consistent Sagittal 37-Limit

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Re: consistent Sagittal 37-Limit

Post by cmloegcmluin »

Thanks for digging it up, @Dave Keenan , and thanks for breaking it down, @volleo6144 !

I'll just go ahead and use this as the consolidated complexity metric we've been looking for to objectify the decisions we're trying to make about primary commas for the Insane precision level's symbols over on the Magrathean accents topic.

I'll give it a little time before I pull the trigger on unsplitting the 75th mina, eliminating the 47S from the Extreme precision level (on the JI precision level diagram, JI notation spreadsheet, etc.). Just in case anyone does think of any objections. But I think we've done about as much as we can reasonably expect ourselves to do at this point in order to find a good reason not to do so.
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Re: consistent Sagittal 37-Limit

Post by cmloegcmluin »

I went back and added the column to the chart in my earlier post (here) showing which -- between the default and primary commas considered when developing the Extreme precision level -- had the lower SoPF>3. I suspect the 499S there is a typo because 499 is a relatively huge prime. Otherwise, as you can see all but the 69th mina worked out so that the primary comma had the lower SoPF>3, and there the default comma was 29 while the primary comma was 30 so it only beat it out by 1. So I guess that's another tiny argument in favor of 11:23S over 47S.
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Re: consistent Sagittal 37-Limit

Post by volleo6144 »

cmloegcmluin wrote: Sat May 30, 2020 2:30 pm I suspect the 499S there is a typo because 499 is a relatively huge prime.
The 499S is a pretty simple ratio (499:512), and it is in the range of 91 minas (at 91.28 in 2460-EDO and 91.25 in 233-EDA), so if it is a typo, someone must have just gone along with it. And this was in or before 2007, because there's a reference to it in a quote from 2007 here. (Edit now that N2D3P9 has been found: our N2D3P9 for "499" is 499/2 × 499/9 = 249001/18 = 13833.)

The 49S is also a particularly simple ratio (even simpler, at 48:49), but it's only 73 minas, and I can't think of any other ratios of 91 minas that could be confused with "499S"...
Last edited by volleo6144 on Wed Jul 21, 2021 7:08 am, edited 2 times in total.
I'm in college (a CS major), but apparently there's still a decent amount of time to check this out. I wonder if the main page will ever have 59edo changed to green...
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Re: consistent Sagittal 37-Limit

Post by Dave Keenan »

I'm not here. But @volleo6144 is right, that 499S is not a typo. Ratios of 499 weirdly occur a few times in the Scala archive. I think that George and I held on too long to the idea that we should give priority to commas that occur more often in the Scala archive. In hindsight, once you get beyond the 100-or-so most common 2,3-reduced ratios (maybe only the first 40-or-so) then the numbers of occurrences are so small as to be due to historical accidents that are unlikely to be predictive of future use.

Part of the reason we held on so long, is that in the early days, in order for the Sagittal idea to survive politically, we had to be seen to be basing the comma choices on something objective like the Scala archive stats, not some "arbitrary complexity function" that, it would be argued, merely suited our biases. The only reason SoPF>3 had any respectability, at least in the minds of George and I, is that it gives a ranking that is similar to the Scala archive ranking, for those first 40-or-so 2,3-reduced ratios. And it is easy to calculate mentally.

But perhaps the time has come, to try to find a better complexity function for our purposes. One that matches the Scala archive stats even better, but filters out the historical noise. And it need not be easily mentally calculated. For starters, the weighting of each prime need not be the prime itself.

But more importantly, one area where SoPF>3 has always fallen down, is that it ignores the relative signs of the different prime exponents. e.g. it treats 5:7 as having the same rank as 1:35, and 5:11 the same as 1:55, and 7:11 same as 1:77, which they are not. The ratios where the primes are on opposite sides are more common.

Even more complicated: In order of decreasing frequency of occurrence, we have the ratios 11:35, 7:55. 5:77 and 1:385, which are all treated equally by SoPF>3.

I challenge readers to come up with a simple function that treats them differently, and whose coefficients can be fitted to the Scala stats. But beware Von Neumann's elephant (second-last paragraph).

Compare:
viewtopic.php?p=259#p259
and
viewtopic.php?f=4&t=99#rank

Thanks @volleo6144 for reverse-engineering George's "weighted complexity". It does seem to have some redundant multiple uses of the 3-exponent.

@cmloegcmluin, Thanks for the table. I've rearranged things a bit and included "other" as an option for the comma with the better SoPF>3 below. I also corrected the current sagittal for the 69th mina.

	old	old
# of	default	primary other	current	 better
minas	ratio	ratio   ratio	Sagittal SoPF>3
-----	-------	-------	-------	-------	 -------
  1	31:49n	455n		primary	 primary
  2	13:37n	65:77n		primary	 primary
 11	11:31k	605k		default	 primary *
 22	5:53k	5:161k	13:49k	other	 other
 31	5:47C	7:143C		primary	 primary
 42	19:73C	19:169C	253C	other	 other
 59	13:47C	605C	325C	other	 other
 66	53C	19:49C	11:91C	other	 other
 69	29S	13:17S		default	 default
 75	47S	11:23S		both	 primary *
 91	499S	17:49S		primary	 primary
 98	83M	5:187M	11:325M	other	 primary *
108	7:29M	2375M	7:275M	other	 other
110	47M	11:85M		primary	 primary

This shows that George and I moved away from using Scala archive stats ("default") and towards a complexity measure that was dominated by SoPF>3, at least in deciding the primary commas for such unpopular minas as these.

* I find it conceivable that a better complexity measure than SoPF>3 could justify all the asterisked choices, except for the choice to retain 47S as an option in a split 75th mina. The splitting of the 75th mina is not justified by having two popular commas as options, nor is it justified by the DEfLL property (Drop Elements for Lower Level, previously DAFLL = Drop Accents For Lower Level) .
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Re: consistent Sagittal 37-Limit

Post by volleo6144 »

So after drifting away from this forum for reasons I don't remember after my last post here, I came back (also for reasons I don't remember even though it was literally a few days ago...) and decided to polish up a few of my old posts, because I couldn't find a place to contribute that was remotely recent and therefore (I thought) relevant.

...except it apparently notifies you if someone just edits a post that you were quoted in (just why?). Anyway, @Dave Keenan then PM'd me suggesting an updated version of the above chart with our new LPEI badness metric (assuming each capture zone is actually 1\2460 wide and not 3\8539 or 4\8539 or whatever, and that the mina is based on 2460edo and not 2459.427edo or 2459.985edo). Blue indicates that the apotome-complement is simpler and is therefore listed instead.
	old	old
# of	default	primary other	current(?) better LPEI
minas	ratio	ratio   ratio	Sagittal badness (comma ratio, margin)
-----	-------	-------	-------	-------	 -------
  1	31:49n	455n		primary	 primary (4095:4096, 2.551547 over  784:837)
  2	13:37n	65:77n		primary	 primary (2079:2080, 0.948918 over  104:111)

 11	11:31k	605k		default	 primary (16335:16384, 1.231842 over   31:33)
 22	5:53k	5:161k	13:49k	other	 other     (49:52,   0.487336 over  160:161)

 31	5:47C	7:143C		primary	 primary  (567:572,  0.817269 over 1269:1280)
 42	19:73C	19:169C	253C	other	 other    (253:256,  1.856199 over  169:171)
 59	13:47C	605C	325C	other	 other    (975:1024, 1.309843 over  576:605)
 66	53C	19:49C	11:91C	other	 other   (22113:22528, 0.049588 over   53:54)

 69	29S	13:17S		default	 default  (232:243,  0.501173 over   51:52)
 75	47S	11:23S		both	 default   (47:48,   0.247220 over   22:23)
 91	499S	17:49S		primary	 primary   (49:51,   6.763374 over  499:512)

 98	83M	5:187M	11:325M	other	 other   (2600:2673, 0.492058 over  180:187)
108	7:29M	2375M	7:275M	other	 other    (550:567,  0.838989 over   28:29)
110	47M	11:85M		primary	 primary   (85:88,   0.560484 over  729:752)
The lowest LPEI badnesses for the 222nd, 202nd, 167th, 158th, and 135th minas aren't the same as the apotome-complements of the 11th, 31st, 66th, 75th, and 98th minas.
I'm in college (a CS major), but apparently there's still a decent amount of time to check this out. I wonder if the main page will ever have 59edo changed to green...
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Re: consistent Sagittal 37-Limit

Post by Dave Keenan »

Thanks for that, @volleo6144.

The capture zones are currently 1/233 of an apotome, so that the zone boundaries are mirrored about the half-apotome. Or putting it another way: so that the zone boundaries for an apotome-complement are the apotome-complements of the zone boundaries of the original symbol. That's equivalent to 1\2359.427234.

But this:
viewtopic.php?p=1495#p1495
reminds me that the boundaries are slightly irregular, as they are first rounded to the nearest odd half-mina (mina = 1/233-apotome) and then rounded to the nearest odd half-tina (tina = 1/809-apotome).


What do the strike-throughs mean?

That was good thinking, to consider the badness of the apotome-complements. I'd forgotten about that. But I think maybe we only want to do that when the apotome complement has a single-shaft symbol. i.e. if it rounds to 140 minas or less.

Apotome complements should have the same N2D3P9, the same absolute slope and the same absolute error. They should only differ in absolute 3-exponent.

That's surprising that 47:48 has lower badness as 75 minas than 22:23 as 158 minas.
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Re: consistent Sagittal 37-Limit

Post by volleo6144 »

Dave Keenan wrote: Tue Jul 27, 2021 9:18 pm The capture zones are currently 1/233 of an apotome, so that the zone boundaries are mirrored about the half-apotome. Or putting it another way: so that the zone boundaries for an apotome-complement are the apotome-complements of the zone boundaries of the original symbol. That's equivalent to 1\2459.427234.

But this:
viewtopic.php?p=1495#p1495
reminds me that the boundaries are slightly irregular, as they are first rounded to the nearest odd half-mina (mina = 1/233-apotome) and then rounded to the nearest odd half-tina (tina = 1/809-apotome).
Yeah, I think I remembered that, but that just leads to the question of whether that means a 1-tina error is more acceptable if the capture zone happens to be 4 tinas wide instead of 3...
What do the strike-throughs mean?
The final column there has "[first-place comma], [margin] over [second-place comma]", which isn't exactly very conducive to rows with three commas, for which I crossed out the third-place comma.
That was good thinking, to consider the badness of the apotome-complements. I'd forgotten about that. But I think maybe we only want to do that when the apotome complement has a single-shaft symbol. i.e. if it rounds to 140 minas or less.
Yeah, I suppose. The blue commas listed are just the apotome-complements when they have lower odd limits, because otherwise I'd have to include the font size trick for a lot more than just two commas (one of them was even 7 digits on both sides before complementing it); the N2D3P9 margin listed is still for the original commas, and the paragraph at the end is just there.
Apotome complements should have the same N2D3P9, the same absolute slope and the same absolute error. They should only differ in absolute 3-exponent.
Yeah, though my analysis used 2460edo instead of 2459.427edo for the definition of the mina, so they were slightly different. Ignoring the tina rounding (because, again, 3-tina and 4-tina capture zones are even more ambiguous), this doesn't affect any of the choices, though, just the margins, which I didn't bother to edit. [Actually, some minas beyond the half-apotome are changed.]
That's surprising that 47:48 has lower badness as 75 minas than 22:23 as 158 minas.
The 47S's apotome slope is only -1.24426, while the 11:23SS has -4.73849, which is a somewhat substantial difference in the badness calculation that just barely beats all of the other factors (which are all really close; 11:23 has an N2D3P9 of 107.76 and 47 has 122.72).
I'm in college (a CS major), but apparently there's still a decent amount of time to check this out. I wonder if the main page will ever have 59edo changed to green...
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Re: consistent Sagittal 37-Limit

Post by Dave Keenan »

Thanks for explaining. That all makes sense. Unfortunately, it doesn't bode well for the title of this thread.
volleo6144 wrote: Wed Jul 28, 2021 12:28 am Yeah, I think I remembered that, but that just leads to the question of whether that means a 1-tina error is more acceptable if the capture zone happens to be 4 tinas wide instead of 3...
No, I think the error should be measured in minas (1/233-apotome) no matter how wide the capture zone is. But it might make sense that the nominal value that the error is measured from, should be rounded to the whole tina (1/809-apotome) that is nearest to the whole mina.
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Re: consistent Sagittal 37-Limit

Post by volleo6144 »

	old	old
# of	default	primary other	current  better LPEI
minas	ratio	ratio   ratio	Sagittal badness (comma ratio, margin)
-----	-------	-------	-------	-------	 -------
01 :`::|:	31:49n	455n		primary	 primary (4095:4096, 2.595913 under  784:837)
02 :``::|:	13:37n	65:77n		primary	 primary (2079:2080, 1.179357 under  104:111)

--- 3.70 minas, schismina/schisma boundary ---
--- 9.22 minas, schisma/kleisma boundary ---

11 :,::|(:	11:31k	605k		default	 primary (16335:16384, 1.424515 under   31:33)
22 :``::)|(:	5:53k	5:161k	13:49k	other	 other     (49:52,   0.642317 under  160:161)

--- 24.0 minas, kleisma/comma boundary ---

31 :`::~|(:	5:47C	7:143C		primary	 primary  (567:572,  0.688236 under 1269:1280)
42 :,,::/|:	19:73C	19:169C	253C	other	 other    (253:256,  1.752773 under  169:171)
59 :,::'::|):	13:47C	605C	325C	other	 other    (975:1024, 0.494992 under  576:605)
66 :`::|\:	53C	19:49C	11:91C	other	 default   (53:54,   0.162018 under 22113:22528)

--- 68.4 minas, comma/S-diesis boundary ---

69 :`::(|:	29S	13:17S		default	 default  (232:243,  0.697298 under   51:52)
75 :,::.::(|(:	47S	11:23S		primary	 default   (47:48,   0.235615 under   22:23)
91 :,::'::/ /|:	499S	17:49S		primary	 primary   (49:51,   6.604224 under  499:512)

--- 92.5 minas, S-diesis/M-diesis boundary ---

98 :,,::/|):	83M	5:187M	11:325M	other	 other   (2600:2673, 0.405932 under  180:187)
108 :,::/|\:	7:29M	2375M	7:275M	other	 other    (550:567,  0.546074 under   28:29)
110 :`::/|\:	47M	11:85M		primary	 primary   (85:88,   0.403643 under  729:752)

-- 116.5 minas, M-diesis/L-diesis boundary, half-apotome --
(note to self, if this needs more revisions: 159:160, 4617:4672, 416:423, 1539:1568, 249:256, 2304:2375)

...oh dear, now it's not even 47-limit. edits 4 and 5: yes, half of this was because I wondered if it would notify anyone and potentially start forum activity back up, and I don't understand why size=0 just doesn't work
Last edited by volleo6144 on Fri Dec 10, 2021 12:46 pm, edited 5 times in total.
I'm in college (a CS major), but apparently there's still a decent amount of time to check this out. I wonder if the main page will ever have 59edo changed to green...
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Re: consistent Sagittal 37-Limit

Post by Dave Keenan »

Thanks for that. Cute signature [:@1: :@2: :@3: :@4: :@5: :@6: :@7: :@8: :@9:] BTW. :-)

Yikes. 53-limit.

I noticed the tiny question mark after "current".
You can find both old and new versions of the JI Notation Spreadsheet in this thread:
viewtopic.php?f=4&t=86
It isn't current in the case of the 75th mina. That was originally split and had 47S in one part and 11:23S in the other. Then we unsplit it and got rid of 47S. And now this badness metric is telling us we should instead have ditched 11:23S and kept 47S.

I find it strange that you say "47:48, 0.235615 over 22:23". Shouldn't it be "under", since 47S has a lower badness?

If you wanted to turn this into a full-on review of the extreme precision level (Olympian) comma assignments, and/or a review of the LPEI badness metric, the spreadsheet given here may be useful:
viewtopic.php?p=2636#p2636

But you have made me realise that I was remiss in not including any single-shaft symbols past the half-apotome in that spreadsheet.

There is a degree of circularity in this process, in that the badness measure was justified by matching the maximum number of existing comma assignments, and is then used to suggest changes to the comma assignments. The fact that I didn't include symbols past the half-apotome in that spreadsheet means that I may not have found the best badness metric. Only one existing comma of a complementary pair needs to match the badness metric (i.e. have the lowest badness of all the candidates for that mina). Its complement then gets a free ride.
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