Re: consistent Sagittal 37-Limit
Posted: Thu May 14, 2020 3:44 pm
...I mean the mina and schisma. *facepalm*
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You beat me to it by the skin of your teeth. I had already written:cmloegcmluin wrote: ↑Thu May 14, 2020 3:39 pm Oh, right. I still get the mina and tina confused sometimes, clearly.
cents monzo ratio comma "rank" 36.53357970860179 | -13 15 -1 -3 > 14348907/14049280 1/1715C 26 36.5888086148658 | 22 -12 -1 0 1 0 -1 > 46137344/45172485 11/85C 33 36.35693218545814 | -10 11 1 -1 -2 > 885735/867328 5/847C 34 36.542244404772084 | 20 -5 -2 0 0 -2 > 1048576/1026675 1/4225C 36 36.52933627696398 | 1 -4 1 1 -1 1 > 910/891 455/11C 36 36.5259897559322 | -21 11 0 1 -1 0 0 1 > 23560551/23068672 133/11C 37 36.39991273195046 | 7 -5 1 -2 > 12160/11907 95/49C 38 36.36675894371364 | -6 6 -1 0 0 1 0 0 0 -1 > 9477/9280 13/145C 47 36.568970302424745 | -4 -5 0 0 1 0 0 2 > 3971/3888 3971/1C 49 36.52575261336313 | 16 -13 0 0 0 -1 1 1 > 21168128/20726199 323/13C 49 36.3958168885402 | -23 12 -2 0 0 1 0 0 0 0 1 > 214170723/209715200 403/25C 54 36.42993229295411 | -8 9 0 1 0 0 -1 0 0 0 -1 > 137781/134912 7/527C 55 36.429695150384504 | 29 -15 0 0 1 -1 0 0 0 0 -1 > 5905580032/5782609521 11/403C 55 36.51580850736912 | 20 -10 0 -2 -1 0 0 0 0 0 1 > 32505856/31827411 31/539C 56 36.45875309521124 | 12 -9 -1 0 1 -1 0 0 0 1 > 1306624/1279395 319/65C 58 36.458990237780995 | -25 15 -1 1 0 0 -1 0 0 1 > 2912828121/2852126720 203/85C 58 36.38937629765928 | -14 13 0 1 0 0 0 0 -1 -1 > 11160261/10928128 7/667C 59 36.466364671724484 | 15 -2 -1 0 0 0 0 0 -1 0 -1 > 32768/32085 1/3565C 59 36.50671892301813 | 9 -11 0 0 -1 2 0 0 1 > 1990144/1948617 3887/11C 60 36.38176472114606 | -17 6 0 1 1 -1 0 0 0 0 1 > 1740123/1703936 2387/13C 62 36.469881961561335 | -24 12 1 0 1 0 1 0 0 -1 > 496897335/486539264 935/29C 62 36.49542261655127 | -2 4 -2 0 0 0 0 0 -1 1 > 2349/2300 29/575C 62 36.33008942237331 | 8 -7 0 0 1 0 0 0 1 -1 > 64768/63423 253/29C 63 36.56576865573409 | 24 -15 0 0 -1 1 1 0 -1 > 3707764736/3630273471 221/253C 64 36.49083601109001 | -26 7 0 1 2 0 0 0 0 0 0 1 > 68538393/67108864 31339/1C 66 36.54687033562636 | 5 -3 0 -1 1 0 1 0 0 0 -1 > 5984/5859 187/217C 66 36.359147367200194 | -9 -1 -1 0 1 0 0 0 1 0 1 > 7843/7680 7843/5C 70 36.51043795685604 | -18 8 1 0 1 0 0 0 1 0 -1 > 8299665/8126464 1265/31C 70 36.567854565336546 | 24 -7 0 -1 0 0 0 0 0 -1 0 -1 > 16777216/16426557 1/7511C 73 36.437799045682624 | 14 -4 0 -1 0 -1 1 0 0 0 0 -1 > 278528/272727 17/3367C 74 36.57951194405419 | -27 12 0 0 0 2 0 -1 0 1 > 2604592341/2550136832 4901/19C 74 36.37010546474512 | 16 -9 0 0 0 2 0 -1 0 -1 > 11075584/10845333 169/551C 74 36.328003512771055 | 8 -15 0 1 0 1 1 0 0 0 0 1 > 14653184/14348907 57239/1C 74 36.48101673474406 | -6 4 0 -1 1 0 0 1 0 0 0 -1 > 16929/16576 209/259C 74 36.48972027400201 | 2 5 0 0 1 0 0 -2 0 -1 > 10692/10469 11/10469C 78 36.5038897891344 | -12 13 0 0 -1 0 1 -1 0 0 -1 > 27103491/26537984 17/6479C 78 36.401366666912025 | -9 7 1 0 0 -1 0 0 1 0 0 -1 > 251505/246272 115/481C 78 36.403857591873766 | 26 -10 0 0 0 1 -1 0 -1 0 0 -1 > 872415232/854261883 13/14467C 90 36.3778862349868 | -12 6 0 0 -1 0 -1 0 0 1 0 1 > 782217/765952 1073/187C 94 36.58577773698017 | -11 7 0 0 0 0 1 -1 0 -1 1 > 1152549/1128448 527/551C 96 36.33733023969181 | -18 10 0 0 -1 0 0 0 -1 0 1 1 > 67729203/66322432 1147/253C 102 36.560480131393085 | -16 10 0 0 0 0 -1 0 1 0 1 -1 > 42101937/41222144 713/629C 108
Ah, yes you are right! Cutoff for C to S being 33.382¢ = L/2 = | 13.5 -8.5 >.Dave Keenan wrote: ↑Fri May 15, 2020 3:43 am I think they are all small dieses not commas, so "S" not "C".
Let that be a lesson to you, kids: this is what happens when you decide to manually remove the trailing zeroes when you notice them after getting the tabs just right, instead of just going back and regenerating things the way you want programmatically. See I'm going to have to do it anyway now.95/49S = 12160/11907 is 19-limit and its monzo is [ 7 -5 1 -2 0 0 1 >, but you have its "rank" correct at 5+7+7+19 = 38. So it is still a contender.
Yes, apotome slope was what I was hinting at when calling out the smaller count of fifths for the 95/49S. Here you go.Could you filter to those less than 910/891 and calculate their apotome-slopes.
cents monzo ratio comma limit apotome slope "rank" 36.35693218545814 | -10 11 1 -1 -2 > 885735/867328 5/847S 11 1.2792165082625901 34 36.5259897559322 | -21 11 0 1 -1 0 0 1 > 23560551/23068672 133/11S 19 1.285164788879173 37 36.39991273195046 | 7 -5 1 -2 0 0 0 1 > 12160/11907 95/49S 19 -3.8421863287452656 38 36.36675894371364 | -6 6 -1 0 0 1 0 0 0 -1 > 9477/9280 13/145S 29 -0.3198905655426278 47
I will do this soon.Dave Keenan wrote: ↑Fri May 15, 2020 11:58 am Great work. Thanks. Could you please edit-in the values for the existing primary comma, 47S = 48/47 for comparison above.
Dang! They were wrong.I agree that the absolute value of the 3-exponent is probably a better metric than the slope. But slope is still worth considering. Are you sure those slopes are correct?
apotome_slope = exponent_of_3 - 7 × untempered_size_in_cents/113.685
A comment you made earlier makes me worry that you might have misread this as:
apotome_slope = (exponent_of_3 - 7) × untempered_size_in_cents/113.685
I found one further example! The case of . Other than and , everything seems to be hunky-dory.cmloegcmluin wrote: ↑Fri May 15, 2020 6:25 am I will write a script sometime soon to verify that we don't have any other secondary commas akin to the 455/11S which warrant doing a boundary shift.
| -2 2 1 0 -1 > | 15 -8 -1 > | -12 2 1 1 0 1 > | 1 -4 1 1 -1 1 >And:
| -4 -1 0 2 > | 5 -3 1 -1 -1 1 > | 1 -4 1 1 -1 1 >So what does that mean? I think it means that it was a mistake to make both and valid symbols, since their sum-of-elements values are identical. I think we need to retire one or the other of them, and replace it with an alternative.
I want to get this soon.Can someone please generate a list of the commas that are sum-of-elements or sum-of-subsets for the symbol we are considering redefining . i.e.
+ + +
+ +
+ +
+ +
+ +
+
+
+
+
Did I miss any?
I also want to get this soon.In the above sum-of-subsets, we should also consider the common secondary commas for and .
I'm pretty sure they exist — one for each diacritic — but I can't readily find what they are.
They can be found by subtraction. i.e. take the primary comma for a symbol with a mina diacritic, and subtract the primary comma the same symbol with the mina diacritic removed, and see if you get something other than 455n or 65:77n.
[Continued in next post]George Secor in email, 2-Oct-2007 wrote:
--- Dave Keenan <d.keenan@...> wrote:
> Hi George,
Hi Dave,
Here's my reply to your last long e-mail, which I completed on Friday
and reviewed over the weekend. It's a good thing I took the time to
review it, because after going over all the latest things we said, a
"what-if" idea occurred to me:
<< As I've worked it out, dropping right accents from olympian symbols
will almost always result in the correct maximum-split herculean (which
I'll subsequently refer to as herculean-X) symbol: out of the 116 minas
in the half-apotome, only 5.5 minas do not follow DAFLL [drop accents
for lower limits/levels/whatever] (so few that they could easily be
memorized -- and most of these exceptions could be eliminated fairly
easily; I'll leave the details for later). >>
Rather than leaving the details for later, on Saturday I investigated
exactly what it would take to eliminate *all* of the exceptions and
found that there was no good reason not to do it! (Robert Walker will
be delighted, to say the least!) Thus there are two levels of Sagittal
which we can highly recommend for JI: one with right accents and the
other without. I've attached a new zip file (JI-Nota.zip) to replace
the one I previously sent.
I've gone through this message to update those changes (a search for
the word "update" will find all of these). I was thinking of
reordering parts of it to flow more logically (and editing out the
parts that are repetitive), but that would take a lot more time &
effort, and it's really not worth the trouble. I suggest that you read
the entire message (25 pages!) to get everything in context before
replying to anything.
> At 11:22 PM 4/06/2007, you wrote:
>> Anyway, I now agree that it [Assignment of 49M to (/| ] shouldn't be
> changed.
>
> That's good. If your plan was to do a reductio ad absurdum on
> whatever rules you were using before, then showing that they assigned
> 49M to .)/|\ was a good way to go about it.
>
> And it still seems to me that mina boundaries and DAFLR were causing
> the problem.
After all this time I don't even remember what the "problem" was, and
as much as I'm tempted to want to go back and figure it out in order to
argue that mina boundaries probably weren't causing the "problem", I'm
biting my tongue and forcing myself to forget it and go on, because
there's no point in pursuing that any further.
In the rest of this discussion I'll continue to refer to minas, because
I still think that they're the solution to, rather than the cause of,
problems of notating the less popular commas. (I'll be making a case
for this as I go along.) For one thing, when I needed to identify
those regions where there are only very unpopular commas, I found it
very useful to use mina boundaries in order to locate them. Therefore,
I'll be referring to them by number of minas.
>> This isn't merely about detail, but rather that it's an example that
>> points up the question concerning how far we should go in defining
>> commas. I was making the point that, just because it's *possible* to
>> define a very unpopular and complex comma (rank 322, complexity 36.780)
>> with a relatively complicated symbol, I don't think that's a sufficient
>> reason to do it, since the comma could always be notated in a secondary
>> role using a simpler symbol.
>
> Agreed. But IF there are other commas less popular that have a symbol
> OR there is not much of a gap between it and the previous more
> popular comma that has a symbol THEN it should get a symbol if
> possible.
That first condition isn't a good reason for requiring that a symbol be
assigned. Since we've already agreed that *all* minas must have at
least one symbol assigned (so as not to have half-cent gaps in the
notation), there will be situations where the only possible assignments
will be very unpopular and/or high-prime-limit commas, and for many of
these it will not even be clear which of several competing commas
should be assigned.
Here's my list of the half-dozen minas (in order of size) that are
regions of lowest popularity. For each I give the simplest symbol,
followed by several comma-candidates for defining that symbol. For
each candidate I give (in parentheses) the following information:
popularity rank, prime limit (if high), and weighted complexity. I
also indicate the ones that are SoCA or alternate SoCA. (If you need
more data, such as distance from SoCA and alt. SoCA of a comma, you can
consult NotDeriv.xlsx, by which I determined these figures.)
)|. at 6 minas: 23:43s (393 pop. rank, 43 limit, 69.193 cplx.) vs.
29:61s (509 pop. rank, >47 limit, 90.852 cplx.) vs. 35:187s (677 pop.
rank, 40.039 cplx.) vs. 13:625s (705 pop. rank, 52.842 cplx.) vs.
19:4375s (Alt. SoCA, 1618 pop. rank, 103.390 cplx.)
/|.. at 42 minas: 19:73C (396 pop. rank, >47 limit, 93.365 cplx.) vs.
19:169C (454 pop. rank, 54.060 cplx.) vs. 625:2401C (662 pop. rank,
49.167 cplx.) vs. 19:4375 (674 pop. rank, 82.861 cplx.) vs. 253C (763
pop. rank, 47.634 cplx.)
)/|. at 50 minas: 5:187C (384 pop. rank, 39.899 cplx.) vs. 420175C (537
pop. rank, 108.230 cplx.) vs. 65:77C (552 pop. rank, 42.968 cplx.) vs.
13:125C (782 pop. rank, 39.302 cplx.)
(|. at 67 minas: 42875C (344 pop. rank, 79.301 cplx.) vs. 17:37C (411
pop. rank, 37 limit, 61.170 cplx.) vs. 4235C (426 pop. rank, 62.782
cplx.) vs. 23:125C (630 pop. rank, 53.241 cplx.)
'//|. at 91 minas: 499S (546 pop. rank, >47 limit, 598.932 cplx.) vs.
17:49S (562 pop. rank, 37.911 cplx.)
/|).. at 98 minas: 83M (289 pop. rank, >47 limit, 99.815 cplx.) vs.
5:187M (438 pop. rank, 45.286 cplx.)vs. 49:85M (650 pop. rank, 36.697
cplx.) vs. 11:325M (SoCA, not in pop. lists, 47.958 cplx.)
Of these, the 91st mina, with '//|. as symbol, is the one with the
least popular most-popular comma, 499S (I hope you understood that!).
Considering its >47 prime limit and huge complexity, 17:49S would seem
to be a better choice. Whichever is assigned, the conclusion follows
that it would be mandatory to assign all commas with better than 556 or
562 popularity rank. This amounts to no cutoff point at all, because
it would include so many commas as to require that nearly all possible
symbols be assigned commas.
Consideration of how frequently the gaps of unassigned commas occur is
a much better criterion.
> So the question is "How big do we let the gaps get (or the sum total
> of gaps) before we stop". Lets just see if something naturally
> suggests itself. I think I'd want to stop if we got to the point
> where there were as many unsymbolised commas as symbolised. So if
> we're assigning 150 or 200 symbols then we wouldn't want to look
> further than about the 300 or 400 most popular commas. Except maybe
> for a few Geneisms.
>
> And for "popular" above you can substitute "simple" if you prefer.
>
> A few that have occurred to me in the past as natural stopping places
> are:
> Rank
> 283 just includes 455n
> 346 just includes 65:77n
> 454 just includes 7:121C
I found your stopping places to be overly generous; the gaps start
occurring way before that (see below).
>> I've come to this conclusion as a result of the following:
>> 1) 25:49M (rank 62, complexity 27.332) *requires* '/|\ in a secondary
>> role, and
>
> Yes I see that is so, because it is only 0.02 cents from 55M.
>
>> 2) We've agreed to exclude 13:17M (rank 117, complexity 31.869) from a
>> separate symbol definition because it's closer to 1/2-apotome than
>> )/|\, which is defined as 5:49M.
>
> Yes I see the denial of a symbol to 13:17M as the result of a (new)
> rule whereby accents cannot be used where they cause a core to cross
> the half-apotome (or M/L diesis) boundary, except in the single
> special case of )/|\'
I chose )/|\. as the simplest symbol for 595M (or 17:37M, if you prefer
that one; it's not very clear which comma is simpler) over '(/|. and
'/|\'', the only other possibilities. Its apotome complement would
then have to be )/|\'' for 595L (or 17:37L), as opposed to .|\)' or
.(|).. . (Another reason for choosing .)/|\ is that this promotes
DAFLL.) Since this double-right-accented symbol crosses the M/L
boundary, I would restate your rule in a way that would not require any
exceptions: "A left-accent cannot be used where it causes a core to
cross the half-apotome (or M/L diesis) boundary."
> At least 13:17S gets to be symbolised by (|' along with 29S.
Yes, and smart defaults would easily distinguish 13:17M from 5:49M.
>> If we're excluding the above from symbol assignments, then why bother
>> to assign significantly less popular, more complex commas with more
>> complicated symbols?
>
> But those two presumably just result in small gaps in the list.
Yes, with 13:17M we're only out to pop. rank 117. (At this point, I'm
repeating 3 paragraphs from my message of 31 July, because it directly
addresses your foregoing remarks concerning gaps.) However, I found
that the gaps are no longer small nor infrequent once you pass rank 133
(where the next 3 out of 4 commas are >23 limit). After that point, I
ended up with more gaps than assignments, even though there were still
plenty of regions [i.e., minas] where commas still needed to be
assigned. Above rank 150, no more than 20% of the commas were assigned
to symbols. By this point the weighted complexity varies all over the
place (e.g., consider these two 11-limit commas: 25:77S is pop. rank
143 & wc 29.162, while 385C is rank 144 & wc 60.529). In giving
priority to one obscure comma over another I found it very difficult to
attach much significance to differences in pop. rankings.
The less popular comma-symbol assignments therefore seem a bit
arbitrary, which is not a good thing if boundaries between symbols are
to be based on the commas assigned. I concluded, therefore, that it
would be safer (and much more straightforward) to stick with strict
233-EDA boundaries, which would guarantee that the yet-unassigned
regions would be treated equitably.
While I did find it necessary to split some minas, there were not very
many (only 6 up to 1/2-apotome in olympian, where there were two fairly
popular commas within the same mina boundaries). For super-olympian, I
split 38 more minas, in most cases to notate either the next most
popular commas (through rank 133) or as many remaining 7-limit commas
as possible.
Mina 6 is an excellent example of a region in which assignment of )|.
to a comma is not at all clear. Popularity, weighted complexity, prime
limit, and proximity to SoCA of the symbol each suggest different
orders of preference, so it's largely a judgment call.. I settled on
19:4375s for two reasons: it's the alternate SoCA, and it's only 0.055c
from the midpoint of this mina (hence is a good nominal value for the
symbol).
To summarize, once popularity drops beyond a certain point in the list,
it becomes much more important to fill out the remaining gaps by comma
*size* than by popularity. Using minas to set symbol boundaries has
two distinct advantages:
1) The boundaries are much more objective in that they do not depend on
the (sometimes unclear and therefore somewhat arbitrary) choice of
comma to define a given symbol; and
2) It's a straightforward (objective) way to determine how many (and
where) symbols remain to be defined (for the olympian level, i.e., the
finest one that will be required for most computer-based applications).
Super-olympian is a superset of olympian that distinguishes a few dozen
more (very obscure) commas with additional symbols (by splitting minas
halfway between the symbol definitions; these are held in reserve, in
the event someone requires separate symbols for some of these.
>> Who will ever use them?
>
> Probably only Gene, but who can say?
>
>> I would assign relatively unpopular, complex commas to symbols only if:
>> 1) They're 7-limit, or
>> 2) They're defined as a higher-prime-limit comma (generally as SoFA) so
>> that they can be used in a secondary role by a 7-limit comma, in order
>> to distinguish that comma from another 7-limit comma having the same
>> number of minas.
>
> I had trouble with the referents of your two "they"s above (in the
> two "they're"s. I'll assume the first refers to the commas and the
> second to the symbols.
Yes (sorry!).
> I find the phrase "having the same number of minas" to be redundant
> here. Couldn't it simply be "to distinguish that comma from another
> 7-limit comma <period>".
Yes, you're right.
> Now I certainly think that is a good thing to do. And a good way to
> approach it. But I wouldn't rule out assigning a symbol to a comma
> where the symbol has neither primary nor secondary 7-limit role, if
> the (non-7-limit) comma was more popular than the least popular of
> the other primary roles.
As I pointed out above, that would justify assigning anything with a
popularity rank better than 556 or 562 to some complicated symbol for
no reason other than that we have a bunch of obsure symbols that would
otherwise go unused. For example, do we really need '|(.. for 13:55k
(SoCA, pop. rank 385) and .)|(.. for 13:35k (pop. rank 347), when there
are many *more popular* commas that are unassigned? I expect that
there would be instances where, because of an unfavorable size order,
this might result in assignment of a more complicated symbol to a much
more popular comma in order that a simple symbol could be reassigned to
a highly unpopular comma (the alternative would be to dig deeper down
the popularity list to come up with something for the more complicated
symbol).
>> I looked at many of these less-popular 7-limit commas last night, and I
>> was delighted when I observed that most of them will get symbols that
>> are only moderately complicated, e.g.:
>>
>> 625S ~|\..
>> 3125C '|)'
>> 15625k '|('
>> 15625C .(|'
>> 49:625k ~|..
>>
>> Some of the above are in secondary roles.
As it turns out, all of above commas have been assigned the above
symbols in *primary* roles, but there are other, more complicated
7-limit ones that get secondary roles. For example, in the 27th mina
78125C is a secondary role for .~|(' (primary role of 11:23C), while
)~|''. is defined as 1715C.
>> It's not required that each
>> one be defined in a primary role, only that 7-limit commas be capable
>> of being notated uniquely up to a certain exponent level, e.g., all
>> 7-limit commas having the sum of the 5 and 7 exponents equal to 6 (or
>> possibly 7) or less be distinguishable from one another by different
>> symbols.
>
> This is a _great_ thing to do. I've argued for it before. To see how
> far you can push notation of the 5^n*7^m plane before it starts
> getting too frayed around the edges. Clearly higher powers of 5 are
> of more interest than higher powers of 7 (in about a 7:5 ratio I'd
> guess).
>
> We should publish this as a separate list. This would be "the maximal
> Sagittal 7-limit notation". No precision level need be mentioned. No
> boundaries need be defined. If you need to notate a 7-limit comma
> that doesn't have a symbol you just reuse the symbol whose 7-limit
> default is the closest.
FYI, the following are the simplest (most popular, or most notable)
pairs of commas *not distinguished* from one another within the 7 limit
in super-olympian:
1715k (2^25:3^9*5*7^3), 54.537 cplx, 206 pop. rank (vs. 78125k 151 pop. rank)
7:3125M (7:5^5), 68.457 cplx., 581 pop. rank (vs. 5:49M, 19 pop. rank)
40353607s (7^9), 188.689 cplx., 611 pop. rank (vs. 5s, 4 pop. rank)
7s (3^14*7:2^25), 224.798 cplx., no pop. rank (vs. 49:15625s, 1446 pop. rank)
We could distinguish 7s using .|(, which is SoCA, if you think anyone
might want that. (The symbol is certainly simple enough.) I've seen
the 7-schisma mentioned once or twice before, and this would
distinguish it from another 7-limit comma, 49:15625s in a theoretical
discussion. It's not really needed for notating 7/4 in JI until you
take F# as 1/1, and even then it's needed only for Fb'!(, which is the
*2nd alternate* spelling.