## consistent Sagittal 37-Limit

Dave Keenan
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### Re: consistent Sagittal 37-Limit

[Reverse-engineered from browser cache after disaster. Originally posted Mon May 25, 2020 6:50 pm]

My frazzled mention of 27-EDA was purely because it appeared on that old diagram of George's, and yet I had no memory of it ever being mentioned. Thanks for making it clear that it had to be a typo.

Dave Keenan
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### Re: consistent Sagittal 37-Limit

[Originally posted by cmloegcmluin Tue May 26, 2020 8:52 am (East Australian time)]
[Unformatted text recovered from browser cache after disaster.]
Dave Keenan wrote: ↑
Mon May 25, 2020 2:11 pm
cmloegcmluin wrote: ↑Sun May 17, 2020 2:02 pm
... I did go ahead and [find] every value that and  could be said to take when subtracting the rest of the symbol out from the Extreme Precision (Olympian) JI notation. Here those all are. There are actually 25 different values for and 13 different values for :
This blew me away. I had no idea there were so many. Or that they came so close to crossing over. It seems bad. But I think simply noting it and moving on, is the right approach.

I think it's probably a good thing, actually. Doesn't it just mean that a great variety of commatic alterations are captured at the Extreme level, leveraging the fact that so many different prime exponent vectors are close to the primary commas for and ?
Dave Keenan wrote: ↑
Mon May 25, 2020 5:45 pm
NotDeriv was short for "Notation Derivation".
Yeah.... right... I knew that! I was just, uh, testing you.
But which of the two valid (by mina arithmetic) symbols was assigned to the whole 75th mina before it was split? I can only assume it must have been because if it had been there would not have been any DAFLL violation, which George clearly states as the reason for the split.

Then we must ask, why was George so resistant to assigning to 47S? Surely the pre-existing boundary at the VH precision level (one level down), between and would have either been at 18.5/58 of an apotome, or midway between their commas 49S and 11S, which you have shown amount to practically the same thing, which is very near 74.5 minas, which should have predisposed George to using for the 75th mina.
Okay, that would make sense. That convincingly answers my question about where the boundary probably was before it was moved as part of splitting the mina.

Except I think in the last sentence you meant "which should have predisposed George to using for the 75th mina."
Or else we have to answer, why was the boundary at the VHP level not set near 18.5/58 apotomes or the midpoint between the commas?

Basically, I need a convincing explanation of how someone as smart of George could have got this so wrong.
I agree that we should not execute on this change until we can convince ourselves what George's intentions were. Uncertainty is not good enough.
On the other questions: I agree that slavishly assigning symbols in order of frequency in the Scala archive doesn't make a lot of sense when we get to such rare ratios. Such frequencies of occurrence can be accidents of history which are unlikely to reflect the probability of future notational needs.

One such effect I've noticed in the past is some historical scales that appear to be "tempered by ratios". Or at least ratios seem to sometimes be used merely to describe approximate pitches, possibly because that's the tool the author was most familiar with. It might be worth searching the archive to look for these 47's to see if it looks like their use might be predictive of future notational requirements or not.
That's an excellent consideration to surface. The stats are not everything. We have no frequency of use data to weight them by. They're noisy and a lot of junk in there. And they're only historical.

The Scala scale archives do indeed correspond with what you've noticed.

So we've got a bunch of mere catalogs of harmonics / primes:
• Kyle Gann from Anatomy of an Octave, edited by Kristina Wolfe (2015)
• Roger Dean's 91 primes non-octave scale (2008) (which is just a listing of primes from 3 to 467)
• Roger Dean's 81 primes non-octave scale (2008) (up to 419)
• Harmonics 2 to 256, Johnny Reinhard
• Warren Burt, primes until 251. "Some Numbers", Dec. 2002
• First 30 harmonics and subharmonics
• Harmonics 32 to 64
• Bruce Kanzelmeyer, 18 harmonics from 32 to 64. Base 388.3614815 Hz
• Harmonics 30 to 60
rational "temperings" of logarithmic tunings:
• Franck Jedrzejewski continued fractions approx. of 96-tet
• Franck Jedrzejewski continued fractions approx. of 66-tet
• Franck Jedrzejewski continued fractions approx. of 26-tet
• Franck Jedrzejewski continued fractions approx. of 13-tet
• Franck Jedrzejewski continued fractions approx. of 16-tet
• Franck Jedrzejewski continued fractions approx. of 41-tet
• Franck Jedrzejewski continued fractions approx. of 42-tet
• Franck Jedrzejewski continued fractions approx. of 43-tet
• Franck Jedrzejewski continued fractions approx. of 21-tet
• Franck Jedrzejewski continued fractions approx. of 54-tet
• Franck Jedrzejewski continued fractions approx. of 55-tet
• Franck Jedrzejewski continued fractions approx. of 17-tet
• Franck Jedrzejewski continued fractions approx. of 78-tet
• Franck Jedrzejewski continued fractions approx. of 84-tet
• Franck Jedrzejewski continued fractions approx. of 90-tet
• Franck Jedrzejewski continued fractions approx. of 60-tet
• Franck Jedrzejewski continued fractions approx. of 30-tet
• Franck Jedrzejewski continued fractions approx. of 20-tet
• Franck Jedrzejewski continued fractions approx. of 15-tet
• Franck Jedrzejewski continued fractions approx. of 10-tet
• Franck Jedrzejewski continued fractions approx. of 5-tet
• 10-tET harmonic approximation, fundamental=27
• 14-tET harmonic approximation, fundamental=27
• 10-tET harmonic approximation, fundamental=27
• 79 MOS 159-tET Just Intonation Ratios
impositions of ratios on traditional tunings that do not traditionally use them:
• Gamelan Udan Mas (approx) s6,p6,p7,s1,p1,s2,p2,p3,s3,p4,s5,p5
• Lou Harrison´s Kyai Udan Arum, pelog just gamelan tuning
• Slendro Udan Mas (approx)
• Gamelan selunding from Kengetan, South Bali (Pelog), 1/1=141 Hz
• AEU extended to quasi-cyclic 41-tones in simple ratios (the ‘national theory of Turkish Music’ in use today known as the Arel-Ezgi-Uzdilek (AEU) System)
similar to the above category, perhaps (see: http://www.tonalsoft.com/enc/s/schlesinger.aspx):
• Conjunct Tonos-25 Enharmonic (tonos!)
• Conjunct Tonos-33 Enharmonic
• Conjunct Tonos-25 Chromatic
• Tonos-25 Chromatic
• Tonos-25 Enharmonic
• Inverted Enharmonic Conjunct Phrygian Harmonia
• Phrygian Enharmonic Tonos (from The Mathematical Theory of Tone Systems by Jan Haluska)
• 1+2 rat. hexachromatic/hexenharmonic genus derived from K.S.'s 'Bastard' Hypodo
• 1+1 rationalized hexachromatic/hexenharmonic genus derived from K.S.'Bastard'
• 1+1 rationalized hexachromatic/hexenharmonic genus derived from K.S.'Bastard' (twice somehow? and I think there were more duplicates)
• 1+3 rat. hexachromatic/hexenharmonic genus from K.S.'s 'Bastard' Hypodorian
• 1+3 rat. hexachromatic/hexenharmonic genus from K.S.'s 'Bastard' Hypodorian (twice)
• 1+4 rat. hexachromatic/hexenharmonic genus from K.S.'s 'Bastard' Hypodorian
• 1+5 rat. hexachromatic/hexenharmonic genus from K.S.'s 'Bastard' Hypodorian
• Rationalized Schlesinger's Hypophrygian Harmonia in the enharmonic genus
• Schlesinger's Phrygian Harmonia in the enharmonic genus
• Inverted Schlesinger's Enharmonic Phrygian Harmonia
• Inverted Schlesinger's Enharmonic Hypophrygian Harmonia
• Hypophrygian Enharmonic Tonos
• HarmE-Hypophrygian
• Hyperenharmonic genus from Kathleen Schlesinger's enharmonic Phrygian Harmonia
and then a few that may actually be indicative of future needs:
• George Secor's 24-triad proportional-beating well-temperament (24e) (maybe George was just tying to slip the 1/47S by us so he could notate this scale exactly... )
• M. Schulter, scale from mainly prime-to-prime ratios and octave complements (Gb-D#) (another friend! Hi @mschulter )
• Sparschuh's epimoric two- and one-7th part of syntonic comma (2010) (epimoric!)
• Modified Sparschuh temperament with a'=419 Hz by Tom Dent
• Andreas Sparschuh WTC temperament. 1/1=250 Hz, modified Collatz sequence
• E scale (link here: https://github.com/cmloegcmluin/Scala-s ... iter26.scl)
• 31-tone Tetraphonic Cycle, conjunctive form on 5/4, 6/5, 7/6 and 8/7
Now of course we might audit any one of the commas in this way and be similarly disappointed with the quality of the results. The point of the Scala stats is to keep us honest. As more of a heuristic.
If not, we should probably shift to using the sum-of-prime-factors (SoPF) metric that seems to be a reasonable match for the ranking of commas for notating more popular ratios. Is 11:23S the 37-limit comma with the lowest SoPF in the 75th mina (say with absolute slope ≤ 12)?
There are two commas in that range with SoPF < 11:23S's SoPF of 34, however, they both have absolute apotome slope > 12. They are the 1/1715S and 11/85S, which might be familiar from my last catalog of commas in this vicinity on this topic: viewtopic.php?p=1352#p1352

The 5/847S — whose details are also in that list — has the same SoPF as 11/23S, and is lower limit, but it has worse apotome slope. I don't think that's enough to depose 11/23S. Incumbency has some benefits.

36.706¢, [ -2 0 -1 -1 1 1 ⟩, 143/140, 143/35S, 13 limit, -2.260 slope, 36 SoPF


This is the only other interesting one, I think, if you're willing to take a dent in SoPF. It's opened up now that I widened the scope for my script to the entirety of the 75th mina (it was in the upper half before).
I would also need to do some archaeology to answer your question as to how we went from simple Scala archive occurrence counts for ratios with their factors of 2 and 3 removed (which I think is what's in the ratio popularity spreadsheet) to assigning notional occurrence counts to the commas that might be used to notate those ratios in various spellings (which I think is what's in the comma popularity spreadsheet).
I don't know exactly what you mean by notional occurrence counts, but that does seem to be so. I look forward to whatever you can find.
Dave Keenan wrote: ↑
Mon May 25, 2020 6:50 pm
My frazzled mention of 27-EDA was purely because it appeared on that old diagram of George's, and yet I had no memory of it ever being mentioned. Thanks for making it clear that it had to be a typo.
So... George is fallible.

But I do think we should assume he knew what he was doing, and it was for better reason than exactly notating his 24-triad proportional-beating well-temperament. Perhaps if we can find a reason why the 11/23S would also have been really important to him, that would be enough.

[Originally posted by cmloegcmluin]

Dave Keenan
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### Re: consistent Sagittal 37-Limit

[Reverse-engineered from browser cache after disaster. Originally posted Tue May 26, 2020 10:43 am]
cmloegcmluin wrote: Mon May 25, 2020 2:47 am *I sometimes see it as 59-EDA but I don't understand what it would mean to be 58-EDA/59-EDA. Neighboring ED's are in some senses as different as possible. Does this mean actually 58 * 59 = 3422-EDA where some steps are 58 and some are 59? Or does it mean that both EDA's co-exist and bounds can be snapped to one or the other of them?
You'd be referring to this table of JI precision-level approximate EDAs and EDOs.

I'm not quite sure what I mean by "58/59 eda". It's possible I'm simply mistaken about there being any 59-EDA aspect to the VHP Herculean level at all. I may have been seduced by the fact that both 612edo and 624edo have highly-accurate JI approximations. I'm pretty sure 624edo corresponds to 59-EDA. You might check the VHP boundaries to see if any of them are closer to odd half steps of 59 than odd half steps of 58.

Dave Keenan
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### Re: consistent Sagittal 37-Limit

[Reverse engineered from browser cache after disaster. Originally posted Tue May 26, 2020 11:52 am]

cmloegcmluin wrote: Tue May 26, 2020 8:52 am I think it's probably a good thing, actually. Doesn't it just mean that a great variety of commatic alterations are captured at the Extreme level, leveraging the fact that so many different prime exponent vectors are close to the primary commas for and ?
You have probably seen by now, that Ash has come up with a new use for that table — deciding tina commas. Would it be difficult for you to go back and add an occurrence count for each such comma, and add commas that correspond to zero minas (preferably without requiring horizontal scroll bars).
Okay, that would make sense. That convincingly answers my question about where the boundary probably was before it was moved as part of splitting the mina.

Except I think in the last sentence you meant "which should have predisposed George to using for the 75th mina."
I did indeed. I copied and pasted the wrong smiley conglomeration. I've added an edit note to the original. Thanks.
That's an excellent consideration to surface. The stats are not everything. We have no frequency of use data to weight them by. They're noisy and a lot of junk in there. And they're only historical.

The Scala scale archives do indeed correspond with what you've noticed.

So we've got a bunch of mere catalogs of harmonics / primes:
...
rational "temperings" of logarithmic tunings:
...
impositions of ratios on traditional tunings that do not traditionally use them:
...
similar to the above category, perhaps (see: http://www.tonalsoft.com/enc/s/schlesinger.aspx):
...
Wow! All those had 47's in them? Thanks for doing this survey.
and then a few that may actually be indicative of future needs:
• George Secor's 24-triad proportional-beating well-temperament (24e) (maybe George was just tying to slip the 1/47S by us so he could notate this scale exactly... )
Hah!
• M. Schulter, scale from mainly prime-to-prime ratios and octave complements (Gb-D#) (another friend! Hi @mschulter )
• Sparschuh's epimoric two- and one-7th part of syntonic comma (2010) (epimoric!)
• Modified Sparschuh temperament with a'=419 Hz by Tom Dent
• Andreas Sparschuh WTC temperament. 1/1=250 Hz, modified Collatz sequence
• E scale (link here: https://github.com/cmloegcmluin/Scala-s ... iter26.scl)
• 31-tone Tetraphonic Cycle, conjunctive form on 5/4, 6/5, 7/6 and 8/7

Now of course we might audit any one of the commas in this way and be similarly disappointed with the quality of the results. The point of the Scala stats is to keep us honest. As more of a heuristic.
Agreed.
36.706¢, [ -2 0 -1 -1 1 1 ⟩, 143/140, 143/35S, 13 limit, -2.260 slope, 36 SoPF


This is the only other interesting one, I think, if you're willing to take a dent in SoPF. It's opened up now that I widened the scope for my script to the entirety of the 75th mina (it was in the upper half before).
Thanks for that survey too. Let's keep 11:23S.
So... George is fallible.
Yes. He wasn't really a Greek god.
But I do think we should assume he knew what he was doing, and it was for better reason than exactly notating his 24-triad proportional-beating well-temperament. Perhaps if we can find a reason why the 11/23S would also have been really important to him, that would be enough.
And failing that, we should try to at least find a plausible story about how the sequencing of events left him in a cul de sac.

I hadn't considered that he might have felt it was important to notate 11:23. Good point! Perhaps he felt we should be able to notate the entire 23-limit diamond exactly, equivalent to notating all pairs of opposing primes from 5 to 23 exactly. Are there any other 11:23 commas? Are all the other pairs exactly notatable?

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### Re: consistent Sagittal 37-Limit

Dave Keenan wrote: Tue May 26, 2020 10:18 pm Would it be difficult for you to go back and add an occurrence count for each such comma, and add commas that correspond to zero minas (preferably without requiring horizontal scroll bars).
I added the occurrence count. That was easy (the different symbols were already enumerated right there). But I don't know what is meant by the zero minas request. (I asked for clarification on the Magrathean thread too, so of course you only have to explain in one or the other place)
Let's keep 11:23S.
Aye-aye.
I hadn't considered that he might have felt it was important to notate 11:23. Good point! Perhaps he felt we should be able to notate the entire 23-limit diamond exactly, equivalent to notating all pairs of opposing primes from 5 to 23 exactly. Are there any other 11:23 commas? Are all the other pairs exactly notatable?
11/23C is 13.269¢, [ 9 -5 0 0 1 0 0 0 -1 ⟩, yes. (There's also the 1/11.23C, 20.408¢, [8 0 0 0 -1 0 0 0 -1 ⟩, but that's not what we're looking for.) So that theory's shot.

Oy. What happens when you search your email history for "11:23"??

Dave Keenan
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### Re: consistent Sagittal 37-Limit

cmloegcmluin wrote: Wed May 27, 2020 12:17 pm I added the occurrence count. That was easy (the different symbols were already enumerated right there). But I don't know what is meant by the zero minas request.
Thanks for that. Very useful. And yeah, forget the zero thing.
11/23C is 13.269¢, [ 9 -5 0 0 1 0 0 0 -1 ⟩, yes. (There's also the 1/11.23C, 20.408¢, [8 0 0 0 -1 0 0 0 -1 ⟩, but that's not what we're looking for.) So that theory's shot.
Your shot missed. 11.23C doesn't notate a ratio of the diamond. The two primes have to be on opposite sides of a slash or colon for that.
Oy. What happens when you search your email history for "11:23"??
You asked for it! These are only the "From George", not the "To George".

Feel free to format them better. e.g. Wrap [pre]...[/pre] tags around each email, so the tables stand out.

And delete the extraneous carriage returns that result in lots of lines with one or two words on them, in quotes.

First one done for you.

Dave Keenan
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### Re: consistent Sagittal 37-Limit

Re: An Olympian proposal
From George Secor 21/05/2005

--- Dave Keenan <d.keenan@bigpond.net.au> wrote:

> At 03:04 AM 20/05/2005, you wrote:
>> ...
>> Following is an explanation of how I arrive at a herculean symbol set,
>> following the above principles.
>>
>> It's essentially a superset of athenian-level JI, ...
>> ...
>> In order to distinguish pitches and intervals differing by a 5-schisma,
>> we need the higher resolution of the herculean level, which divides the
>> apotome into 58 parts (~2 cents each).  In order to accomplish this,
>> left-accent marks, amounting to a 5-schisma (~2 cents) are introduced.
>> These may modify any of the athenian-level symbols either upward or
>> downward:
>> '   up
>> .   down
>>
>> Used in combination with the above athenian-level symbol cores, left
>> accents triple the number of possible symbols (and thus the number of
>> ratios represented exactly) without overly complicating the learning
>> curve.  These exactly represent the ratios in the following table,
>> shown opposite the corresponding number of herculean degrees, adjusted
>> so that they coincide with 233-EDA boundaries (the distance between
>> boundaries always being 4 minas, except for those delimiting the
>> natural sign, which are 5 minas apart).  This, BTW, has the desirable
>> consequence of having 17C fall below the 7deg upper boundary, 4 minas
>> below 23C, making '~|( the same number of minas as |~, and conversely
>> .|~ the same number of minas as ~|(, which neatly eliminates acouple
>> of virtually useless right-accented symbols from both the herculean and
>> olympian sets.
>>
>> Here's the table, with intervening commentary regarding which symbols
>> are the best candidates for each herculean degree:
>>
>> degrees symbol comma   ratio
>> ------- ------ -----   ----------------------------
>>    1     '|    5s      32768:32805
>>    2*    .|(   7k      3^14*7:2^25 ***
>>    3      |(   5:7k    5103:5120
>>                11:13k  351:352 (approximated) |(.. in olympian
>>
>> For degrees 1 and 3 there is only one candidate apiece.
>> (13-approximated ratios in this table are given only for reference.)
>
> I note that with these boundaries 11:13k actually falls within the range of
> 2degHerc, not 3.

Yes, there would have to be a downward adjustment of the lower 3deg
boundary.

> ...
>>    6    ')|(   11:35k  2816:2835
>>         .~|(   5:17C   135:136
>>
>> For degree 6 there are two possibilities.  I expect that 11:35k (with
>> the lower prime limit) would be the more popular of the two, hence
>> should be awarded the position.
>
> Deemed occurrences:
> 11:35k 69
> 5:17C  63
> So it could go either way, and )~| could be used for 11:35k too. But of
> course only ')|( gives the desired result in athenian by dropping the
> accent.

First off, I believe that the primary role of a symbol should be
*independent of the resolution* and that all primary roles should
therefore be *defined at the olympian level*.

Next, I would like to establish a "doctrine of reasonable expectation"
(DRE):  If two reasonably popular ratios differ by a 5-schisma, and if
one of the two has as its primary role an unaccented symbol, then we
would *reasonably expect* the other to be the primary role for the same
symbol with the appropriate left-accent added.  7:11k and 11:35k fit
these conditions.  (If one of the two ratios were of very little
popularity or usefulness or overly complex, we would then reasonably
expect a simpler ratio to be used in its place.)  This, taken together
with DAFA, is why I view an accented athenian core as being less
complex than a non-accented non-athenian core.

Besides, 11:35k isn't even the right number of minas for )~| to be its
primary ratio.  You were suggesting a one-mina downward shift in order
to accomplish this, but I think I've given enough justification to use
a different comma-definition (143C) for that symbol.

I might add that DRE would also mandate that the most popular ratio one
mina higher than )~| get .~|( for 5:17C, so I anticipate that the )~|
might never be used with any accents (except to subdivide a mina,
possibly for 11:35C).

> ...
>>    8*   '~|(   85C     2^27:3^13*5*17 ***
> ...
>> Degree 8 has a complex ratio, hence will be an opportunity to consider
>> a non-athenian (unaccented) symbol (guess which one!).
>
> There is of course only one. |~ as 23C.
>
> The most popular commas for 8degHerc are
> 7:25C   45 deemed occurrences   |~..
> 23C     39                      |~
> 1225C  38                      |~.
>
> So it would be difficult to argue for .~~|
>
> However, it looks like |~ should have primary role 7:25C in herculean but
> 23C in olympian.

1) I think that primary roles should be independent of resolution, so I
would regard |~ as approximating 7:25C in herculean;
2) For 25/14 of C, 7:25C would be used as an alternate spelling A#!~,
after the preferred spelling Bb'|(, whereas |~ is used in the main
spelling of 23/16.
See my remark below about slope.

>> Degree 9 will require a new non-athenian symbol, since no
>> athenian-cored symbol is possible.
>
> Most popular commas for 9deg58-EDA
> 125C    63 do (deemed occurrences)
> 11:49C  30 do
> 11:25C  30 do
> 95C     23 do
>
> 125C is [26 -12, -3>, the comma least likely to be used to notate 125. My
> formula distributes the 2,3-reduced occurrences of 125 in the Scala archive
> as follows
>
> 125S 326 do
> 125M 103 do
> 125C  63 do

You'll have to explain how you came up with that, because at this point

> You may well disagree, but there's a factor of 2 to be bridged before you
> can claim 11:49C or 11:25C as more popular than 125C. Fortunately they are
> all 36 minas, which is notatable by ~~|  |~''  .)|~.  and  '~|('' (triple
> accent).

I don't think we'll need a separate olympian symbol for 125C -- see
below.

> Of those, the only ones still in range when you drop the right accents are
> ~~| and .)|~
>
> No prizes for guessing you're gonna go for ~~|

Yep, you win the non-prize!

> ...
>>   20     (|(   5:11S   44:45
>>                7:13S   1664:1701 (approximated) (|(.. in olympian
>
> 5:11S  315 do
> 23S     78 do
>
> But 7:13S falls into the 19 degree range.

Yep, the boundary needs to be tweaked.

> ...
>>   22    //|    25S     6400:6561
>>                5:13S   39:40 (approximated) //|'' in olympian
>
> Agreed. But if |~) is ever used, I think its primary role should also
> be 5:13S.

We'll be discussing that when we get to olympian.

>>   23*  '//|    5S      2^23*5:3^16 ***
>
> 5S is off the radar, we have
>
> 13:19S   33 do    '//|  or |~)''
> 13:25M   18 do    '//|'' (triple accent) or  '|~)
> 25:77M   17 do    '//|' or '|~).
>
> 13:19S is significantly ahead. I expect you will propose '|~) as 13:25M for
> 23 degrees, despite the fact that |~) does not appear in the set.

I thought about something like that for a while, but then came to my
senses.

>> For degrees 19, 20, and 22 there is only one (obvious) candidate. (The
>> herculean-level lower boundary for (|( must be tweaked downward at
>> least ~0.248c to the olympian sub-boundary halfway between 5:23S and
>> 7:13S in order for 7:13S to fall within its range.)
>
> If that's your solution, you'll have to do a similar thing for 11:13k and
> the 2-3 degree boundary. But you probably knew that.

Yes.

> ...
>> At this point candidates will need to be considered together with their
>> apotome-complements.
>
> When I calculated deemed occurrences for commas, I'm pretty sure I added
> the corresponding M and L ocurrences together and gave them to the M, since
> I don't look at L's separately. But so far I can't find the spreadsheet in
> which I did this, to check.
>
> So when I give occurrences for Ms below, take them as really M + L. Let me
> know if the numbers I give look really unlikely.

Okay.

>>   24    ./|)   175M    127575:131072
>>                5:13M   6480:6566 (approximated) ./|). in olympian
>
> 5:13M  72 do
> 7:23M  41 do
> 5:77M  29 do
> 37M    25 do
> 175M   24 do
>
> So it looks like the primary role of ./|) in herculean should be 5:13M even
> though it will be 175M in olympian, unless we use a new unaccented
> non-athenian for 5:13M, like the one I half-heartedly proposed as a
> replacement for |~\, namely )//| . But that would be just too ugly.
>
> So how do you feel about ./|) changing primary roles between herc and
> olymp? That's not nice either.

I'm not in favor of a symbol changing primary roles at all -- ever!
The primary role of a symbol should be independent of the resolution.
For ./|) 175M is the most popular ratio that's the right number of
minas, so it gets the primary role.  In herculean all of the other
ratios above are notated with ./|), but as approximations.  As long as
everything within the 24-herculean-deg range gets notated ./|), it's
rather irrelevant to the user which ratio defines the symbol.  (And we
seem to be agreeing on the symbols in spite of our apparent difference

>>    9*          nothing athenian-cored available
>>         ~~|    11:25C  99:100
>>                  11:49C  98:99 (approximated) |~'' in olympian
>>
>> ~~| is the obvious choice!
>
> For the symbol yes, but 125C [26 -12, -3> is the more popular comma.
>
> 125C     63 do  [26 -12, -3>
> 11:49C  30 do
> 11:25C  30 do

However, observe the following, which puts 125C at the very *edge* of

125/64 of C can be notated only as C'\\! or B/|) -- 125C can't be used!
125/64 of E may be notated as E'\\! or D#/|) or Fb~~! -- 125C is the
*least* likely spelling!
44/25 of C may be notated as Bb~~! or A#'\\! or Cbb|( -- 11:25C is the
*most* likely spelling!
49/22 of C may be notated as D~~! or Cx'\\! or Ebb|( -- 11:49C is the
*most* likely spelling!

So I don't imagine that we'll need an exact olympian-level symbol for
125C.  You've taken this (slope, I believe) into account in your
comma-popularity rankings, but I think you've given much too much
weight to third-place spellings that will hardly ever be used.

On the basis of usefulness, the contest for the primary ratio for ~~|
is between 11:49C and 11:25C.  I would select 11:25C, because it has a
lower combination of primes.  Subdividing the mina would give 11:49C
|~'' as its exact symbol.  Then let 125C be approximated by one of
these.

>>   13*    .|)   35C     3^10*5*7:2^21 ***
>>         )/|    5:19C   40960:41553
>>                13C     6561:6656 (approximated) .|). in olympian
>>
>> This one is more difficult.  While admitting symbols for primes 19 and
>> 23 was okay, I'm not very excited about 5:19C.  The alternate spelling
>> of 13/8 of C is G#.|). in olympian, and it would make sense if the
>> right-accent were simply dropped in herculean, so I would go with 35C
>> and .|) for 13deg herculean.
>
> Again, I agree with the symbol, but 77C is the more popular comma for it in
> herculean.
> 77C   96 do  [-11   3,  0  1  1>
> 35C   64 do  [ 21 -10, -1 -1>

Okay.  I couldn't tell that from your olympian.xls table, since you
only ordered them *within* each mina.  I don't remember whether you
gave me an extensive popularity listing.

Anyway, 77C isn't the right number of minas for the exact definition of
.|), so I would list it this way:

13*    .|)   35C     3^10*5*7:2^21 ***
77C     2048:2079 (approximated) .|)' in olympian
13C     6561:6656 (approximated) .|). in olympian
)/|    5:19C   40960:41553

>>   18*   '(|    11:35S  2^27*11:3^16*5*7 ***
>>                25:49S  49:50
>>          ~|)   49S     48:49
>>
>> This one is also a bit difficult.  As it turns out, there is a much
>> simpler ratio for '(|, 25:49S, which is almost exactly the same size as
>> 11:35S (same number of minas), and which could be assigned the exact
>> definition of that symbol.  This competes with 49S for the 18deg
>> herculean position (1 mina larger).  On the one hand an accented
>> athenian core may be regarded as slightly simpler than an unaccented
>> non-athenian core, but, on the other hand, a ratio with two primes 3
>> would be expected to be slightly more popular than one with two
>>
>> As I see it, an accented 7-limit athenian vs. an unaccented 7-limit
>> nonathenian symbol is a tossup, so I'm inclined to award the position
>> to 49S on the basis of its simpler combination of primes >3.
>
> It's easy. 49S is way more popular (by a factor of 3) than anything else in
> the range. And it is more popular than the primary commas of many
> athenians, and it has an unaccented symbol (albeit 1 mina different from
> its SoF), namely ~|) . The fact of its popularity is far more important
> than any pro-athenian prejudice against it.
>
> So I agree.
>
>>   Should
>> you agree, I would then advise that we make .~|) the exact symbol for
>> 25:49S
>
> No that doesn't work out. Since ~|) is 49S (73 minas),
> ...
> I'm totally confused as to what you intend here.

Sorry!  I used a left-accent when I meant a *right* one.  This is the
sort of olympian sequence I had in mind:

minas symbol exact ratio
----- ------ -----------
70   (|''  17:25S
71   ~|).. ???     instead of '(|.
72   ~|).  25:49S  instead of '(|
73   ~|)   49S
74   ~|)'  5:17S
75  .(|(.  11:23S  instead of ~|)''

This allows one to drop accents for herculean (from olympian, DAFH) for
the above range of symbols (plus quite a bit more above and below).

>>   23*  '//|    5S      2^23*5:3^16 ***
>>                13:19S  38:39
>>                13:25S  5^2*3^12:2^20*13 (approximated) '//|. in olympian
>>                13:25M  53248:54675 (approximated) '//|'' in olympian
>>          '|~)  7:115M  112:115
>>                187M    187:192
>>
>> We're looking at the lower half of that no-man's land ranging from 91
>> to 98 minas that's filled with garbage ratios, so finding a better
>> primary ratio for '//| doesn't look very promising.  The best I can do
>> is 13:19S -- not very popular, I imagine, but at least it's something
>> reasonable.  Do you agree that this should be the primary ratio?
>
> Yes. Poor though it is.
>
>> The only non-athenian symbol available to compete for the position is
>> '|~), which has a couple of possible ratios: 7:115M and 187M.
>
> Huh? What about 13:25M for '|~) ?

If //|'' is 5:13S, then I would reasonably expect 13:25M (a 5-schisma
larger) to be '//|'', which could become '//| in herculean.

>>   However,
>> it gets rejected, because an accented non-athenian symbol is more
>> complicated than an accented athenian one,
>
> Piffle.
>
>> and the '//| symbol gets the
>> position by default.
>
> The '//| symbol gets the position because of its more popular comma
> 13:19M.

For whatever reason.

>> I was wondering how often this symbol might actually be used at the
>> herculean level.  There are a couple of 13-limit ratios approximatedby
>> '//|: 13:25S, the closest, is arguably more complicated than 5S (and
>> would hardly ever be used),
>
> I have it that 5S would never be used and 13:25S would be used almost never
> (5 times out of 29,400).
>
>>  but 13:25M is reasonably simple for an
>> accented athenian.  However, the olympian version of '//| for 13:25M
>> (with right accents) would be '//|'' -- more accents than you've been
>> willing to accept.  I notice that the olympian degree one mina higher
>> has ./|). for 5:13M and that the upper herculean boundary for 23deg
>> separates 13:25M and 5:13M.  This works out extremely well when it
>> comes to adding/dropping right accents with change of precision, so I
>> think I've found a good reason for a double-right accent in combination
>> with a left accent.
>
> Maybe you've found a good reason for a new unaccented symbol for this
> region. With primary role 5:13M. e.g. )//|

Then ')//| would be exactly 13M.  Were you also thinking of having )//|
instead of ./|) in herculean?  I'm questioning whether there's
sufficient justification for another symbol that introduces even more
redundancy in that:

13:25M could be either )//|. or '|~), which are equivalent to '//|''

Double-right accents show up only in olympian, and it's simple enough
to understand and remember how they work, and this is a region where
they can be dropped quite easily for olympian-to-herculean conversion.
If '//| and ./|) will cover the herculean range in this area, why
introduce a new symbol core?

> Then 13:25M could be )//|.  (right accent).
>
> But why wouldn't 13:25M be '|~) in olympian?

Only if |~) is 5:13S.  That, plus the question of how (and whether)
5:13S and 7:23S should have separate (exact) symbols, is an issue we'll
have to take up when we discuss olympian.

>> I haven't read the details of your 3-flag symbol proposal(s) yet, but
>> I'll be keeping the above in mind when I do (very shortly).

> ...
>> So here's what I get for the herculean symbol sequence:
>>
>> '|  )|  |(  '|(  )|(  ')|(  ~|(  |~  ~~|
>>   ./|  /|  '/|  .|)  |)  '|)  |\  (|  ~|)
>>   .(|(  (|(  .//|  //|  '//|  ./|)  /|)  '/|)
>>   /|\  (/|  '(/|  |\)  (|)  .(|\  (|\  '(|\

Evidently we agree (whew!).

>> And here's the resulting table:
>> ...
>>
>> How does this look to you?
>
> Here's mine. The main reason for any differences is that I've gone for the
> most popular comma in each range, even if the symbol must have a different
> comma meaning in olympian.
> ...

Then only thing about which we disagree, then, as far as herculean is
concerned, is whether symbol definitions should be independent of
resolution (which is more of a theoretical than a practical issue).
And the only thing remaining is to finalize the boundary tweaking for
those couple of degrees that require it.

Thank God it's almost the weekend -- I desperately need a break from
all of this!

Best,

--George

Dave Keenan
Posts: 1990
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: consistent Sagittal 37-Limit

Re: An Olympian proposal
From George Secor 26/05/2005

--- Dave Keenan <d.keenan@bigpond.net.au> wrote:

> At 06:03 AM 21/05/2005, George Secor wrote:
>> First off, I believe that the primary role of a symbol should be
>> *independent of the resolution* and that all primary roles should
>> therefore be *defined at the olympian level*.
>
> I agree, if primary role is defined as the role to be assumed in the
> absence of any other information. I recently took the use of "other
> information" to an extreme level in defining the athenian smart
> defaults.
>
> But the mere fact of knowing that the symbol is being used in the
Herculean
> manner is additional information. And so
slightly-smarter-than-primary-role
> defaults exist without even knowing what nominal and sharps or flats
are
> involved. It is these that should be used in the sag_jiX.par files
for
> converting from notation to ratio. Do you agree?

Yes.

> It is these that I'm referring to. But you're right, I shouldn't call
them
> primary roles. Herculean defaults?

Yes, that's good.

>> Next, I would like to establish a "doctrine of reasonable
expectation"
>> (DRE):
>
> I don't see much value in a name like that. There's no objective way
of
> determining what is a "reasonable expectation" other than by surveys.

Oh, come on! If we first introduce "e" (meaning any athenian symbol)
and then later ' and . as 5s-accents as something that can be added to
alter e, then it's reasonable to expect that 'e and .e will be the
notation for 5s-altered athenian-level commas. You don't need a survey
for that -- but perhaps I need a better, more specific name.

>> If two reasonably popular ratios differ by a 5-schisma, and if
>> one of the two has as its primary role an unaccented symbol, then we
>> would *reasonably expect* the other to be the primary role for the
same
>> symbol with the appropriate left-accent added. 7:11k and 11:35k fit
>> these conditions. (If one of the two ratios were of very little
>> popularity or usefulness or overly complex, we would then reasonably
>> expect a simpler ratio to be used in its place.)
>
> I agree it is a good principle that a 5-schisma difference between
> most-popular commas should be represented by a left accent difference
> between their symbols (the "5sC=LAS" pronounced "five sclas"
principle?),

Okay, that's a better name than DRE, but your abbreviation is too long.
(What does "LAS" stand for? "Left-accented something"? Oh, I finally
managed to figure it out a day later: "left-accented symbol" -- duh!

> but it clashes with others.

Yes, I know some of these principles are going to conflict. But we
possible in a consistent manner.

> At the olympian level at least, it applies
> equally to these non-athenians:
>
> )| 19s .)| 5:19n
> ~| 17k .~| 85k
> ~~| 125C .~~| 25C
> )|~ 19C .)|~ 95C
> )/| 5:19C .)/| 19:25C
> ~|) 49S '~|) 245S
> |~) 5:13S '|~) 13:25M
> )//| 5:13M .)//| 13:25S (if these exist)
> (/| 49M '(/| 5:49M
>
> But applying it consistently in this way will lead to non-monotonic
cores
> (core crossovers).

Yes, I agree that avoiding core-crossovers is also a desirable
principle.

> Even among athenian cores we have
> .|\ .(| |\ (|

Yes, I appreciate your point. I would say that 5sC=LAS would not be
applied to any symbol core if it resulted in a core-crossover with a
nearby athenian-cored symbol (thus excluding most non-athenian cores
from application of the 5sC=LAS principle). I think that will address
most of the examples you gave above.

I was thinking that there's no crossover with 5:49M, but I see that
/|\ 11M '/|\ 55M
would cross over (/| -- only 55M isn't very simple, so I don't think it
would be needed very much. Try notating 55/32 and you'll see that you
can't use '/|\ with 1/1 from C to C# -- only with G# as 1/1 do you
begin to encounter it as F'/|\, the principal spelling. So I would
recommend using an olympian sequence that excluded '/|\, such as this:
... /|\ /|\' /|\'' or (/|. (/| (/|' (/|'' '(/|. '(/|
...
with the choice of the 3rd symbol in that sequence depending on where
the herculean boundary between /|\ and (/| was set, in order to achieve
DAFO (drop accents for herculean -- from olympian). (Sorry, but I'm
not finished with the issue of herculean boundaries, as you'll see
below.)

But I see that this also requires choosing .(|)' in preference to |\),
since 49/32 won't require |\) until 1/1 is D#. Hmmm, looks like .(|)
should be in the herculean set instead of |\), but that (/| should
stay, which could give us this olympian sequence:

... '(/| .|\) .|\)' .(|). .(|) .(|)' (|)'' (|)' (|)
...

Although this would upset our penchant for symmetry, it has the
most-probably-used non-right-accented symbols. Perhaps
complement-symmetry should be mandatory only in ET symbol sequences.

>> This, taken together
>> with DAFA, is why I view an accented athenian core as being less
>> complex than a non-accented non-athenian core.
>
> The above are each the most popular commas for their respective
minas. I
> submit that the application of this principle at the olympian level
should
> have little to do with athenianity and much to do with popularity.
>
> If you want to eliminate a core crossover, the symbol for the least
popular
> comma should go.

Yes, I would tend to agree with that, and I think that would eliminate
most left-accented non-athenian cores.

> So in fact isn't DAFA the _only_ reason you view an accented athenian
core
> as being less complex than a non-accented non-athenian core? (which
seems
> to me to be circular reasoning).

No, there are two other reasons:

1) Every non-athenian core introduced is one more core for which the
meaning must be remembered, thus, the more non-athenian cores
introduced, the more complex the notation. If you introduce a new
symbol core into herculean, you should have an extremely good reason
for doing so, as with the following five:

(/| is required, because accented /|\ and (|) symbols will not fill out
a complete sequence between the two unaccented symbols.

|\) was added, because it's the complement of (/| -- and now I'm
questioning whether it's really necessary (or even desirable) in
herculean.

~~| is required, because an accented athenian symbol will not fill out
a complete sequence two degrees below /| .

|~ was added because 5sC=LAS would have given '~|(, which represents a
rather complex ratio.

~|) was added because 5sC=LAS would have given '(|, which represents a
rather complex ratio, and also to avoid a fairly complicated
combination of accents '(|' for an important notational comma, 49S.

2) I said that 5sC=LAS was a reason that was to be "taken together"
with DAFA for regarding accented athenian cores as less complex.

>> Besides, 11:35k isn't even the right number of minas for )~| to be
its
>> primary ratio.
>
> I remind you that 11:35k isn't the right number of minas for ')|( to
have
> it as primary comma either. We shifted )|( too.

Well, sure, if the (unaccented) core gets shifted, then so do its
accented versions.

So then what? Do we then shift whatever else might strike our fancy,
such as )|~ down one mina to use for 25C, or )~| *up* one mina for
5:17C, or /|~ down one mina for 23S?

Oops, we *did* do that last one some time ago -- tee hee! But,
then again, 23S doesn't have an athenian-cored symbol 4 minas distant
that could have been accented (and we didn't have accents at the time
we shifted its symbol).

My point is that left-accented athenians are *perfectly* good for
notating commas such as 11:35k, 5:17C, and 25C and since 5sC=LAS in
each of these instances, then there's a perfectly good reason *not* to
introduce a new symbol core for any of these. Now there is one
exception: |\. will be used for 7:55C instead of .(| to avoid crossing
over the unaccented athenian |\ symbol used for the more popular 55C.

If we replace |\) with .(|) in herculean, then we've limited the number
of non-athenian symbol cores to four, and only one of the four needs to
be accented.

>> You were suggesting a one-mina downward shift in order
>> to accomplish this,
>
> Yes. Its the same shift as required for ~|) for 49S. So we
effectively have
> the same alternative definition of ~: in both cases.
>
>> but I think I've given enough justification to use
>> a different comma-definition (143C) for that symbol.
>
> 143C is the most popular for 25 minas but it is far less popular than
> 11:35k. 23 deemed ocurrences versus 63. What's wrong with ')|(' for
> 143C?

There's nothing at all wrong with that. But the only reason we need
)~| is for promethean and to notate some ET's without using accents, so
I would give it only a token appearance in olympian: in only a single
mina (sequence shown below), without any accents. It would also appear
in promethean, where its default value would be 11:35k. That could
then serve as a basis for notating certain ET's "promethean-style",
without accents. ET's that require at least some left accents could be
notated "herculean-style", and ET's that require right accents would be
notated "olympian-style".

Herculean and promethean are opposite approaches to expanding the
spartan and athenian symbol sets: one simplifies by keeping the number
of symbol cores down, while the other simplifies by not requiring the

> I'm sorry, I've forgotten your justification.
>
> When we add new unaccented symbols, shouldn't we use the most popular
ratio
> that's within one mina of its sum of flags. The only reason ')|( is
11:35k
> is because we moved )|( to 7:11k.

Before the shift, 11:35k would have been ')|(', which is a bit more
complicated than ')|(, so the shift simplified the symbols for both
7:11k and 11:35k. So what's wrong with ')|( for 11:35k? In addition
to 5sCLAS, it would allow both DAFA and DAFO.

>> I might add that DRE would also mandate that the most popular ratio
one
>> mina higher than )~| get .~|( for 5:17C,
>
> 5sCLAS would have .~|( for 5:17C but DAFA and MTC (monotonic cores)
> would not.

You could have both MTC and DAFH for both 11:35K and 5:17c if the
olympian sequence were:
... ')|( )~| .~|( .~|(' ~|(.. ~|(. ~|( ~|(' |~.. |~. |~ ...
and the herculean step for ')|( were subdivided into ')|( and .~|( --
something I'm seriously considering (among other things) as a result of
our latest discussion.

You could also have DAFA if the althenian lowerbound of ~|( were
lowered about 2 minas, but I'm not recommending that at this point,
because it's probably too much. I'll just accept the idea that you
can't have everything everywhere. You also won't have DAFO if 11:35k
is )~| and ')|( is used in herculean; however, if you have )~| for
11:35k in both olympian and herculean, then you won't have DAFA.

>> so I anticipate that the )~|
>> might never be used with any accents (except to subdivide a mina,
>> possibly for 11:35C).
>
> Note that I'm saying that both ')|( and )~| should be 11:35k. The
main use
> I see for )~| is for ETs that would otherwise have only one or two
accented
> symbols, to avoid accents entirely.

suggestion above for )~| as primary ratio of 143C but with 11:35k as
its *promethean default*. This minimizes the number of symbol cores in
other (herculean-style) contexts, but allows this (and other) symbol
cores to come to the fore with their own (not necessarily primary)
promethean default ratios, as needed.

BTW, I happened to notice that )~| is valid as 11:35k, 143C, 5:17C, and
as the sum of its flags in 270, 306, 364, 388, 400, and 494. It's also
valid for all of these less one in 288 (not for 143C) and 311 (not as
sum of flags, which is not a problem). So it looks as if this is going
to be a very useful symbol, albeit within a somewhat limited scope.

>>> ...
>>>> 8* '~|( 85C 2^27:3^13*5*17 ***
>>> ...
>>>> Degree 8 has a complex ratio, hence will be an opportunity to
consider
>>>> a non-athenian (unaccented) symbol (guess which one!).
>>>
>>> There is of course only one. |~ as 23C.
>>>
>>> The most popular commas for 8degHerc are
>>> 7:25C 45 deemed occurrences |~..
>>> 23C 39 |~
>>> 1225C 38 |~.
>>>
>>> So it would be difficult to argue for .~~|
>>>
>>> However, it looks like |~ should have primary role 7:25C in
herculean but
>>> 23C in olympian.
>>
>> 1) I think that primary roles should be independent of resolution,
so I
>> would regard |~ as approximating 7:25C in herculean;
>
> OK. But 7:25C is the most likely value for it in Herculean and so
should be
> the default listed in sag_ji3.par. But I agree not to call this a
"primary
> role" in future, but rather the "herculean default".

Agreed.

>> 2) For 25/14 of C, 7:25C would be used as an alternate spelling
A#!~,
>> after the preferred spelling Bb'|(, whereas |~ is used in the main
>> spelling of 23/16.
>
> Agreed. But even so, 7:25 is so much more popular than 23 that this
> alternate spelling of 7:25 could well be a more popular use of the |~
> symbol than 23 in herculean. But it is close.

Yes, that's possible. I see 3 possible cases of alternate spellings:
1) When there are two possible spellings, each of the two will likely
be used, but generally one more often than the other.
2) If there are three possible spellings, and one of them is clearly
preferred far ahead of the other two, then the other two will rarely be
used.
3) If there are three possible spellings, but none is clearly preferred
far ahead of the other two, then one of them will almost never be used,
and this will, for all practical purposes, be the same as case 1.

For 28/25 you have the following three olympian spellings with C=1/1:
Cx!/' D.!( Ebb|~..
I would classify this one as case 2. For 25/14 it's case 3, with
Cbb|\. being the almost-never-used spelling. The picture may change
slightly when you use a different nominal as 1/1, but you get the
general idea.

>>>> Degree 9 will require a new non-athenian symbol, since no
>>>> athenian-cored symbol is possible.
>>>
>>> Most popular commas for 9deg58-EDA
>>> 125C 63 do (deemed occurrences)
>>> 11:49C 30 do
>>> 11:25C 30 do
>>> 95C 23 do
>>>
>>> 125C is [26 -12, -3>, the comma least likely to be used to notate
125. My
>>> formula distributes the 2,3-reduced occurrences of 125 in the
Scala archive
>>> as follows
>>>
>>> 125S 326 do
>>> 125M 103 do
>>> 125C 63 do
>>
>> You'll have to explain how you came up with that, because at this
point
>
> Darn. I was hoping you weren't gonna ask me that. 'Cause I've
lost the
> spreadsheet and can't remember. I'll look some more, but in the
meantime,
> why not tell me how you would distribute them.

With C=1/1, there are only 2 spellings: C'\\! and B/|) -- case 1,
second spelling perhaps more likely, but only slightly.

With E=1/1, there are 3 spellings: E'\\! D#/|) Fb~~! -- case 3, third
spelling almost never used.

With Db=1/1, there are 3 spellings: Db'\\! C/|) B#.|)' -- case 3,
third spelling almost never used.

I imagine it would translate into something like 54% for 125M, 45% for
125S, and 1% for 125C.

>>> Agreed. But if |~) is ever used, I think its primary role should
also
>>> be 5:13S.
>>
>> We'll be discussing that when we get to olympian.
>
> I think we have to do olympian, herculean and promethean (if we have
it at
> all) more or less simultaneously.

Yes, you're right! And I hope I've strengthened the justification for
the existence of promethean by what I said above.

> ...
> Is it conceivable that 1 in 10 occurrences of 125 (with various
factors of
> 2 and 3) might be notated by using 125C? I guess you're saying no.

Yes, that's a "no".

>> So I don't imagine that we'll need an exact olympian-level symbol
for
>> 125C. You've taken this (slope, I believe) into account in your
>> comma-popularity rankings, but I think you've given much too much
>> weight to third-place spellings that will hardly ever be used.
>
> That's funny, because when I designed the distribution formula I
remember
> thinking that although I personally wouldn't bother with third-place
> high-slope spellings, you seemed to often think them worthwhile,

I did? Well yes, I guess I did say that symbols should be provided for
all possible spellings, but that doesn't mean that third-place
spellings have to be the *primary* roles for their symbols.

> so I'd
> better not weight them too low. It seems I may have overdone it.

Caution: Our thoughts are subject to oscillation. :~}

> But be aware that if I change it in some uniform way, you may not
like the
> effect it has on other (e.g. second-place) alternate spellings.

You'll have to look at some examples and figure out what sort of
adjustment is appropriate for the relative powers of 3. When you see
double-sharps or double-flats, then I would say you're somewhere around
the 1-percent category.

>> On the basis of usefulness, the contest for the primary ratio for
~~|
>> is between 11:49C and 11:25C. I would select 11:25C, because it has
a
>> lower combination of primes. Subdividing the mina would give 11:49C
>> |~'' as its exact symbol. Then let 125C be approximated by one of
>> these.
>
> OK. So I'll fiddle the formula (if I ever find it) to make 125C come
out as
> just slightly less popular than both 11:25C or 11:49C. I expect
11:25C will
> win then too.

Wait a minute -- 11:25C and 11:49C should each be much more popular
than 125C! Those would each be used for the preferred spellings of the
ratios they notate (case 2).

>> ...
>> Okay. I couldn't tell that from your olympian.xls table, since you
>> only ordered them *within* each mina. I don't remember whether you
>> gave me an extensive popularity listing.
>
> I'll send you one when I fix the 125C thing.

I just might want to see an extensive popularity listing for the ratios
taken collectively, without any allowance for powers of 3. Thus 125
would be listed without specifying whether it's M, S, or C.

>>> ...
>>> I'm totally confused as to what you intend here.
>>
>> Sorry! I used a left-accent when I meant a *right* one.
>
> Oh dear. I thought no one would ever do that. But of course it
only
> happened because it was a non-athenian core. Right accented athenians
are
> so much simpler.

Yes, especially in herculean, where you can't see the accents. d-;

>> This is the
>> sort of olympian sequence I had in mind:
>>
>> minas symbol exact ratio
>> ----- ------ -----------
>> 70 (|'' 17:25S
>> 71 ~|).. ??? instead of '(|. 847S = 7*11*11S
>> 72 ~|). 25:49S instead of '(|
>> 73 ~|) 49S
>> 74 ~|)' 5:17S
>> 75 .(|(. 11:23S instead of ~|)''
>>
>> This allows one to drop accents for herculean (from olympian, DAFH)
for
>> the above range of symbols (plus quite a bit more above and below).
>
> That's OK, but when we get to olympian,

The above symbols *are* olympian.

> what about the 5sCLAS principle and the DAFA principle
> and the MTC (monotonic cores) principle?
>
> 77 '~|) 245S = 5*49S

Well, with a non-athenian symbol core you can't have DAFA no matter how
you look at it. My sequence *does* have MTC, although that eliminates
5sCLAS. So that's one out of three.

If it's any consolation, note that 254S is not the preferred spelling
for 49/25 (taking C=1/1):
preferred: B~|(.
second: Cb.(|(' or Cb'~|)
third: Ax)!('

> For what it's worth, I have 847S = 7*11*11S for ~|).. at 71 minas.

Yes, that's nice. I was looking for something simpler than 17:19S.

>>>> The only non-athenian symbol available to compete for the
position is
>>>> '|~), which has a couple of possible ratios: 7:115M and 187M.
>>>
>>> Huh? What about 13:25M for '|~) ?
>>
>> If //|'' is 5:13S, then I would reasonably expect 13:25M (a
5-schisma
>> larger) to be '//|'', which could become '//| in herculean.
>
> What has that got to do with whether '|~), if used, should have
primary
> role 13:25M or not?

I'm having trouble with two different symbols having the same primary
role. If you have a spreadsheet to give you the olympian notation,
then there would be more than one result. There's a solution to this,
but I don't have time to elaborate on it now.

>>>> think I've found a good reason for a double-right accent in
combination
>>>> with a left accent.
>>>
>>> Maybe you've found a good reason for a new unaccented symbol for
this
>>> region. With primary role 5:13M. e.g. )//|
>>
>> Then ')//| would be exactly 13M. Were you also thinking of having
)//|
>> instead of ./|) in herculean? I'm questioning whether there's
>> sufficient justification for another symbol that introduces even
more
>> redundancy in that:
>>
>> 13:25M could be either )//|. or '|~), which are equivalent to '//|''
>>
>> Double-right accents show up only in olympian, and it's simple
enough
>> to understand and remember how they work, and this is a region where
>> they can be dropped quite easily for olympian-to-herculean
conversion.
>> If '//| and ./|) will cover the herculean range in this area, why
>> introduce a new symbol core?
>
> It's the same reason as for the existence of other non-athenians (and
now
> non-herculeans) - to notate some large ETs which would otherwise have
one
> or two accented symbols.
>
>>> Then 13:25M could be )//|. (right accent).
>>>
>>> But why wouldn't 13:25M be '|~) in olympian?
>>
>> Only if |~) is 5:13S. That, plus the question of how (and whether)
>> 5:13S and 7:23S should have separate (exact) symbols, is an issue
we'll
>> have to take up when we discuss olympian.
>
> Since we've agreed on herculean, it looks like we're now discussing
either
> olympian or promethean. I think both of them require a decision on
this.
> Then there's the possibility that |~) is not used in any JI notation,
but
> only in large ETs. We'd still have to decide what comma it is. And
why
> wouldn't it be the simplest in range?

I've been thinking about that and have an answer, but since this is
still subject to oscillation, I'll tell you later.

>>>> So here's what I get for the herculean symbol sequence:
>>>>
>>>> '| )| |( '|( )|( ')|( ~|( |~ ~~|
>>>> ./| /| '/| .|) |) '|) |\ (| ~|)
>>>> .(|( (|( .//| //| '//| ./|) /|) '/|)
>>>> /|\ (/| '(/| |\) (|) .(|\ (|\ '(|\
>>
>> Evidently we agree (whew!).
>
> It isn't pretty, and it isn't memorable. It would be nice to have a
> reasonably short account of why it's the best. A few principles
> consistently applied. Even if it's not exactly how we arrived at it.
> Because people are going to ask. I want to be able to make a
> that comes up with it automatically. The boundary tweaking seems
> particularly difficult to justify.

I have an idea for herculean boundaries that gives up equidistance in
order to gain more 5sC=LAS, DAFA, DAFH, and MTC, and also fewer
instances of symbols having both left and right accents when you add
accents for olympian. Rather than elaborate on it now and run the risk
of oscillating back and forth on the details, I need to work on it and
see how well it comes together before I say any more. I'll need at
least a few days (it covers everything from athenian through olympian),
so it will probably not be till next week before I've settled most of
it.

>> Then only thing about which we disagree, then, as far as herculean
is
>> concerned, is whether symbol definitions should be independent of
>> resolution (which is more of a theoretical than a practical issue).
>> And the only thing remaining is to finalize the boundary tweaking
for
>> those couple of degrees that require it.

So I thought -- before my latest brainstorm.

> Yes. Will you put together the relevant sag_ji3.par file?

Okay.

> Let's not send it to Manuel until we have the olympian and possibly
> promethean too.

Yes, I agree.

>> Thank God it's almost the weekend -- I desperately need a break from
>> all of this!
>
> Yes. Congratulations, we have agreed on herculean symbols and
boundaries
> for the first time ever.

No, not the boundaries yet. (Groan!) Not till my latest brainstorm
runs its course.

> I'm sorry you had to drag me kicking and screaming
> for so much of the way.

I hope I don't have you kicking and screaming too much next week.

> How about a 455nC=RAS principle and a ???nC=DRAS principle?

Huh? What? Duh -- now I get it! I don't know how often we could
apply that. There might also be a 4375nC=RAS alternative where 455n
doesn't work. And that other one might be a 65:77nC=DRAS principle.

--George

Dave Keenan
Posts: 1990
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: consistent Sagittal 37-Limit

Re: New comma popularities
From George Secor 22/06/2005

Hi Dave,

I thought I'd give you a progress report, just to let you know that
there's been a lot of progress -- slow, but sure.

I've gone through the popularity list, comma by comma, up to 49:125C,
which was a little more than enough to assign ratios to all of the
(unaccented) symbol cores (and almost all of the accented herculean
ones). I then copied the ratios in the list to another worksheet,
added a column that calculates the number of minas for each, and then
sorted it in order of number-of-minas + popularity. I then looked at
the minas that didn't already have ratios assigned and determined
primary ratios and olympian defaults (if different) for each.

>>> 2. Since there's such a thing as a Herculean default (58-EDA) for
a symbol,
>>> which may be different from its once-and-for-all primary role,
then might
>>> there not also be Olympian defaults (233-EDA) that are different
from
>>> primary roles derived from considering all possible accented
symbols.
>>
>> I hope not. But I do think that there should be promethean defaults
>> for symbols that are different from their primary roles.
>
> I can accept that there might be a few promethean defaults that
differ from
> their primary roles, but there had better be a simple rule explaining
why,
> not a bunch of random exceptions.

<< There will be -- with no exceptions! >>
but that was without thinking that we had tentatively agreed that all
primary ratios would be 23-limit. If the most popular ratio for a
particular number of minas is >23-limit, then that would be the
olympian default, but the primary ratio would be the most popular
23-limit ratio.

Here are the instances in which I think that this principle can be
applied without controversy:

# of default primary
minas ratio ratio
----- ------- -------
1 31:49n 455n
2 13:37n 65:77n
11 11:31k 605k
22 5:53k 5:161k
31 5:47C 7:143C
42 19:73C 19:169C
59 13:47C 605C
66 53C 19:49C
69 29S 13:17S
75 47S 11:23S
91 499S 17:49S
98 83M 5:187M
108 7:29M 2375M
110 47M 11:85M

role of slope in the popularity list, and I've found a few instances
where it does result in an undesirable popularity placement for certain
ratios. I have only one of these handy:

I don't think it's appropriate to exclude 7:23S in the popularity list
in favor of 7:23C, since the only two possible spellings for 23/14
(with C=1/1, without using a new symbol core) are A'\!)' and G#//|'',
requiring 7:23M and 7:23S, respectively. I think we're going to have
to re-evaluate the comma-diesis-kleisma distribution, because it may be
shifting the popularity of other ratios in ways I didn't expect.

I also found ratios that, unexpectedly, are *completely absent* from
the popularity list:

1) 7:19C is in the list, but why is there no 7:19-anything else? I was
looking at ratios for 122 minas and found that 7:19L and 23:25L seem to
have the lowest numbers in their ratios. (I think that this will be
7:65L, anyway, but I thought I needed to ask the question.)

2) Search the popularity list for 7:115-anything and you won't find it,
even though 7:115M has a very simple ratio, 112:115. What you do find
are the ridiculously obscure ratios 67:115C and 67:115M. There's no
way you're going to convince me that you can make a ratio more popular
(in order to make the list) by replacing prime number 7 with prime
number 67, so I can't help thinking something got overlooked!

On another note: It doesn't seem appropriate that the pythagorean
comma, 1C, be given equal popularity with the unison, 1n. In fact, I
would think that the unison should be at the very top of the list,
since it's in each and every scale, and the pythagorean comma (a much
less-preferred spelling) should be somewhat lower. I'd find it
appropriate (and convenient to have it somewhere between 5s and
5:7C.

I'm going to be reviewing the rules I've been applying for assigning
ratios, and I'll also be reading your recent messages that I haven't
answered yet, and perhaps taking several days to think about all of
this before I show you what I have.

Best,

--George

Dave Keenan
Posts: 1990
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: consistent Sagittal 37-Limit

Re: Comma lists
From George Secor 13/04/2007

--- Dave Keenan <d.keenan@bigpond.net.au> wrote:

> .~|('
> ')~|..
> seem to be in the wrong order.

Yep, it would seem so. However, I again determined both a SoCA (using
455n or 65:77n) and an alternate SoCA (using 4375n or 13:125n) for the
3 possible symbols in the 27th mina, which gives the following figures:

)~|'' as 12.897c or 12.883c
.~|(' as 13.199c or 13.172c
')~|.. as 13.186c or 13.200c

Averaging out the figures for each symbol, .~|(' comes out lower than
')~|.., but I don't remember exactly why I split this mina into 3
parts. Since ')~|.. is more complicated than the other two symbols,
and since it's so close in size to .~|(', I now see that it would be
better not to use it. (Thanks for spotting that.)

So here's how I would now assign the symbols.

The five least complex commas for this mina are (in order of size):

Cents Name Complex Pop. Rank
------ ------- ------- ---------
12.943 19:121C 49.462 962
13.066 11:133C 44.456 817
13.074 1715C 46.916 110
13.189 35:247C 44.117 924
13.269 11:23C 34.857 129

Taking both complexity and popularity rank into account, 11:23C gets
priority for a symbol assignment, with 1715C in 2nd place. They both
compete for .|(', but 11:23C gets it, since it's closer. The choice
for )~|'' is clearly 19:121C. With these two symbol definitions, the
symbol boundary would be at 13.106c, which puts 1715C in the )~|''
range, giving the two most important commas in the 27th mina separate
symbols.

You assigned ')~|.. to 1715C, so if we eliminate that symbol, we can
delete that assignment.

You assigned .~|(' to 78125C -- whoa, that's 5^7, which is much more
complicated and less popular than 11:23C. (However, if you cut the
prime limit below 19, then this one would get it.)

You assigned )~|'' to 5:847C (SoCA); however, this is 26 minas, which
is not atomically correct. Since there are at least 3 or 4 commas less
complex and/or more popular than 5:847C in the 26th mina, we shouldn't
be obligated to give it a symbol, which would then allow )~|'' to be
used in the 27th mina.

I'll be looking at the rest of your spreadsheet over the next several
days to see what else needs attention.

--George