, thanks to my newly acquiring skills of profiling Node (thanks again, Dave), I was able to figure out where my script was crashing!

It was spending most of its time computing monzos from prime exponent extremas. Yes, I know that's a double plural: "extremas" in my app are like monzos except each term is a pair of extrema (a min and a max) between which that term must lie (inclusively, on both ends). Given a maximum N2D3P9, such an "extremas" is just what the code needs; it's essentially the structure Dave first presented here for max N2D3P9 = 136). It then checks each possible monzo given said extremas — which in other words is every monzo which has any hope of its N2D3P9 being less than the max — and if its N2D3P9 is found indeed to be within the max, then that monzo makes it into the output table (where it gets ranked, converted to a ratio, matched up with notating Sagittal symbols, etc.)

The problem was that when the max N2D3P9 gets big, the count of monzos to check gets huge. By max N2D3P9 = 307, we'd apparently already crossed the threshold where an object storing every monzo to check was beyond Javascript's comfort level. So I needed to find a cleverer way of doing a bit of work for each possible monzo given prime exponent extremas. I needed to find a way to do it without precalculating and storing every monzo, but rather doing them one a time and transforming the current monzo to the correct next monzo. Once I identifying the problem, it wasn't hard to get this working, and the code is better for it.

Another problem this turned up was that some monzo-from-extremas work was happening over and over wastefully, in a different corner of the code: whenever it tries to find the max exponent for a denominator prime, it has to know what the max numerator can be for the given max N2D3P9. Each denominator prime was calculating it all over again. I plan to change it to memoize whatever it finds. Potentially I could even find the breaking points between each max numerator based on the max N2D3P9 and hardcode that, but again, that can be taken care of whenever I get to it. For now, for 307, btw, the max numerator is 21875, 55 * 7.

No, I still haven't done the work to hardcode the sorted N2P's and N2's. That would help performance too, but apparently wasn't the bottleneck to finding our lovely metacommas*. It still took a while for it to check the N2D3P9 for the tens of millions of possible monzos (unfortunately I didn't get an exact number).

So I was finally able to get it to spit out the ranked list of top 307 ratios. Since that's such a bulky table, I'm actually going to share it in another post, and actually probably just back on the N2D3P9 thread (update: here).

I found that what we really wanted from this new script was for it to give a list of commas, sorted by increasing size in cents, grouped by which tina it'd be assigned to (as we were structuring our tables earlier in this thread).

And then for each of these tina comma candidates, which of the not-yet-exactly-notated commas it would enable Sagittal to exactly notate given the commas it already has.

And since each tina comma candidate could potentially enable notation of multiple not-yet-exactly-notated commas, and each not-yet-exactly-notated comma might be able to be exactly notated by more than one tina comma candidate, we needed the table to have a column for each not-yet-exactly-notated comma, so we'd in the end have a grid of all the possibilities, so we could choose the metacommas most efficiently.

And that I should sort the columns by decreasing popularity, so we can optimize for ones closer to the left.

And rather than a simple "x" in the cells where there is a match, use the cell to display the existing comma in Sagittal which when combined with this metacomma gets you the

Said table, it turns out, it just over the max allotted character count per post on the forum. I might be able to fiddle with the forum settings, but that limit is probably there for a reason. So I'll start the table on the next post, splitting it up by tina, and spread it across two posts.
Dave Keenan wrote: Thu Sep 03, 2020 10:09 am So we don't need to go any further in N2D3P9, but I'd be glad if you would write a script to check these results.
Should be a simple matter to verify using the tables I'm about to drop.

Next thing I'd like to do is cross-reference these with the tables we were working with before with some yellow hiliting and see how in tune they are with each other.

*I keep meaning to ask about this. I have a vague memory of reading somewhere at some point that if a comma is defined as a small interval between between two pitches, then a [????] is a small interval between two commas. I can't figure out what I'm thinking of, or even if it was on Tonalsoft or the Xen wiki or XPSC or Untwelve camp materials or what. Does that ring any bells for you? I want to say that whatever those ????s are, they were one of the commatic interval terms Sagittal uses as a name for one of the size categories in its systematic comma naming scheme, in which case the collision would prevent us from getting much mileage out of the term in this context. Besides, I rather like "metacomma"!

cmloegcmluin
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### Re: Magrathean diacritics

Alright, well due to some formatting struggles, I'm just going to break it into two chunks. Also I have to go make dinner but I want to get this out there in some form.
metacommatinas25/11121/129/195/135/13119/1121/5125/4923/7125/11133/1625/7115/129/5539/11715/155/13143/5121/7187/131/5605/1343/2541/135/1785/7119/5175/11425/1169/5121/2543/1161/1221/11375/129/7209/119/17637/12401/1145/135/1995/7133/577/1325/2347/131/7475/137/523/1365/49715/1847/129/25247/1155/115625/1
123201/1232000.1385/1
1990656/19906250.1923125/1
655371/6553600.20631/1
17826237/178257920.31317/11
37180/371790.32717/13
20736/207350.591143/1
19684/196830.62637/1
213003/2129920.64343/1
3408075/34078720.73385/125/13
14400/143990.85477/2525/17
426496/4264650.89765/7
13312/133110.92517/13
196625/1966081.06325/1143/1275/1
10241/102401.20377/519/549/577/1
9801/98001.25249/11175/111/749/1
184549376/1845281251.42325/155/1
4100625/41000961.58791/5
7696/76951.60119/13
7425/74241.658275/1275/711/1
163863/1638401.72917/1
1146880/11467171.75811/713/565/7
6656/66551.8513/11
1703936/17036731.90719/13
6273/62721.96449/17
376893/376832223/11
6076/60752.02849/5
5985/59842.06419/1135/11
5832/58312.11349/117/5
655473/6553602.128343/591/5
5440/54392.2749/17
5376/53752.291125/7
250047/2500002.313125/749/25
5265/52642.33465/7
5104/51032.41211/135/11275/7
61965/619522.58385/117/1
32775/327682.62623/1
1879453125/18790481922.65455/7
4375/43742.81849/1
14417920/144145172.90313/11
4225/42242.911169/1
4096/40953.003169/149/137/5343/1
3969/39683.117/1245/1
67392/673753.1149/13875/1
152361/1523203.31617/11
3565/35643.45123/11
33564375/335544323.651625/1
3136/31353.92849/11275/711/7
557056/5568754.00617/5
3025/30244.07825/7125/735/1149/25
50193/501764.17755/49
2926/29254.21365/1125/19
1393119/13926404.24191/1
20007/200004.312625/1
46592/465754.4925/1391/525/7

cmloegcmluin
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### Re: Magrathean diacritics

metacommatinas25/11121/129/195/135/13119/1121/5125/4923/7125/11133/1625/7115/129/5539/11715/155/13143/5121/7187/131/5605/1343/2541/135/1785/7119/5175/11425/1169/5121/2543/1161/1221/11375/129/7209/119/17637/12401/1145/135/1995/7133/577/1325/2347/131/7475/137/523/1365/49715/1847/129/25247/1155/115625/1
2640/26394.66891/11
28431/284204.76849/13
2097152/20963254.86385/1275/1
2500/24994.92417/5125/7
17415/174084.95317/5
194560/1944815.003343/1
2432/24315.06719/1319/11143/1
2431/24305.067143/725/17
2401/24005.138125/77/125/1
2376/23755.18811/5
11781/117765.2323/11
2304/23035.34449/1
11016/110115.59317/13
2176/21755.66425/1717/5
91168/911255.80777/25
256000/2558795.82149/1365/7325/1
2107/21065.84949/13
2080/20795.928143/7455/125/75/111/7
188416/1883255.95631/123/5
2058/20575.98549/17
2025/20246.08477/2523/1111/1
2016/20156.11365/113/7
20185088/201750756.11377/25
10491039/104857606.19813/5
160083/1600006.38349/25
9477/94726.50413/5
5635/56326.56149/1135/11
79734375/796917766.58219/7
57375/573446.661625/125/17/5
1835008/18339756.938275/7
1729/17287.12391/513/191/1119/765/191/25
1716/17157.1849/11143/1343/1
1702/17017.23737/1
1701/17007.24449/2525/175/17/1
21888/218757.315625/1
72171/721287.34411/7
1665/16647.40125/13
2097152/20958757.50825/1
11271/112647.6517/11
1600/15997.70725/13
7344675/73400327.78565/7
20493/204807.81477/5
361088/3608557.95691/11
13640319/136314887.9771/1
1540/15398.00611/1
1521/15208.105169/119/1
43904/438758.13465/125/7
8394435/83886088.561245/1
2460375/24586248.7747/5
505197/5048328.90917/11
131072/1309778.9381225/149/1755/4949/1191/1
1377/13768.94517/1
16395939/163840008.97317/1
1365/13649.02365/11
41503/414729.208343/149/1
89667/896009.208175/1
6669/66649.24449/13
1328125/13271049.4793125/1
1879453125/1879048192
So, you'll probably want to copy this out into Excel to play with it. I added a struckthrough bit at the bottom to get the first column widths to agree between the two posts. Feel free to fiddle with these posts however you like. I've got the data secured on my end so go wild.

Dave Keenan
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### Re: Magrathean diacritics

Thanks for that table. It is certainly of interest, but I don't see how it lets me verify which metacomma occurs first for each tina. I was not taking any notice of whether commas were exactly notated (without using tina accents), as that is something that could change.

It would make more sense to me if you did it for commas that are not yet exactly notated without using minas (or tinas), since minas are essentially triple-tinas. For example, the table above fails to suggest the obvious definitions for 3 tinas and 6 tinas.

The word you are looking for is "schisma". THE schisma is 5s, the difference between 5C and 3C, the syntonic and Pythagorean commas.

cmloegcmluin
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### Re: Magrathean diacritics

Huh. Well, I’ve read over what you’ve said a few times and unfortunately I can’t make much sense of it. I don’t really even know what to ask either. I tried to make sure I understood exactly what you wanted and why before I spent time getting it, but there must have still been some major miscommunication. If there’s anything further you can think of to write to help clarify before I get up tomorrow I may have some time to work on it then.

Dave Keenan
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### Re: Magrathean diacritics

I think you've run with what I wrote here:
Dave Keenan wrote: Tue Sep 01, 2020 8:23 am In this post, I noted that the first 2,3-free ratio in the above list that does not yet have a sagittal symbol, 25/11, is a candidate to be notated by a 1 tina accent applied to the existing symbol for 49/11  . That makes 1 tina = 1225/121n which is what George Secor suggested for it. There may be other un-notated but moderately popular ratios that could be notated with tina accents. This might be the best way to assign commas to the accents for 1 thru 9 tinas.
and that's fine. The more approaches the better.

But I'm going with what I wrote later in that post:
... it would be interesting to see that single lowest-slope comma, and its size in cents, listed against each ratio in the above list. Then search for cases where two such commas differ by 9 tinas or less, and consider assigning each tina accent to that correctly-sized difference whose pair of 2,3-free ratios have the lowest maximum N2D3P9.
I've checked my results above, and have attached the spreadsheet I used, with bold outlines on the cells corresponding to the meta-commas I listed.

I planned to give that list of tinas and their lowest-N2D3P9 metacommas again here, but with the pairs of ranks converted to meta-comma names, so we could compare them to the yellow-highlighted rows in the tables you mentioned from way back. But I ran out of time tonight.

I'd be pleased if you can generate another list of 131 2,3-free ratios, this time with the notational comma for each, that has lowest abs3exp (LATE) (and smallest size, in the case of a tie), instead of lowest absolute apotome slope (LAAS).
Attachments
N2D3P9metacommas.xlsx