## consistent Sagittal 37-Limit

Dave Keenan
Posts: 1024
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: consistent Sagittal 37-Limit

[Continued from previous post]

George Secor, 2-Oct-2007 wrote: Dave wrote:
> However there could well be other 7-limit Sagittal notations where
> the range of exponents of 5 and 7 are reduced and a lot of accents
> are dropped.
>
> This is how I think we should be viewing JI notation, in the way its
> own proponents view it, not as "precision levels" but as prime limits
> and prime exponent limits. So I'd like the best part of DAFLR, the
> DAFL part (which was always a good idea) to be applied not to
> resolutions but to limits. Drop accents for lower limits (both prime
> and exponent) DAFLL or perhaps more generally drop accents for
> smaller subsets DAFSS.

You can call it whatever you like. I would suggest either DAFS, drop
accents for simplification, or DAFLL (drop accents for lower levels (of
resolution, limits, simplicity, or whatever). Thus we can still refer
to athenian, promethean, herculean, etc. as "levels" (of whatever).
I'll give reasons for keeping this number of levels (with these names)
below.

A DAFLL scheme works best when monotonic symbol cores are maximized.
Since two symbols within the same mina are virtually guaranteed to have
different symbol cores, DAFLL will work best if mina-splitting is
minimized. This is exactly what I've done in olympian: strict 233-EDA,
with only 6 split minas <1/2 apotome. Thus olympian is much better for
DAFLL than super-olympian. I would reserve super-olympian only for
emergencies, where someone absolutely insists on separate symbols for
distinguishing between two complicated commas.

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 (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). Dropping
all accents from either olympian or herculean-X symbols will in most
instances result in the proper athenian (if the symbol set is allowed
to change according to the prime limit) or promethean symbol (where
some core changes are inevitable -- the price you pay for making a
schismatic distinction without accents).

Update: As I said at the beginning, I've redone this so that there are
now no exceptions.

Therefore, I recommend that the two principal JI levels should be
olympian (allowing right accents) and herculean-X (no right accents).

These are the 6 minas that I split in olympian:

36th mina:
11:49C (89:99), 30.054 cplx, 77 pop. rank as ~~|
11:25C (99:100), 25.307 cplx, 75 pop. rank as |~''

51st mina:
5:19C (40960:41553), 25.550 cplx, 53 pop. rank as )/|
13C (6561:6656), 23.054 cplx, 36 pop. rank as .|).

72nd mina:
25:49S (49:50), 24.049 cplx, 32 pop. rank as '(|
31S (243:248), 38.901 cplx, 73 pop. rank as ~|).

78th mina:
7:13S (1664:1701), 30.237 cplx, 31 pop. rank as (|(..
5:23S (45:46), 28.228 cplx, 78 pop. rank as /|~

105th mina:
625M (16384:16875), 36.078 cplx, 29 pop. rank as '/|)'
5:11M (10935:11264), 26.115 cplx, 56 pop. rank as ./|\

113th mina:
55M (1048576:1082565), 26.875 cplx, 45 pop. rank as '/|\
31M (31:32), 37.311 cplx, 48 pop. rank as (/|'

Most (i.e., 5 out of 6) of the boundaries between these split-mina
comma-pairs coincide with herculean-X symbol boundaries, so that DAFLL
is maintained in spite of the olympian splitting.

I didn't split the 36th mina for two reasons: 1) it made more sense to
give 11:25C the ~~| symbol, rather than |~, in herculean, because it's
much closer in size; 2) the boundary placement is less irregular.

Update: Here are the changes I made to remove the DAFLL exceptions. I
unsplit one olympian mina in the above list and split two others, so
now olympian has 7 split minas.

14th mina:
I added the symbol.~| 85k, (30.841 cplx., 108 pop. rank) to the
herculean-X symbol set.

36th mina:
This is no longer split in olympian (only in super-olympian), so that
the boundary between the |~ and ~~| cores coincides in olympian and
herculean-X.

37th mina:
I changed the 37th-mina olympian symbol from .)|~ 95C (rank 91, SoCA)
to ~~|' (rank 89).

47th mina:
I added the symbol .)/| 19:25C (34.869 cplx., 122 pop. rank) to the
herculean-X symbol set.

49th mina:
In olympian I split this mina between )/|.. 5:77C (rank 86) and '/|'
4375C (rank 249) and moved the herculean-X boundary between '/| and )/|
to coincide with the split.

75th mina:
In olympian I split this mina between ~|)'' 47S (rank 72) and .(|(.
11:23S (rank 113) and moved the herculean-X boundary between ~|) and
.(|( to coincide with the split.

Athenian, promethean, and super-olympian remain the same as before. (I
included a new Sag_ji1.par file in the update package, because I found
a mistake for one of the athenian-level boundaries.)

>> Is there any need to clutter the notation with a bunch of ugly symbols
>> representing >7-limit commas that nobody's ever going to use? IMO it
>> would only add to the freak-out factor. (I believe something you said
>> below indicates that you agree with this.)
>
> Well we'd certainly have to do the maximal 11-limit notation and the
> maximal 13-limit. These might have exponent limits of 2 for the
> primes above 7. Then we might want to do 17, 19, 23 limits with
> exponent limits of 1. And we have to notate harmonics up to 31 to
> keep Ben Johnston fans happy.

I think that olympian (with 6 split minas -- now updated to 7 split
minas) should work just fine, but judge for yourself. The most popular
commas that are *not distinguished* are:

7:13C, pop. rank 55 (uses 17C symbol) -- used only for alternate
spelling of 13/7
A super-olympian symbol was assigned.
25:49M, pop. rank 62 (uses 55M symbol) -- no super-olympian symbol is
possible
7:23M, pop. rank 64 (uses 5:13M symbol) -- no super-olympian symbol is
possible
5:49C, pop. rank 65 (uses 7:11C symbol) -- ')|)' is possible, but
rather complicated
Used only for 2nd alternate spelling of 49/40, so super-olympian
symbol not justified IMO.
11:35C, pop rank 68 (uses 11:35k symbol) -- used only for alternate
spelling of 35/22
A super-olympian symbol was assigned.
Update: added 11:25C, pop rank 75 (uses 11:49C symbol) -- used for
preferred spelling of 25/22
A super-olympian symbol was assigned.
43C, pop. rank 79 (uses 7:125C symbol) -- used for preferred spelling
of 43/32
A super-olympian symbol was assigned.
121k, pop. rank 80 (uses 1225k symbol) -- used for preferred spelling
of 121/64
A super-olympian symbol was assigned.
5:11C, pop. rank 81 (uses 7:25C symbol) -- used only for 2nd alternate
spelling of 11/10
A super-olympian symbol is available, which also covers 49:625C.
77s, pop. rank 83 (uses 91s symbol) -- used only for alternate spelling
of 77/64
A super-olympian symbol was assigned.
13:35C, pop. rank 87 (uses 23C symbol) -- used for preferred spelling
of 35/26
A super-olympian symbol was assigned.
37S, pop. rank 88 (uses 25S symbol) -- '~|\'' is possible, but very
complicated
Used only for alternate spelling of 37/32, so super-olympian
symbol not justified IMO.
Update: added 95C, pop rank 91 (uses 11:31C symbol) -- used for
preferred spelling of 95/48
A super-olympian symbol was assigned.
Update: removed 11:31C, pop. rank 89 (uses 95C symbol) -- used for
preferred spelling of 31/22
A super-olympian symbol was assigned.
121C, pop. rank 92 (uses 1225C symbol) -- used only for alternate
spelling of 121/64
A super-olympian symbol was assigned.
5:343C, pop. rank 93 (uses 3125C symbol) -- used for preferred spelling
of 343/320
A super-olympian symbol was assigned.
5:29M, pop. rank 95 (uses 31M symbol) -- no super-olympian symbol is
possible

> Or the exponent limit might be expressed as a weighted sum of the
> exponents of all the primes in the limit. The weights might be 1/p^2.
>
>>> Or you could say now: One symbol per tina wherever possible.
>>
>> I don't really want to say that, for reasons given above.
>
> Fine. Forget tinas.
>
>> But I would
>> want every mina to have at least one symbol, at most two, and two only
>> if comma popularity (or 7-limit uniqueness, as described above)
>> warrants it.
>
> Why would minas have anything to do with a maximal 7-limit notation?
> Any more than tinas would? If either of these are used to ensure
> consistency of flag and accent arithmetic that's OK.

Some folks like to have a convenient unit of measure with which they
won't experience frequent rounding errors. The fact that minas are
divisible by 12 is a plus. Do you really think that the JI folks are
going to object to this?

> But I see no
> need for either of them in setting boundaries. I'd find that decidedly
> un-JI.

There's a problem with using the commas to set the boundaries when we
look at those regions that have only very unpopular commas (see my list
of the half-dozen least-popular minas), because the choice of which
comma should be assigned to a symbol is often not clear.

> For flag accent arithmetic consistency purposes, tinas appear
> superior to minas to me at present since they can distinguish more
> symbols.

Unless we were to define an additional type of accent, tinas cannot
give you good DAFLL (on account of poor core monotonicity: two symbols
with SoCA closer than 0.4c will hardly ever have the same core), nor do
they strictly maintain arithmetic consistency if not all tinas have
separate symbols (i.e., allowing the same symbol to cover more than two
or more adjacent tinas is not real consistency).

>>> However,
>>> I think superolympian is really about giving a symbol to the most
>>> popular comma that doesn't already have a symbol, wherever possible.
>>
>> Yes.
>
> Good to agree on that.

>>> But our main disagreement seems to be about the meaning of "possible"
>>> here. It really means "allowable". So the disagreement seems to be
>>> about what we are _allowed_ to do to accomodate commas.
>>
>> I suspect that whatever difference of opinion we have is due more to a
>> lack of clarification on the specifics than disagreement in general
>> principles.
>
> That's good. But precision levels and mina boundaries seem to be
> sticking points, and the principle of using EDAs to determine JI
> notations seems to me to be a fairly general one that we are now
> disagreeing on.

It seems to me that we were agreeing on it being a good principle for a
very long time (you were the one who came up with the idea of setting
boundaries according to an EDA), but now you're the one who's
disagreeing with it. Boundaries based on an EDA (in this case, minas)
enable us to guarantee that, within specified prime/exponent limits,
adding (or subtracting) a comma of n minas to (or from) a comma of m
minas will result in a comma of m+n (or m-n) minas. With unequal
boundaries, a computation involving one or more commas located near a
symbol boundary could result in unpredictable arithmetical
inconsistencies.

>>> But the simple/complex distinction I'm referring to here is not the
>>> number of symbols in the precision level, but the simplicity or
>>> popularity of the _comma_.
>>
>> Yes, this makes sense.
>
> That's good too.

>>> There is also a sense in which all precision levels lower than the
>>> ultimate one are actually _more_ complex than the ultimate one. At
>>> all lower levels you have to worry about symbol boundaries or which
>>> comma a symbol refers to depending on the nominal it is with (smart
>>> defaults). At the ultimate level, every comma just has its own
>>> symbol.
>>
>> But an ultimate precision level without boundaries would require an
>> infinite number of symbols.
>
> Or merely the ability to always add more as we need them, since the
> list of all commas ever used for notation by a human being is at any
> given time finite.

Yes, the number of possible commas is finite but indefinite, whereas
the number of Sagittal symbols is finite and definite, so we don't
always have the ability to add more -- unless we add more symbol
elements. With better-than-1/2-cent resolution there doesn't seem to
be any reason to do that, because the difference between a symbol's
primary role and any possible secondary role is going to be less than a
mina (and usually significantly less). Olympian should fill the bill
(but we have super-olympian up our sleeve, just in case).

> But this was a hypothetical "ultimate level"
> anyway.

Yes, but we need to release something real, not hypothetical.

>> Since we have only a finite number of
>> symbols, boundaries are still required in determining how secondary
>> roles are assigned. Otherwise, we would be unable to supply a symbol
>> for every possible comma.
>
> In a sense, yes. But I would just reuse the symbol with the nearest
> primary role. So in a sense the boundaries are halfway between
> primary roles, but there are no boundaries in the sense that no extra
> information is required apart from a list of symbols and their
> primary commas.

As I stated above, the problem is that we must then be very careful in
our choice of primary commas, because it's going to affect the location
of the symbol boundaries. The more symbols in the notation, the more
complicated (unpopular, obscure) the commas eligible for the
definitions. The more complicated the commas eligible for the
definitions, the less clear it is which of those eligible commas should
be chosen. Mina boundaries solve that problem by making boundaries
independent of comma choices (except where minas are split).

If minas are split, then they're split in order to notate some worthy
comma (which by definition excludes highly unpopular or complex commas)
from a more popular comma with the same mina boundaries. It's a simple
matter to put the sub-mina boundary halfway between these two commas,
assuming that there are suitable symbols available.

Update: two minas are now split in order to have DAFLL from olympian to
herculean-X (a good reason). One of these involves a rather unpopular
comma 4375C (pop. rank 249), but it's a good choice for '/|', since
it's the alternate SoCA for that symbol (which is only moderately
complicated) in comparison to the less-popular 455C (SoCA, pop. rank
565).

> And if you choose to use a smaller subset of those symbols (e.g. for
> a lower limit, possibly with dropped accents) then if necessary you
> reuse the symbol with the nearest _default_ comma for that subset.
> Still no explicit boundaries need be defined.

I think that we would want to maintain a contiguous size range for each
symbol within each symbol subset. Since that can't be guaranteed by
simply dropping accents, it would be advisable to define one or more
symbol subsets. Furthermore, if we wish to maximize DAFLL, then
explicit boundaries (based on the larger symbol set) would seem to be
necessary.

I've already taken all of this into account with the sub-olympian
levels (or symbol sets, if that's what you prefer to call them) given
in the JI-Nota.xlsx and NotDeriv.xlsx files.

>>> For a JI-ist this is as simple as it could be. It's what
>>> Robert Walker wanted.
>>
>> Okay, now that you've made your point, it's time for a reality
> check.
>>
>> Simplicity is a large number of symbols for a large number of
> commas?
>
> Did I say that? I think my point was, simplicity is a one to one
> relationship between commas and symbols for all the commas you choose
> to use, with simple symbols for simple commas.

Okay, and we can achieve that using either midpoint or mina boundaries.

>> Tell that to Aaron Johnson, who said (tuning msg. #70065):
>>
>>> My one
>>> critique would be it's inelegance (meaning the literal sense--lacking
>>> simplicity--it's a lot of symbols to memorize, and it gets bloated
>>> fast---a lot to ask of players who would be fish out of water to begin
>>> with), but that reflects the real-world situation of any JI endeavor
>>> that would 'leave Kansas' --- but we should remember that there are
>>> very few reasons to modulate far in JI, anyway....
>>
>> After you told him about smart defaults used with smaller symbol sets,
>> his reaction (in tuning msg. #70102) was:
>>
>> << Great...this makes sense, and I would guess, holds the future
>> sink-or-swim status of the adoption of Sagittal. >>
>
> That's Aaron. We need to keep him happy too, but he's not a JI
> fundamentalist or infinite precision guy.

Nevertheless, we should keep our options open, because we can't assume
that all JI-ists are going to react alike.

>> And now you're telling me (below) that we don't need lower-resolution
>> JI?
>
> Not lower resolution, but lower limit. With DAFLL.

This thing about lower limit(s) is rather difficult to pin down, since
there are so many combinations of prime and prime exponent limits. How
many categories of limits did you have in mind?

With Sagittal there's a natural progression from complexity to greater
simplicity (DAFLL) that results from dropping right accents (to get
herculean-X), then all accents (to get promethean), then disregarding
the 5-schisma (to get athenian). You could call it lower limit or
lower complexity, but since we seem to have made it a requirement not
to have overly large gaps between symbols in any given set (or limit,
level, or whatever you want to call it), then we are in fact
guaranteeing a specific resolution (expressible in cents) for that
symbol set.

[Dave:]
>>> This reminds me of the idea that the infinite God must be the
>>> simplest "being" this side of nothing. Because if he had any
>>> attributes he would be less than perfect.
>>
[George:]
>> Heh, heh :-} -- that strikes me as quite funny! I would think that an
>> infinite God lacking such positive attributes as love, mercy, justice,
>> power, intellect, creativity, goodness, truth, and beauty would be less
>> than perfect. Such a "God" reminds me of an idol made by human hands:
>> one which cannot see, hear, or speak (ref. Psalm 115:4-8); except that
>> this particular Idol would be too big to be made by any human agency,
>> other than the imagination. And since belief in such a "God" would
>> fail to account for the presence of such attributes in humans, might we
>> then need one or more lesser gods, each with one or more of these
>> attributes? Or did this "God" earlier have these attributes, but is
>> now in the process of shedding them on the way to a simpler state of
>> perfection? I have more thoughts on this, but we should leave it for
>> another time.
>
> Yes. I'm almost sorry I included this. Because I now must bite my
> tongue and not respond to any of this. Except to make a correction to
> my earlier statement which hopefully you will find no reason to
> respond to either.
>
> I used the wrong terminology. This paradoxical concept is more like
> what Christian mystics call "The Godhead". Their "God" is still as
> you describe, having that list of attributes we most admire in human
> beings.

I'll bite my tongue and not respond to that, either.

>> In the meantime, you can see that some of Smullyan's way
>> of thinking has rubbed off on me.
>
> Indeed!
>
>> BTW, I've finished reading both Smullyan books (which have convinced me
>> that we should indeed resume our discussion beginning with those things
>> that we both agree on). My daughter is now reading Smullyan's memoirs
>> (very entertaining, BTW), so I'll send that off to you after she's done
>> with it.
>
> There's no hurry. I have too many other things to do, and a stack of
> interesting books to read. I'm holding off G&T until we have finished
> Sagittal JI notations. I just finished "Breaking the spell: Religion
> as a natural phenomenon" by Daniel Dennett. Although I thought it was
> very good, I don't think it would be of much interest to you.
>
> However there was one section in there, a little over a page, that
> explained something I wanted to explain to you. It's about why there
> is no necessary connection between scientific or philosphical
> materialism and materialism in the everyday sense. i.e. between
> materialism in the sense of "the universe is made of only one kind of
> stuff" (as opposed to dualism where matter and spirit are separate
> realms) and materialism in the sense of "whoever dies with the most
> toys wins" or "morality is whatever I can get away with". I must type
> this in and send it sometime. Of course he supports the former but
> abhors the latter. He explains it way better than I could.

Yes, I agree that would be good for me to read.

> Now I've decided, if you're willing to have your (and G&T's) ideas
> about neo-darwinian evolution challenged, in the nicest way possible,
> then you need to read Dennett's earlier book "Darwin's Dangerous
> Idea".

I already have the book, since you recommended I get it a while back,
and I put it on my "must read sometime" pile and didn't look inside
until just recently. But just before that I saw a new release, _God,
the Failed Hypothesis_ by physicist Victor J. Stenger, in a bookstore.
I thumbed through it and decided I should get it and read it (which I
did). It contains a broad range of arguments (scientific,
philosophical, moral, religious, etc.) addressing some of the points
made in G&T (although, in some instances, in not as much detail as I
would have liked). Some of his arguments have gaping holes in them
(hopefully due to ignorance and not deliberate disregard of facts), but
I didn't find that a good reason to dismiss the entire book, because he
had some very good things to say.

I've come to see that facts that are deemed important from one point of
view are may not be mentioned by the other side, because they're not
regarded as particularly important. I'd include G&T's failure to
mention "junk" DNA in this -- and I now have a plausible two-part
answer for this point, which I'll leave till later. VJS brings up
quite a few of the points you've mentioned, so I think that I now have
a much better idea of how you might have gotten to where you're at.

I'd encourage you to read G&T all the way through, making notes as you
go along, as I did with VJS, and am now doing with Dennett's book,
which I started about 2 weeks ago. I must say it's absolutely
brilliant -- and indeed very dangerous!. After 3 chapters Dennett
thinks that he already presented enough to thoroughly demolish John
Locke's argument with Darwin's "universal acid"; but I must disagree!
I'm hoping that he'll later address a couple of things I think he might
have overlooked, but I won't be completely surprised if he doesn't.
It's too easy to misjudge the significance of some things when there
are so many areas of knowledge involved in attempting to answer the
ultimate questions of existence and meaning. We can't be experts in
everything, but we can consider the various bits & pieces of data that
reputed authorities having differing viewpoints believe are relevant,
and then evaluate their insights and opinions as best we can.

I look forward to discussing some of this with you -- but not till
after I've finished the book (and perhaps then some). Our future
discussion could turn into a really fascinating dialogue that could be
adapted for use in an epilogue to the mythology, in which Hermes
attempts to convince Dave that polytheism is more reasonable (!) than
atheism. Here's a part of the plot you'll undoubtedly chuckle over.
Hermes reveals a hitherto unknown fact about the origin of Darwinism:
Biological evolution was originally Zeus's idea! His plan was to
retaliate against (the monopolistic) God by having one of the Muses
plant the idea in Darwin's mind. As you can see, the strategy worked
extremely well -- too well, in fact, because now the problem is to make
it more reasonable to believe that the olympian gods are real. (Hermes
will do his very best to convince you!

>>> So I say, rather than forgetting about super-olympian, lets forget
>>> about precision levels and boundaries entirely,
>>
>> Sorry, but the part about not needing boundaries doesn't follow
>> logically, because:
>> 1) Since, we don't have an infinite number of symbols,
>> 2) There will always be the possibility that someone will need to
>> notate a comma not defined by one of our symbols; therefore,
>> 3) We'll have to re-use symbols in secondary roles for those undefined
>> commas;
>> 4) Therefore we will need to have boundaries for the purpose of
>> determining which secondary roles go with which symbols.
>
> Yes. I'm sorry I didn't explain that sooner. No boundaries = pick the
> closest symbol -- via either its primary role or its current default.

That's assuming that one has determined at what point to stop assigning
symbols. Gaps in the popularity list don't fill all the gaps in the
size continuum.

>> Furthermore, there are several problems with forgetting about precision
>> levels:
>> 1) We already said in our Xenharmonikon paper that Sagittal has various
>> symbol sets corresponding to several levels of precision (search for
>> the word "precision" in the paper), in line with our philosophy of
>> providing options for users, according to their needs and
> preferences.
>
> The relevant section is:
> "In accordance with this, we have also defined symbol sets to be used
> for several levels of precision in just intonation. While
> medium-precision (athenian-level) JI, which uses twelve pairs of
> single-shaft symbols (without any accent marks), should be adequate
> for most purposes, we have also defined high-precision (in versions
> with or without left accents) and extreme-precision (utilizing both
> left and right accents) symbol sets for those with more exacting
> requirements, in which case a trade-off of simplicity for precision
> must be taken into consideration."
>
> Can I get away with "A foolish consistency is the hobgoblin of little
> minds"? Seriously, most of that could be satsified by basing
> these on different prime and exponent limits rather than EDAs. So I
> guess I'm not really saying we should forget precision levels, but
> forget basing them on EDAs.

Actually, I've followed the principles you've been advocating for
herculean-X and promethean, which have only a superficial resemblance
to any sort of EDA. (Continued below.)

>> 2) The needs of some JI composers will be fairly simple, and if we tell
>> them that accented symbols and/or non-athenian cores are required for
>> some of their accidentals, they may think that this is unnecessarily
>> complicated and then decide to look elsewhere.
>
> So define sets for limits, or even just for common JI scales like
> harmonic series, diamonds, cross-product sets. Just like we do for
> ETs.

How would that be done? By mapping the tones to some ET and then using
the notation for that ET? Conventional (tuning-list) wisdom would say
"yes" (as we also suggested in the Sagittal paper in footnote 18, page
23), but lately it seems to me that you're thinking that JI types would
not take kindly to this approach. Or do we just give the JI folks the
symbols and not tell them how we got them? The problem with this, of
course, is that symbols for some of the simplest commas may change
according to which ET is selected; e.g., if Partch's tonality diamond
is mapped to 58-ET (the smallest ET that's both 11-limit consistent &
11-limit unique), then 7C will be /|, which is IMO an unnecessary
simplification.). Even going to 72-ET will simplify the symbols for
5:11C and 7:11C, so we must ask whether what might be more convenient
today will become a liability in the long run for a composer who
eventually decides to expands his/her tonal horizons.

In going through this discussion of ours, I've become increasingly
convinced that herculean-X should be the default choice for JI, because
it notates virtually all (update: delete "virtually") of the most
popular commas the same way as in olympian, with right accents dropped.

While herculean-X roughly corresponds to 58-EDA (with a bunch of
degrees split), it has enough differences from a strict EDA that you
could say it's based at least as much on identifying the simplest
symbols (no right accents) and assigning them to the simplest commas.

Promethean, with its highly irregular step sizes, has even less
resemblance to 47-EDA (with a few degrees split), but it follows the
same symbol-to-comma principle as herculean-X (but with no accents).

>> 3) We would have to tell Prent Rodgers that ratios of 13 are now
>> properly notated with accented symbols (contrary to what we said in our
>> paper), and he should now use accented symbols in order to conform to
>> our latest guidelines.
>
> No. I agree that dropping accents for smaller subsets (DAFSS) is
> good.

Good!

Dave, please take a close look at herculean-X. It's detailed in
NotDeriv.xlsx, sheet 1, columns G thru I (when cell H2 is set to "X",
for maXimum splitting; the symbols in sheet 2, column O will also
change according to this setting). It gives the same symbol set that
Prent's using for the 15-limit tonality diamond, so we've got him
covered.

>>> and just concentrate
>>> on giving as many unique symbols as possible to the most popular
>>> commas. i.e. getting as far down the popularity/complexity list as
>>> possible with no gaps (of unsymbolised commas).
>>
>> There will be gaps sooner that we would like, starting with 25:49M,
>> occurring more frequently as we go down the popularity list. It's a
>> question of how and where to quit.
>
> Yes indeed. That is the question.
>
> I now agree we shouldn't stop at the first gap, but neither should we
> go past any really big gaps, or a large number of accumulated gaps.

Yes, that's exactly what I was after.

>>> At the same time we
>>> must try to ensure that the most popular/simple commas get the
>>> simplest symbols, and that flag and accent arithmetic is not too
>>
>> Agreed.
>>
>>> We pretty much obeyed this principle in assigning the values to the
>>> cores. I'm saying let's continue it.
>>
>> Yes.
>>
>>> I say, forget DAFLR because there are no LR (lower resolutions) in
>>> JI.
>>
>> No.
>
> Fair enough. How about DAFLL or DAFSS?

How about DAFLL, where LL = lower level (without specifying exactly
what a level is)?

>> Yes, I see what you're driving at, but I don't agree that eliminating
>> levels of precision, DAFLR, and boundaries solves anything. Instead,
>> it creates a problem in that we would now have to agree on some new
>> formula(s) for making assignments in both primary and secondary roles.
>
> Yes. I don't think that is beyond us.

No, no! You were supposed to say, "Golly, that could get pretty
involved, and we're wanting to get this whole thing over with as soon
as we can."

>>> Some formula for trading off pitch-distance against
>>> complexity-distance (between symbol and comma) would be required. And
>>> sometimes a symbol will just have to be eliminated from consideration
>>> entirely. "If I can't have it then no one can", says the simple comma
>>> of the complex symbol, "because it's too close to me and I already
>>> have this simpler symbol that's further away". Or rules to that
>>> effect.
>>
>> Yes, I agree with that, in principle.
>>
>> But I don't think you need a complicated formula
>
> Nor do I.
>
>> -- the mina boundaries
>> will work quite well to determine that. There should be a symbol
>> assigned to a comma within each mina.
>
> Why do we need minas? Why can't we just use SoCA and popularity? I've
> now modified my spreadsheet to include the two different
> interpretations of the single and double right accents. I have
> temporarily used the characters ` and , to allow me to distinguish
> the alternatives from the standard ' and .

Popularity becomes more meaningless the farther we go down the
popularity list, failing us just when we most need it. Minas, on the
other hand, give us objective boundaries.

>> If all commas within a mina are
>> relatively unpopular and complex, then the most popular comma gets the
>> simplest symbol. If there are two worthy commas within a mina, then
>> assign them the two simplest symbols. I doubt that you'll find 3
>> worthy commas within a mina, but if so, then (if possible) assign them
>> the 3 simplest symbols.
>
> I'm starting to feel that you're like the man whose only tool was a
> hammer (mina) so everything looked like a nail.

It is an easy way to nail down the boundary problem.

> Try to imagine that minas don't exist and you can't invent them.
> That's how the world looks to a JI fundamentalist. Now what would you
> do?

... And on the eighth day the gods secretly created minas. And they
saw that this was very good.

Now it came to pass that JI fundamentalists, in search of the finest
notation, began to devise symbols for all manner of commas. After
naming and symbolizing many commas (in order of worthiness), they
lamented that there were gaps in various and sundry places. The most
adventurous sought to fill these gaps with symbols to satisfy their
inner longing to modulate to places where none had ever gone before.

But as they began to face the prospect of a multitude of increasingly
obscure and "unworthy" commas (symbolized by a multitude of complicated
and "ugly" symbols), they began to quarrel with one another over which
(and how many) of these could be characterized as less obscure or
complicated or ugly or unworthy. After not many days it became
painfully evident that they had reached an impasse.

The gods, eagerly waiting for an opportunity to give direction,
intervened to provide the necessary inspiration, whereby minas were
revealed to the JI world. These minas almost magically provided fixed
boundaries for a finite number of reasonably simple symbols, whereby
even the most obscure or seemingly undeserving comma would be notated
by a symbol.

Some in JI-land praised the gods for their magnanimous gift, while
unbelievers attributed the seemingly miraculous orderliness of minas to
some sort of "self-organization" process. The debate goes on, but all
agree that, however they came into being, minas are indeed a god-send.

Oh, yes, there were a few minas which had two relatively popular commas
within their boundaries, but it was a simple matter to subdivide these
with separate symbols, so as to give each the honor to which they were
entitled.

And they all made music happily ever after.

Dave, lest you think that this story is too far-removed from reality,
consider the process by which Harry Partch created his monophonic
fabric. The 14 tones he added to fill in the gaps in the 11-limit
tonality diamond are exactly the number required when the diamond is
mapped onto 41-EDO (with two split degrees). Partch even changed a few
of the non-diamond ratios a couple of times, but he always stuck with
the same total number of tones in roughly the same places.

There's something inherently right about the EDO (or EDA) as a
governing principle, and I think it would be a mistake to abandon that.

>> Put the boundaries (used to define secondary
>> roles) within multiple-symbol minas midway between the values of the
>> commas they represent.
>
> If you can do this _within_ minas, why not _between_ minas?

Dealt with above.

>>> So let's forget minas and tinas and herculean and promethean. Let's
>>> even forget Athenian (It has had its effect in deciding the values of
>>> the cores). Let's forget DAFLR. Let's just give Robert Walker (and
>>> others) what he wanted -- unique symbols for as many commas as
>>> possible.
>>
>> Then how does one notate the rest of the commas -- the ones that
>> haven't been assigned to symbols? You need boundaries of some sort.
>
> True. As I've explained above.

And, as I've explained above, boundaries that depend on one's selection
of commas are too subjective.

>>> You asked recently whether we really had to throw away the EDA
>>> ladders or scaffolding once we had used them to climb up to the
>>> ultimate JI precision level. I now say "Yes. We must".
>>
>> Okay, I've reached ultimate precision and have now become like the Most
>> High. Hang on to your bootstraps! I'm about to kick away the
>> scaffolding!
>>
>> Ooooooooooooooooooooooooh ... <CRASH!> (groan) ;-(
>
> Hee. Hee. Sorry about that.
>
> Try it again now that I've installed some safety nets.

Noooo, I'm toooo skeeeered! ;>O

>> There shouldn't be any missing minas, because:
>> 1) There's no problem in coming up with at least one reasonably simple
>> symbol for each and every mina, to which we can assign the most
>> popular, least complex comma within its boundaries; and
>
> OK. That's good.

Even if the boundaries don't depend on one's choice of comma, it's
still required that a comma definition be selected, because a computer
program will require some particular ratio in order to calculate a
pitch for a symbol. It's advisable that the comma not be too near a
mina boundary, so as not to differ by too great an amount from a
secondary comma near the other boundary.

>> 2) Should we neglect to assign at least one symbol per mina, then we
>> can no longer guarantee a discrepancy not exceeding ~0.5 cent for all
>> commas notated in secondary roles.
>
> Good point.
>
[Dave:]
>>> We could spend a lot of time designing herculean and promethean
>>> systems and we could well find that no one ever used them. Then all
>>> this agony over DAFLR would be for nought.
>>
[George:]
>> Most of the time has already been spent. All that's needed is to iron
>> out a few remaining wrinkles and get it over with!
>
[Dave:]
> OK. Well, could Hercules and Prometheus please go away and stop
> bothering us gods up here on Olympus with their DAFt ideas about us
> having to fit in with their Lower Resolutions. And could they please
> stop insisting that we have to base our deliberations on Equal
> Divisons of the Apocalypse.

Why don't you look at what I have and tell me what's wrong with it?

>> a lot of time working out symbol definitions; I would only have to
>> review these, retaining the good (popular-comma-to-simple-symbol)
>> assignments, discarding the unpopular-comma and complicated-symbol
>> assignments, and reassigning to reasonably simple symbols whatever
>> worthy commas remain. This shouldn't take very long.
>
> OK. Well that sounds good. You really had me worried there with that

It turned out that distinguishing 13:17M from 5:49M is not that
important in light of the number of more popular commas that can't be
used as primary roles.

> Maybe we won't have too many disagreements after all. Be sure to tell
> me how far down the popularity list you ended up going (or the least
> popular comma symbolised) when you send me your list of symbols and
> commas for the highest level.

I did that above. You can also determine this from sheet 2 of
NotDeriv.xlsx. Sort by Pop2 (Col. G), and then move the cursor so that
Col. J displays immediately next to Col. G (so you can see all the
symbols at all of the levels). The commas are color-coded by the
lowest level in which the comma is used. If you page down you'll find
the first comma not distinguished in olympian is 7:13C (row 68), as
indicated by cyan (a super-olympian symbol definition). The first
comma not distinguished in super-olympian is 25:49M (row 77), as
indicated by a white background. (Green in the promethean & herculean
columns indicates that the comma is included in those sets only when
degrees are split; I recommend always splitting so that all of these
are included.

I think that from this point on, I should be dropping the "X" from
herculean-X, forget about the "N" and "Y" options, and simply call it
"herculean".

>>> I submit that no one is
>>> even really using Athenian. They are just using the simplest set of
>>> unique symbols for the 15-limit diamond, or whatever.
>>
>> This is an unmistakable reference to Prent Rodgers, who is essentially:
>> 1) Using athenian-level JI for the 15-limit diamond,
>> or, if you insist that no one actually uses sub-superolympian JI, he's
>> essentially:
>> 2) Using superolympian, with the accents dropped, for the 15-limit
>> diamond.
>
> Yes. 2) is what I had in mind. Except maybe I was forgetting that he
> had to drop accents (for ratios of 13). Anyway that's fine.

Yes, we could also say that he's using herculean.

>> Either way, the "simplest set of unique symbols for ... whatever" is
>> going to be greatly facilitated by lower limits of JI resolution and/or
>> dropping accents for a simpler notation (a variation on DAFLR).
>
> Yes!
>
>> I disagree, because dropping accents is *simple*. The only instances
>> in which I would avoid DAFLR is if a symbol assignment is SoCA (there
>> will be quite a few of these) or if a symbol would be too complicated
>> for a comma. Assigning symbols as SoCA is simple in that flag(s) and
>> accent(s) often provide an indication of the prime factors in the
>> comma, so I would give that precedence over DAFLR; a notable example is
>> .)| for 5:19n, which would be the only symbol assignment for 3 minas.
>
> Sounds fine.
>
>>> We should replace it with the principle that we should try to
>>> ensure that simpler commas have symbols with fewer and more leftward
>>> accents. SSSC -- simple symbols for simple commas.
>>
>> I agree with the principle of SSSC (of course!), but disagree that
>> left-accents are simpler than right-accents. Dropping a right accent
>> (either single or double) does less damage than does dropping a left
>> accent, and, I repeat, dropping accents is *simple*.
>
> OK.
>
>> At this point I'd love to abandon tinas. I'd want to retain minas,
>> however, because some folks like the convenience of expressing pitches
>> or intervals in units that minimize rounding errors (with
>> 12-divisibility as a bonus). If a JI person doesn't care about that,
>> then so what? It won't hurt them.
>
> I have no objection to listing the minas _and_ tinas against the
> symbols and commas. I only object to using them to determine
> boundaries, particularly at the highest level.

>>> OK. Let us instead say that such visually complex symbols (as
>>> accented 3-flaggers) can only be assigned to very complex commas.
>>
>> Are 25:77M, 13:25M, and 17:25M complex enough to qualify?
>
> I guess so.
>
>> When you get down to olympian precision, the simplest symbols within a
>> mina don't always match up with the simplest commas. The 114th mina is
>> a case in point: the only way to give the simplest symbol to the
>> simplest comma (none of which is very simple) is to assign only one
>> symbol to one comma and forget about the rest.
>
> Right. But can't this be framed in terms of distance to SoCA rather
> than minas?

>> As I see it, the objective is to assign one symbol (preferably the
>> simplest one) in each mina to the most popular/least complex comma, and
>> then, where possible, assign no more than one additional symbol (as
>> long as its comma isn't overly unpopular or complex). I don't think
>> you're going to find 3 worthy commas within a single mina.
>
> Forget minas. Just work down the popularity/complexity list assigning
> to the symbol whose SoCA is nearest without being too visually
> complicated.
>
>> Since I've proposed (above) uniquely notating fairly complex 7-limit
>> commas, this would raise the total to perhaps 20 to 25. The great
>> majority of minas would still have only one symbol. This could be our
>> olympian (i.e., top) level of precision -- the one that uses *all* of
>> the defined symbols. (Forget about the "superolympian" label). If we
>> later find that there's a comma that someone *must have* in aprimary
>> role, then if it's possible to do it, we could add it.
>
> That suits me, split a hundred minas if you want. Then it should be
> really clear that mina boundaries don't matter.

I was driven to the opposite conclusion when I got to those regions
where no comma was worthy of being notated. (But I've already said

>>> All right then, lets not call it super-olympian. But please can we
>>> get away from the obsession with minas. That's just producing a
>>> notation for an EDA. We're supposed to be producing a notation for
>>> ratios (JI).
>>
>> Minas don't *prevent* us from producing a JI notation, but rather
>> *allow* us to determine boundaries for secondary roles.
>
> There are other ways to define boundaries.

(Bite tongue -- ouch!)

>> The mina
>> boundaries are useful in that they *guarantee consistency* in providing
>> symbols for commas within the 27 limit, *regardless* of whether or not
>> symbols *have actually been defined for each and every comma required*
>> by a user.
>
> Please explain exactly what it is that is consistent here, and why it
> is important to a JI-ist.

I dealt with that above (somewhere).

>> This covers us in the event we've failed to define some
>> particular comma within a narrow size range that's most important to
>> someone having unanticipated special requirements: for such a user the
>> notation will function *as if* we had actually defined the symbol(s)
>> for the desired commas in a consistent manner.
>

If the hypothetical user (above) should happen to disagree with us
about our selection primary role for some particular symbol, it
wouldn't matter, because the boundaries are where they would have been
had the hypothetical user's comma been selected for the primary role,
because mina boundaries are independent of the selection of primary
commas.

>> Minas are far from arbitrary: they correspond to the smallest
>> difference between accidentals required to notate a 13-limit diamond
>> (and also to 4375n, if you prefer extended 7-limit JI). Likewise,
>> athenian is the best division of the apotome that ignores the
>> 5-schisma; likewise, herculean is the best division that allows the
>> 5-schisma, while promethean is the most complex (i.e., versatile) one
>> that doesn't require accents.
>
> Yes they are far from "completely arbitrary". I shouldn't have said
> that. My point is that, not everyone will consider these properties
> to be the most important ones.

But they're rather obvious criteria for distinguishing levels (or
symbol sets, if you prefer). Everyone should be interested in at least
one of these properties.

>> Of course everyone is going to agree on SSSC, but this shouldn't
>> require the abolition of levels of precision and/or DAFLR. I think I
>> know what you want,
>
> I expect you do. You're a very smart guy.
>
>> and I'm pretty sure now about what I want. If you
>> give me a week or two, I think I'll be able to put something together
>> that will satisfy both of us -- if only to bring this project to a
>> quick and happy conclusion!
>
> Sounds wonderful.

Still waiting to see what you think. (Take your time.)

Whew, this message is over 24 (update: over 26) pages!

--George

____________________________________________________________________________________
Attachments
JI-Nota.xlsx

Dave Keenan
Posts: 1024
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: consistent Sagittal 37-Limit

George Secor, 3-Oct-2007 wrote: Hi Dave,

I need to make a quick correction to something I just wrote you:

<< 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. >>

7s (3^14*7:2^25), 224.798 cplx., no pop. rank (vs. 49:55s, 61 pop. rank)

My statement that "We could distinguish 7s using .|(, which is SoCA" is
invalid, because the size order of 7s (3.804c) and 49:55s (3.930c) does
not correspond to the size order for .|( (SoCA of 3.804c) and )|' (SoCA
of 3.801c, alt. SoCA of 3.774c). So we must have only one symbol in
the 8th mina.

Since .|( is closer to 49:55s than )|', this raises the question of
which symbol should have been chosen for 49:55s in the first place.
Since both symbols are simple (and the two SoCA's are very close to one
another), the answer is determined by deciding which symbol best
promotes DAFLL for herculean. In each of the adjacent minas we have
only a single comma with sufficient popular to warrant a symbol: )|
for 19s and )|'' for 385k. It makes sense to choose )|' for 49:55s,
because when a right accent is dropped, then )| will be the herculean
symbol for all 3 minas.

--George

____________________________________________________________________________________

Dave Keenan
Posts: 1024
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: consistent Sagittal 37-Limit

Wo! George!

Awesome stuff.

I finally read the whole thing. I'm convinced on the point that popularity and all complexity measures fail us at this point. No one will ever agree on them. We are forced to resort to something like odd half-mina boundaries on both sides of any insufficiently popular comma. But when two adjacent commas are very popular, why not use boundaries that are midway between their primary roles?

e.g. What if both commas are in the region of the popularity list prior to the first unsymbolised comma (the first gap)? Does this even happen very often?

This is really a minor nitpick. I'm majorly impressed by what you have done. The drop accents thing is great! Yes. Make the "X" versions the only versions.

I'm not sure I can even follow this stuff any more. I just know I've really pushed you hard on this, and you've bent over backwards to accomodate my wacky ideas. To some degree I'm just running on trust. On behalf of future users, thanks so much for all your hard work.

Just make sure you list all the commas Gene gave us, in whatever you go public with, even when they are only in secondary roles. You should probably also list all commas more popular (or less complex, take your pick) than 65:77n (or 7:121C).

SABounds.gif is awesome!

What sort of spreadsheet do we need to publish this? Something like the character-map spreadsheet? But listing all the commas in size order and giving their up symbol in every JI notation level and showing (by colouring the symbol background?) which is the primary/default comma for that symbol at that level?

-- Dave
Attachments
SaBounds.gif

Dave Keenan
Posts: 1024
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: consistent Sagittal 37-Limit

George Secor, 6-Oct-2007 wrote:
--- Dave Keenan <d.keenan@...> wrote:

> Wo! George!
>
> Awesome stuff.

Thanks! (Yikes! )

> I finally read the whole thing. I'm convinced on the point that
> popularity and all complexity measures fail us at this point.

Whew! I'm relieved that we're not having yet another go-around about
this.

> No one
> will ever agree on them. We are forced to resort to something like
> odd half-mina boundaries on both sides of any insufficiently popular
> comma.

Huh? There's not a single case where a mina must be split between two
insufficiently popular commas, because if there are not two
sufficiently popular commas within a mina, then there's no reason to
split!

> But when two adjacent commas are very popular, why not use
> boundaries that are midway between their primary roles?

I don't know if I'm understanding what circumstances you're referring
to. Are the two commas within the same mina boundaries?

If you're referring to two popular commas within the same mina
boundaries, then I did exactly what you're asking. In fact, I did that
with every single pair of commas within the same mina boundaries,
regardless of the popularity.

If OTOH you're referring to two popular commas in adjacent minas, then
I would argue for keeping the strict (EDA) mina boundary between them,
because it avoids symbol inconsistencies such as the following.
Suppose that pitch A moves upward by interval #1 (or comma #1) to pitch
B, then pitch B moves upward by interval #2 (or comma #2) to pitch C,
then pitch C moves downward by interval/comma #1 to pitch D, and
finally pitch D moves downward by interval/comma #2 back to pitch A (in
a sort of tonal parallelogram). If the boundaries were not EDA, then
the two occurrences of pitch A might end up notated with two different
accidentals. (This could still happen if pitch A is in a split mina,
or if a prime exponent exceeds the point where consistency is
maintained, but the purpose of using EDA boundaries is to be able to
guarantee that it *won't happen* under given sets of circumstances
(i.e., to minimize the chances for inconsistency).

> e.g. What if both commas are in the region of the popularity list
> prior to the first unsymbolised comma (the first gap)? Does this even
> happen very often?

Okay, evidently you're referring to commas in the same mina. The first
gap in olympian is 7:13C (pop. rank 55) and the first in super-olympian
(and 2nd in olympian) is 25:49M (pop. rank 62). The most popular pairs
of commas within a single mina are:

113th mina: 55M (rank 45) & 31M (rank 48)
51st mina: 13C (rank 36) & 5:19C (rank 53)
----- first olympian gap -- 7:13C (pop. rank 55) -----
30th mina: 17C (rank 20) & 7:13C (rank 55) - split only in super-olympian
105th mina: 625M (rank 29) & 5:11M (rank 56)
----- first super-olympian gap -- 25:49M (pop. rank 62) -----
24th mina: 11:35k (rank 43) & 11:35C (rank 68) - split only in super-olympian
72nd mina: 25:49S (rank 32) & 31S (rank 73)
36th mina: 11:49C (rank 77) & 11:25C (rank 75) - split only in super-olympian
78th mina: 7:13S (rank 31) & 5:23S (rank 78)
28th mina: 7:125C (rank 50) & 43C (rank 79) - split only in super-olympian

> This is really a minor nitpick. I'm majorly impressed by what you
> have done. The drop accents thing is great! Yes. Make the "X"
> versions the only versions.

Agreed, and on the exponent-factor notation spreadsheet that's released
I should remove the olympian-split options and simply provide separate
columns for olympian and super-olympian.

I'm wondering whether we really need the prime options in athenian.
Herculean (or if one prefers no accents, promethean) would seem to be
much less of a hassle, because then you wouldn't make the mistake of
using the wrong symbol if the option wasn't set right.

> I'm not sure I can even follow this stuff any more. I just know I've
> really pushed you hard on this, and you've bent over backwards to
> accommodate my wacky ideas. To some degree I'm just running on trust.
> On behalf of future users, thanks so much for all your hard work.

Yeah, I know we were both on the verge of getting burned out after
we've been through.

> Just make sure you list all the commas Gene gave us, in whatever you
> go public with, even when they are only in secondary roles. You
> should probably also list all commas more popular (or less complex,
> take your pick) than 65:77n (or 7:121C).

Do we really need to list everything? I was hoping that the
JI-Nota.xls spreadsheet would take care of it.

Two things that will be required are sag_ji4.par (olympian) and
sag_ji5.par (super-olympian) files for Scala, and the latter will
contain *all* of the primary comma definitions and boundaries.
(Anyway, Scala doesn't yet handle right accents.) Those who write
software will need to know the symbol definitions and boundaries, and
they should be encouraged to use these .par files instead of their own
tables for calculating symbols from commas (and vice versa), because
that will make it easier to implement updates to the notation.

BTW, sag_ji3.par (herculean) and sag_ji2.par (promethean) don't work
quite right in Scala. The mixed-short option can't handle the extended
ASCII characters, and the pure option doesn't interpret left accents in
combination with multiple shafts properly.

> SABounds.gif is awesome!

Thanks. It was very useful in keeping track of the "big picture" as I
went along. I found zooming to 200% best for viewing.

> What sort of spreadsheet do we need to publish this? Something like
> the character-map spreadsheet? But listing all the commas in size
> order and giving their up symbol in every JI notation level and
> showing (by colouring the symbol background?) which is the
> primary/default comma for that symbol at that level?

How about if I revised JI-Nota.xls as follows:
1) Remove the 2460-EDO calculation (since that was only for determining
consistency).
2) Replace the 2460-EDO display (E7:E15) with super-olympian.
3) Redo the symbol calculations so that they're no longer dependent on
options in M7:M10.
4) Remove the contents of cells in rectangle I6:R15, then reformat &
move A17:R24 there.
5) Move the data in Cols. AP and AR:AU to Cols. A thru E, then refreeze
the panes so that Cols. A thru E (and their respective headings) are
always visible.
6) The symbols and lower boundaries for each of the levels (data
beginning at T21) may now be viewed by moving the cursor to the right
and down; the comma definitions (both names & ratios) will be visible
at the left; they would need to be color-keyed according to Cols. A
thru E

I'll make these changes and send it to you.

--George

____________________________________________________________________________________

cmloegcmluin
Posts: 721
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer
Contact:

### Re: consistent Sagittal 37-Limit

Oh my gosh, this is huge!

It's an information overload, of course, which will take some time to fully process. And some handy acronyms, which I could have used earlier. Please accept my apologies having not grokked it all yet.

Also plenty of funny banter in there. I can't get over "Equal Divisions of the Apocalypse". I love it. I may be forced to write a piece one day named that.

For now I'll focus on the matter at hand.
George Secor wrote: Update: Here are the changes I made to remove the DAFLL exceptions. I
unsplit one olympian mina in the above list and split two others, so
now olympian has 7 split minas.

75th mina:
In olympian I split this mina between ~|)'' 47S (rank 72) and .(|(.
11:23S (rank 113) and moved the herculean-X boundary between ~|) and
.(|( to coincide with the split.
So if I'm following this correctly:
1. The Herculean level is determined. and are assigned to 49S and 11S, respectively. The boundary between them is planned around halfway between them, waiting to get snapped to the mina-based grid if possible once the Olympian level gets determined in that vicinity.
2. The mina-based grid in Olympian comes down but unfortunately its proposed boundaries are just about as far as they can be from the boundary between and  .
3. There is a principle that higher levels of precision should not affect decisions already made in lower levels, aside from snapping boundaries to nearby minas. But "nearby" has a threshold. And the amount that would be necessary to move the boundary between and in the Herculean level to snap it to a nearby mina is beyond that threshold. It is not nearby enough. Therefore it is decided to split that mina up into two pieces. They get assigned the symbols and  .
4. Fast-forward to this week: I point out that and have the same SoCA, which is subpar.
Now I don't disagree that waking up in the middle of the night in a cold sweat over WinCompose sequences for short ASCII of conventional symbols is probably not worth it. But this sort of thing is hardcore Sagittal, and important! Last night my subconscious was certainly locked on this problem. At some point I jotted down: why not just set the boundary exactly at 36.529336277¢, so depending on how you looked at it, either both and can represent it, or neither. But then I decided that was a horrible idea. There's actually a kind of beauty to setting the boundaries between capture zones for a JI tuning on irrational EDXs... they'll never be a rational interval themselves, which is good, because if made a rational interval a boundary, it would find itself in a no-man's land, which would be sad for any JI composer who ever came along one day and wanted to find the symbol for that rational interval. I suspect the reason the EDAs were removed from the JI precision levels chart before sharing out was that you were afraid JI hardliners would be allergic to even seeing equal divisions anywhere near their precious just symbols. But if they could only see that the EDAs were their friends, in the service of JI...

So I think I should retract my earlier proposal to assign as the symbol instead of because it'd introduce a DAFLL exception.

But would it be so bad to shift the boundary between and  in the Herculean level over enough to one side or the other so we wouldn't need to split the 75th mina, since finding two symbols for the split parts is so hard?

I also still need to know what people think about the other concerning comma with a SoCAOoB (sum-of-core-and-accents out-of-bounds) exception. Reprinted here:
I wrote:We can't move the upper bound of any higher than 14.191¢, because the next symbol up,  , has a default value of 14.307¢ coming from its primary comma, the 245C. It's sum-of-flags value is 14.307¢ (14.730¢ - 0.423¢), even higher, so we don't need to worry about it. In other words, we need to land it somewhere between 13.897¢ and 14.191¢. It looks like if you move it up by half a mina you get 14.123¢. Good enough?
Last edited by Dave Keenan on Sat May 16, 2020 12:27 pm, edited 3 times in total.
Reason: Attributed quotes. Linked to new anchor for "Equal Divisions of the Apocalypse".

Dave Keenan
Posts: 1024
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: consistent Sagittal 37-Limit

I just noticed that George's Notation Derivation spreadsheet has information on both Sheet 1 and Sheet 2. It opens in Sheet 2, but on Sheet 1 I found the alternative definitions of the 1 and 2 mina accents.

Alt. 4375n (4374:4375)
Alt. 13:125n (255879:256000)

cmloegcmluin
Posts: 721
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer
Contact:

### Re: consistent Sagittal 37-Limit

Dave Keenan wrote:
Sat May 16, 2020 12:20 pm
I just noticed that George's Notation Derivation spreadsheet has information on both Sheet 1 and Sheet 2. It opens in Sheet 2, but on Sheet 1 I found the alternative definitions of the 1 and 2 mina accents.

Alt. 4375n (4374:4375)
Alt. 13:125n (255879:256000)
I am aware of these commas' associations with the mina accents, from the initial New Olympian Diacritics post. But they are described there as secondary commas*.

It may be the case (a) that an "alternative comma" is a distinct concept from secondary commas, only to be used with respect to element arithmetic, when it leads to gnarly situations.

If, on the other hand (b), by "alternative definitions" you mean an "alternative primary comma", or a second primary comma, then I'm concerned we'd have a contradiction in terms. Having such a thing would complicate things pretty badly.

If we're talking (a), then what I think you're suggesting is this, since these two commas are 0.3957558708¢ and 0.8184720367¢, each smaller than their respective primary comma (0.4227161660¢ and 0.8325242041¢), if we used them instead of the primary comma, then we wouldn't find ourselves in the situation where the sum-of-elements for  is the same as the sum-of-elements for , and furthermore, they would diverge by retracting closer to their bare cores, rather than by crossing into each other's territory.

This proves that in some generalizable sense is indeed bigger than But I'm not convinced it solves the problem of the 455/11C being the sum-of-elements for both of them. I still feel like at least one of these symbols should be invalidated.

*Of course I recall this paragraph because I nagged you about rewriting it to be less confusing; I don't have the original phrasing preserved anywhere, unfortunately, but judging from my notes, these were pitched as secondary commas from the beginning.

Dave Keenan
Posts: 1024
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: consistent Sagittal 37-Limit

cmloegcmluin wrote:
Sun May 17, 2020 8:01 am
I am aware of these commas' associations with the mina accents, from the initial New Olympian Diacritics post. But they are described there as secondary commas*.
OK. That's good. Secondary commas is what they are.
It may be the case (a) that an "alternative comma" is a distinct concept from secondary commas, only to be used with respect to element arithmetic, when it leads to gnarly situations.
Nah. Prob just hadn't decided back then. Or looser terminology back then.
If we're talking (a), then what I think you're suggesting is this, since these two commas are 0.3957558708¢ and 0.8184720367¢, each smaller than their respective primary comma (0.4227161660¢ and 0.8325242041¢), if we used them instead of the primary comma, then we wouldn't find ourselves in the situation where the sum-of-elements for  is the same as the sum-of-elements for , and furthermore, they would diverge by retracting closer to their bare cores, rather than by crossing into each other's territory.
Interesting point. I hadn't foreseen it.
This proves that in some generalizable sense is indeed bigger than But I'm not convinced it solves the problem of the 455/11C being the sum-of-elements for both of them. I still feel like at least one of these symbols should be invalidated.
Sorry I still haven't had time to really get into this.

I note that SoE is not the same as SoCA because the value of the core is not necessarily the sum of its flags, although of course it will be very close.

cmloegcmluin
Posts: 721
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer
Contact:

### Re: consistent Sagittal 37-Limit

SoE vs SoCA
I note that SoE is not the same as SoCA because the value of the core is not necessarily the sum of its flags, although of course it will be very close.
Ah. Thanks for making that distinction! The method I used to verify the issue with and identify a new one with was by SoCA, not SoE. Oh the irony — having just pushed for recognizing the concept of element arithmetic over that of flag arithmetic — yet I didn't notice that I was not fully performing element arithmetic, rather, I was doing the kind of weird hybrid that is SoCA.

So I redid my work using SoE this time... and unfortunately the situation is worse with SoE. This shouldn't be surprising, because SoE includes the same divergences from the primary comma involved in SoCA, but further includes divergences involved with summing flags. I mean, I suppose it could have been the case that the divergences introduced by summing flags would always counteract those from summing accents, but the case it is not. In two cases, an unaccented symbol's SoE is not in bounds, meaning that merely by SoF it is not in bounds. That was a head-scratcher for me. But it led to the next insight.

Extreme Precision capture zone vs secondary comma zone

The problem was with the definition of bounds. I had been using the Extreme Precision level's bounds. But I should have been using its secondary comma zone.* In other words, we shouldn't require simpler symbols who have already satisfied our constraints at a lower precision level to keep up with increasingly strict constraints at higher precision levels. This follows from a guiding principle that the more complex stuff in Sagittal should not impugn the simpler stuff. Sure, we allow ourselves to nudge the boundaries ever so slightly as the final step — to lock them to the unit of the highest precision notation — but that's all we allow.

It would not be scalable to say that a symbol's SoE should remain in bounds at each new precision level we introduce. What would happen if we ever complete the Insane precision level, the next level after Extreme? Clearly it would be mad to expect that all symbols' SoE would still be inside the capture zones once they got subdivided to fractions of a cent.

And here's another thought which makes it clear to me that the secondary comma zone is the correct zone to calculate OoBSoE** exceptions with respect to: the SoE of a symbol should be — if not its primary comma — one of its secondary commas. That's a memorable and easy-to-digest principle.

abandon hope?

So I redid my work again, this time using SoE and secondary capture zones. In suspense, I dragged my spreadsheet formula across the field of symbols. And... sadly, the situation was not significantly better than with the Extreme Precision capture zones. It certainly resolved the exceptions for the unaccented symbols, and for some others, but there were still 11 OoBSoE exceptions out of the 149 symbols in Sagittal. still has an exception. is okay, but now is one of those 11 with an exception.

If anyone is interested I can share those exceptions out. I haven't looked into how easy it would be to fix all of them. I'll wait to invest more time in this until I get some feedback on what I've found so far.

My instinct at this point is to reject OoBSoE exceptions altogether as a criterion used for boundary placement / symbol assignment. I'm slightly disappointed about this, because the principle proposed in the previous section about SoE always being at least secondary commas felt so promising. But on the other hand, I hadn't even thought about OoBSoE exceptions before a couple days ago, so maybe they're not that big of a deal.

*I realize now that I've been tossing that term around on the forum but its definition may still be hiding in a correspondence between @Dave Keenan and I that hasn't been shared out yet. Here's Dave:
The concept of a "secondary comma role" predates the invention of diacritics. It originally meant any comma that is not the primary comma of any symbol but is sufficiently close to a primary comma that the symbol can represent it too. The primary comma role is the "default meaning" that the symbol has on playback and is usually the most common comma in the vicinity, although a slightly less common comma may be favoured because it is the sum-of-flags. e.g. 13M was a secondary role for , whose primary role is 35M, before there was a diacritic to give an exact symbol for 13M.

But what is "sufficiently close". Initially it was something like "within a 5-schisma". This was when there were not even 5-schisma diacritics. But now, it would seem more logical to say that a secondary comma must be within the capture zone of the symbol. But which capture zone? Symbols have different capture zones at different JI precision levels. The answer might be: Whatever precision level you need. Or it might be: The lowest precision level at which the symbol is used, but no lower than Athenian. It may be helpful to look at http://sagittal.org/SagittalJI.gif.

And so to be clear, we decided that the "secondary comma zone" was the latter suggestion Dave made, namely, the lowest precision level at which the symbol is used (he disclaimed no lower than Athenian before we decided to throw out the only precision level lower than it: Spartan, AKA Low Precision level).

**On second thought, I switched the order of Out-of-Bounds and Sum-of-Elements, since it makes the same (amount of) sense either way, and OoBSoE is less strange to pronounce.

cmloegcmluin
Posts: 721
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer
Contact:

### Re: consistent Sagittal 37-Limit

I apologize to @Dave Keenan for not having paid closer attention to his words.
Dave Keenan wrote:
Thu May 14, 2020 3:19 pm
What commas do you get for based on summing its components — either sum of flags and diacritics, or sum of core and diacritics, or sum of any kinds of subsets?
Dave Keenan wrote:
Fri May 15, 2020 11:58 am
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
Had I paid closer attention, I may not have failed to appreciate the distinction between SoCA and SoE.

+ + +
+ +
+ +
+ +
+ +
+
+
+
+
Has helped me see that SoCA, which I somewhat denigrated in the previous post as "kind of weird", is actually just one of many possible sum-of-subsets (SoS?). And perhaps any possible SoS is of interest. Certainly it would be when trying to find nearby commas to a given comma (which we have been doing here).

Furthermore, the fact that breaks down into + hadn't occurred to me. But if accents behave no differently to flags with respect to element arithmetic*, why wouldn't it? When I calculate the SoE of I would break it up into and , so why wouldn't I do that with ?
Did I miss any?
You did — just one: + .

In some sense the SoE, which is when you break the symbol down into the smallest possible pieces (no subset with more than one element), is probably the most "important" SoS. I wouldn't want to stipulate that every SoS must be a secondary comma of a symbol. Although it's worth pointing out that if the SoE of a symbol is a secondary comma, it's likely that all SoSs would be secondary commas too, because as you combine up the subsets you may vacillate back and forth some but overall you will be approaching the primary comma for the symbol as a whole.

So I must apologize again, for in failing to recognize that breaks down into + I have still not done my calculations for true SoE. I have redone them... again I didn't expect much to change, because it only involves replacing 0.833¢ cents with 0.845¢ in relevant calculations. It introduces one further OoBSoE exception, so now there are 12 of them.

* In the previous post I mentioned having had a recent insight about flag arithmetic vs. element arithmetic. Here's that:
cmloegcmluin wrote:
Thu May 14, 2020 1:59 am
Frequently but not always it works out that an accented symbol has a primary comma which is equal to the sum of a) the primary comma for the symbol which consists only of the diacritic in question and b) the primary comma for the symbol unaccented. This is quite similar to the well-established fact that frequently but not always it works out that a symbol with two flags has a primary comma which is equal to the sum of a) the primary comma for the symbol which has only the first of those two flags and b) the primary comma for the symbol which has only the second of those two flags.

So, is it not the case that flag arithmetic truly encompasses not only arithmetic of flags, but also of accented, and therefore shouldn't it be referred to as "symbol element arithmetic"?
To which Dave replied:
Dave Keenan wrote:
Fri May 15, 2020 2:07 am
"Element arithmetic" is great. Everything you say in that section is spot on. Yes I usually meant to include diacritics when I have said "flag arithmetic" in the past. "Sum of flags", and now "sum of elements", can refer to the untempered size in cents or the tempered size in EDO degrees, depending on the context.