## consistent Sagittal 37-Limit

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

cmloegcmluin wrote: Thu May 28, 2020 10:11 am
George Secor wrote:
# 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
Here is evidence of a historical difference between default and primary values(/commas/ratios). I thought we had discussed this somewhere and decided there was no meaningful difference between default and primary. But it would seem at some point there was!

I understand the concept of default values per precision level which potentially deviate from the "once-and-for-all" primary comma... but I really don't like it. And it would seem it wasn't retained in the end.

If, at least at some point, 47S was to be merely the default value in Olympian, while 11:23 was the "once-and-for-all" primary comma, I feel like our confidence in dismissing the splitting of the 75th mina to include the 47S has grown that much stronger.

Perhaps we (well, you know, Dave) could now find evidence of where/when/how exactly the concept of precision level default values was eliminated. Then we'd be that much closer to simplifying this bit of the Extreme precision level notation.
I don't think it matters too much where when or how it was decided that default commas would be the same as primary commas. But I do think it would be worthwhile to look at what became of all those examples other than the 75th mina.

You might extend that table with a column that tells us which of the two commas became the present-day combination default/primary comma, "default", "primary", "both" (by splitting the mina) or "other", and another column that tells us which of those commas, "default", "primary", "both" or "other" has the lowest SoPF>3.

cmloegmcluin
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### Re: consistent Sagittal 37-Limit

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


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

Thanks. I look forward to those. A third new column that might be relevant, is one that tells us which comma has the lowest number of notional occurrences in the Scala archive, as per column CD of the popularityOfCommas spreadsheet. [Edit: See more convenient spreadsheet two posts down.]

I reverse-engineered how I apportioned the number of occurrences of a 2,3-reduced ratio in the Scala archive, between the 2 or 3 commas that could be used to notate it:

• Only commas with absolute 3-exponent of 12 or less, and size less than half an apotome, were considered.
• Each comma's apotome slope was calculated.
• Each comma was assigned a weight that was 12 minus the absolute value of its slope, or 0 if its absolute slope was greater than 12.
• The number of occurrences of a 2,3-reduced ratio was apportioned among its two or three commas in proportion to their weight.
i.e. notional_comma_occurrences = ratio_occurrences × comma_weight/(sum of comma_weights)

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

I just realised I have no idea what the complexity is that's being used below. I assume the popularity rank is for the ratio being notated, not the comma itself. [Edit: Wrong. See next post.]
George Secor wrote: >> 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
>
> I think we can ignore those with popularity rank above about 500.

Remember that you said this, because I'll bring it to your attention
below.

>> 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.
>>
>> Now I'll look at your spreadsheet selections and comment.
>>
>> You assigned ')~|.. to 1715C, so if we eliminate that symbol, we can
>> delete that assignment.
>
> If we eliminate that symbol then I'd end up assigning it as a
> secondary comma for .~|(' 11:23C. I don't think that considering
> alternate right flags would change that.

That's not what I had in mind. If at all possible, I would want
different symbols for 1715C and 11:23C, the two commas most worthy of
notating. I don't think this is an unreasonable expectation,
considering that they're ~0.195c apart.

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

I found a more convenient comma popularity spreadsheet. It uses the notional (or deemed) occurrences calculated in the one already posted here, but it has them sorted in descending order of popularity, and has the systematic comma names. It was dated 2-Nov-2005.

The popularity rank in George's table above, is the rank of the comma in this spreadsheet.
Attachments
CommaPopularities2.xlsx

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

This email from George might be useful:
9-Jun-2005
Subject: Re: New comma popularities

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

> At 07:45 AM 8/06/2005, [George] wrote:
>> Sorting it all out takes time, and we can't rush through this as if
>> we're on a deadline.
>
> You're right. Sorry.
>
>> As a result, I've come up with a procedure that, so far, arrives at the
>> entire athenian set (both the symbols and their primary roles, without
>> skipping over any comma-ratios or any other exceptions), with the only
>> "givens" being the comma-popularity list, the Sagittal flags and
>> accents, and athenian-level JI defined as a division of the apotome
>> into 21 (more or less equal) parts.
>
> Sounds brilliant.

It isn't really difficult at all, if you're willing to accept (and
enforce) a one-to-one correspondence between symbols and their primary
roles, such that any symbol will always have the same primary role
(regardless of JI resolution) and a comma-ratio can be the primary role
of one (and only one) symbol. That, taken together with a rule that
new symbol cores should *not* be introduced if a left-accent applied to
an existing symbol core (not previously disqualified from being
athenian-level) will notate a ratio *exactly*, pretty much does the
trick. This has the consequence of maximizing
drop-accents-for-lower-resolution (DAFLR, my latest, and hopefully now
best-and-only acronym for this property) from olympian to herculean to
athenian.

This may also produce some unintended or unexpected consequences as I
get farther down the popularity list, but that remains to be seen.
I've found that the results so far tend to favor the creation of
monotonic core sequences, but I haven't gotten far enough along yet to
formalize any rules for that, other than prohibiting the creation of
left-accented symbols that would result in the unaccented and accented
symbol pair ending up on opposite sides of an unaccented athenian-cored
symbol.

You had earlier said that you would like to avoid having to go through
olympian mina by mina, deliberating over what should be each symbol and
primary role. I'm finding that most of them are pretty obvious and
that there should only be a handful or so left to debate -- not
that a handful won't cause us a lot of trouble. ;-(

>> It also simultaneously derives
>> herculean, promethean, and olympian symbols and their primary roles as
>> far as I've taken it, and I'm hoping that a continuation of this
>> procedure will bring it to completion. I'm having to determine some of
>> the details of the procedure as I go along, but so far (through 245S)
>> it's looking *extremely* good.
>
> That's exciting.
>
>> I've reached a point where I have a question that's stopped me in my
>> tracks: How did the next comma, 23S (16384:16767), get counted as more
>> popular than 23C (729:736)? The size of the numbers in the ratio alone
>> looks suspicious.
>>
>> With C=1/1, 23/16 may be spelled either F#|~ or
>> Gb~|\, which means we're altering a nominal 6 places in a chain of 5ths
> >from C in each case. Looking at it from a composer's perspective, I
>> would probably want to spell it as F-something if I were treating it as
>> a dissonance, which I would most likely resolve by moving up to G.
>>
>> As I see it, everything points to having 23C appearing above 23S, so
>> please explain how you got this.
>
> The comma complexity is misleading. As you say, the absolute exponent of 3
> in both cases is 6. You get the different comma complexities because in one
> case the 23 is on the same side of the ratio as the 3^6 which means it
> takes a larger power of two to counteract it.

Yes, but that's 23S, not 23C.

> You will remember that "slope" is the rate of change of comma size with
> notational fifth size when the comma size is expressed as a fraction of the
> apotome. It's a measure of how useful a comma is likely to be in notating
> temperaments as well as rational tunings.
>
> 23S has the 3^6 on the larger side of the ratio and the apotome has 3^7 on
> its larger side, which means that 23S will remain almost the same fraction
> of the apotome as the notational fifth varies in size, while 23C goes the
> other way more rapidly.
>
> As you can see in the comma-popularities spreadsheet, 23S has an absolute
> slope of 3.5 while 23C has 7.0. So 23C is not as useful notationally
> because it will tend to change its position too wildly relative to other
> symbols.

That's nice to take into consideration when you're notating
temperaments, but what relevance does it have to notating JI?

> If you can think of a simple formula that can take into account knowledge
> as seemingly specific as "... spell it as F-something if I were treating it
> as a dissonance, which I would most likely resolve by moving up to G", then
> please let me know.
>
> Maybe it's just that relative to C as 1/1 we like to have our "compound
> nominals" in the range of Ab to G# (-4 to +8 fifths), so F# is preferable
> to Gb. I could adjust the formula to give that some weight.

Did you intend to have both Ab and G# in that range, or did you mean Eb
to G# or possibly Ab to C#?

Db should also be preferable to C# (for reasons relating to 17/16), so
I think that range should be Db to F#. Yes, you could try something
like that.

> How is this actually impacting on your results (so I can figure out how
> much I need to adjust things to get the results you want)? e.g. Tell me
> what other comma(s) you need 23C to be more or less "popular" than.

I'll have to work down the list through 23C before I know. It's very
possible that there may not be any impact at all.

>> I said above that it doesn't use a strict EDA scheme.
>
> Yes. But even so, it is useful to have an EDA as a reference when
> interpreting this stuff.
>
>> My objective is
>> to a set of unaccented symbols that will convert more smoothly between
>> herculean and promethean (!!!) than those in 42 or 47-EDA, and I've
>> concluded that it doesn't necessarily have to be an EDA to do that.
>
> I agree it doesn't have to be an EDA. But it shouldn't be too uneven.

I've found that there tends to be less unevenness in the region from |(
to /| than outside that.

> ...
>> (or fielding questions about what's linear or
>> MOS, for that matter).
>
> A silly discussion.

It seems to be pretty important to Paul. He's put a lot of time and
effort into the classification of temperaments, and he doesn't want to
be shot down by somebody out there in academia accusing him of sloppy
terminology, and I imagine he's getting rather frustrated by a lack of
progress in building any sort of consensus. You don't have to look any
farther than some the "silly" details we've be discussing the past year
or two to appreciate that.

Anyway, I'll keep you up to date on my progress as the popularity
decreases, and perhaps toss you some more specifics about the rules I'm
using.

Best,

--George

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

This mentions the two cores that compete for the 75th mina.
From George:
28-Jun-2006

[Dave wrote:]
>
> By the way, which core's primary comma has the fewest deemed
> occurrences

That would be (|~ 11:19M and |\\ 11:19L.

> and how many is that?

Before you get too far with that, I'd better warn you that I'm not very
happy with CommaPopularities2.xls, because I recently discovered that
it threw a monkey wrench into my symbol derivation procedure by putting
~|) above (|( in the popularity list.

For two symbols of comparable popularity that fall within the same
athenian boundaries there will have to be some sort of tie-breaker,
e.g., within the set of ratios notated by the two symbols, the chosen
symbol would be the one notating the ratio in that set having the
lowest product complexity.

[Note: "Product complexity" is simply the product of numerator and denominator.]

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

I found the complexity measure that George is using in that table above. He calls it his "Weighted Complexity" elsewhere, including in the attached spreadsheet from 6-Mar-2007.

For those who want to reverse engineer this complexity measure, to explain it to the rest of us, I note that George has left the formulas only in the first row.

But already I can tell you that it is dominated by the sum of prime factors greater than 3 (SoPF>3).

I'm afraid that's all the archaeology you're going to get out of me for many days. I need to do other (non Sagittal) things.
Attachments
CommaEx8.xlsx

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

Dave Keenan wrote: Tue May 26, 2020 9:56 pm • 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... )
I'm starting to think this is a real possibility. Please post the whole scale here.

As you can read above, at one stage, George was trying to keep all primary commas within the 23-prime-limit. I knew he had been forced to go to prime 37 to get commas for some obscure minas, as shown on http://sagittal.org/SagittalJI.gif, but I wasn't aware there was a 47-comma in there until Cml pointed it out recently.

I haven't found any good reason to split the 75th mina. I'm happy for you to unsplit it and eliminate 47S, in the manner you proposed.

volleo6144
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### Re: consistent Sagittal 37-Limit

Dave Keenan wrote: Fri May 29, 2020 3:14 pm For those who want to reverse engineer this complexity measure, to explain it to the rest of us, I note that George has left the formulas only in the first row.

But already I can tell you that it is dominated by the sum of prime factors greater than 3 (SoPF>3).
The weighted complexity is G+J+K+L, where:
- G (">3") is the SoPF>3
- J ("d+n") is the absolute value of (H-I) times G/5, with H = the number of primes >3 (with multiplicity) in the denominator (smaller value) and I in the numerator (larger value)—7:25k (224:225) has G=17, H=1 (actually -1 in the spreadsheet), I=2, J=(2-1) times 17/5 or 3.4. This is zero for any comma that's one prime against another, so 5:7k, 5:7C, 11:23S, etc. are all zero here.
- K ("3-exp.") is 2^(abs(3exp) - 8.5) times ln(G+2), so 5C (80:81) is 2^(4-8.5)×ln(5+2) = ln(7)/sqrt(512) = 1.95/22.6 = 0.086. This ranks commas with high 3-exponents as really complex (3C is at 7.84 here, and 3s is at 17.2 trillion).
- L ("slope") is like K, but with apotome slope instead of 3-exponent. L = 2^(abs(slope) - 8.5)×ln(G+2). 3C (slope = 10.6) is at 2.88 here, and 3s (slope = 52.8) is at 14.8 trillion.

So, for 5C:
G = 5
J = 1×5/5 = 1.0
K = 2^(4-8.5)×ln(5+2) = 0.0860
L = 2^(2.68-8.5)×ln(5+2) = 0.0343
Complexity = 5 + 1.0 + 0.0860 + 0.0343 = 6.12.

The SoPF>3 is factored into all of these, which does seem to make sense, but complex Pythagorean ratios (such as the aforementioned 3s) can often be ranked as very complex anyway due to K (the 3-exponent term) and L (the apotome-slope term, which will usually be very large for very large 3-exponents—well, unless you're well outside the range of a "comma"—a 53-exponent ratio like 3s will only have a small apotome slope if it's large enough to reasonably be considered a sixth).
Last edited by volleo6144 on Sat Jul 24, 2021 8:20 pm, edited 1 time in total.
5-7
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