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primary comma for ,'|( as 1/1225k or 1/121k ?

Posted: Thu Jun 18, 2020 5:32 am
by cmloegcmluin
In case anyone doesn't know about Scala, you should definitely check it out. One of the best pieces of microtonal software out there. A classic.

Recently @Dave Keenan pointed out to me that Scala's configuration files for Sagittal were out of date, especially with respect to the ET notations, due to the developments on bad-fifth ETs that culminated in the Periodic Table of EDOs. But I also took the opportunity to comb over Scala's configuration files for the 12R and JI notations. In the latter case my main objective was updating the ASCII for the new Olympian accents (replacing right ' and , with left ` and , ); as I went along I caught a couple typos, but there was one thing I noticed that sparked some conversation between @Dave Keenan and me that we thought we should surface to the community for further input.

The symbol :,::'::|(: in Scala was assigned the 1/121k as its primary comma, which has ratio 243/242, monzo [-1 5 0 0 -2⟩. However, in George's JI notation spreadsheet (here, since updated here) — which was my primary reference for the updates — :,::'::|(: was assigned the 1/1225k, ratio 19683/19600, monzo [-4, 9, -2, -2⟩.

I'll reproduce the relevant conversation Dave and I had over on GitHub:
cmloegcmluin wrote: seems like we moved from an 11²k to a 5²7²k on this one?
dkeenan7 wrote: That's hard to understand. 121k has a SoPF>3 of 22 and a 3-exp of 5, while 1225k has a SoPF>3 of 24 and a 3-exp of 9. So 1225k looks like the winner to me. I don't remember anything about this one.

I searched email for 121k and 1225k and only found one message, which is already up here:
viewtopic.php?f=4&t=448&p=1373&hilit=12 ... tinguished
It confirms that the primary comma is 1225k.

But 121k has twice as many deemed occurrences, in the spreadsheet here:
viewtopic.php?p=1639#p1639

It is also 1225k in George's JI Notation Spreadsheet.

George's weighted complexity shows 121k as less complex.
viewtopic.php?p=1650#p1650

So every metric seems to be saying it should be 121k, not 1225k.

Is 1225k the SoFLS of ,'|( for some secondary comma of ,| ?
cmloegcmluin wrote: Yes. The difference between '|(, the 25/7k, and the 1/121k is 3024/3025, or the 7/5²11²n, which is about -0.5724 cents, so that's what ,! would represent were the primary comma for ,'|( to be set (back?) to the 1/121k. That value is solidly inside the secondary comma zone for ,! which ranges between 0.211 and 0.773 cents. I agree with your reasoning above that the 1/121k should be preferable to the 1/1225k.

The only evidence we have against it its empirical existence in the current materials (it appears to be one of those missing from SagittalJI.gif). That and its higher limit (11 vs the 7 of the 1/1225k).
For the time being, the comma was standardized to match the more-recently-updated JI Notation Spreadsheet. But we're a ways away from resubmitting the updated Scala configuration to Manuel, the creator and maintainer of Scala, so we have time to discuss this comma assignment. The JI Notation Spreadsheet can very well be changed.

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One thing I've noticed since yesterday is that the Extreme JI notation includes a second comma with >3 prime content of 1225: the 1225C. If we change the 1/1225k, should we change the 1225C as well? Well, first lets go over what we're working with here.

Just as we say that some of Sagittal's commas are "apotome complements" with each other, insofar as they sum to exactly an apotome, these two commas are "Pythagorean comma complements". Analogously to how the apotome complement of a given comma can be found by negating every term in its monzo and then adding the aptome's monzo, [-11 7⟩, you can find the Pythagorean comma complement of a comma by negating every term in its monzo and then adding the Pythagorean comma, [-19 12⟩.

In the case of apotome complements, we see perfect symmetry about the "half-apotome mirror" at [-5.5 3.5⟩ ([-11/2 7/2⟩), or ≈56.8425¢. This position is identical to the size category bound between S (Small diesis) and M (Medium diesis). By perfect symmetry what I mean is that the two sequences of commas moving in either direction from this mirror have the same prime content:

77/25M, 25/13M, 13/5M, 1/175M, 37M, 11/325M, 13M, 1/35M, 125M, 11/19M, 65M, 1/7M, 625M, 11/5M, 17/11M, 5/23M, 7/275M, 11M, 85/11M, 65/7M, 1/49M, 1/31M, 55M, 11/91M, 595M, 5/49M, 
(mirror here)
49/5L, 1/595L, 91/11L, 1/55L, 31L, 49L, 7/65L, 11/85L, 1/11L, 275/7L, 23/5L, 11/17L, 5/11L, 1/625L, 7L, 1/65L, 19/11L, 1/125L, 35L, 1/13L, 325/11L, 1/37L, 175L, 5/13L, 13/25L, 

There is a Pythagorean comma mirror, too, at [-9.5 6⟩ ([-19/2 12/2⟩), or ≈11.730¢. This position is identical to the size category bound between k (kleisma) and C (Comma*). However, it does not exhibit this perfect symmetry. Symmetrical pairs such as the 1/1225k and 1225C do exist. Other examples are the 17k and 17C, the 25/7k and 7/25C, and the 31/11k and 11/31C. It looks like some commas still in the C size category are paired with n's (schisminas).

So this is all to say that I don't feel we are obligated to change the 1225C along with the 1/1225k. There would have to be other compelling reasons to do so.

*Previously Dave had proposed "komma" as a disambiguating spelling when referring to the specific size category rather than the generalized term. I understand that unfortunately this wasn't well received. Has anyone ever suggested a capitalized "Comma" instead, since, after all, the size category bound uses a capital letter?

Re: primary comma for ,'|( as 1/1225k or 1/121k ?

Posted: Thu Jun 18, 2020 11:46 pm
by Dave Keenan
When I wrote (at 2am) "So 1225k looks like the winner to me.", I meant to write "121k looks like the winner to me.". That's because 121k has a lower sum-of-prime-factors-greater-than-3, and a lower 3-exponent.

It's perfectly reasonable (even desirable) for 121 to beat 1225 in the kleisma size category but vice versa in another size category. It would all depend on the 3-exponents.
cmloegcmluin wrote:
Thu Jun 18, 2020 5:32 am
*Previously Dave had proposed "komma" as a disambiguating spelling when referring to the specific size category rather than the generalized term.
Sigh. That's not true. I suggested "komma" for the generic term.
See https://yahootuninggroupsultimatebackup ... 55471.html
And gave up on the idea soon after, due to the extreme hostility it was met with, when I used it (as the generic term) in an exposition on the standardised size categories, in a later thread that I'd prefer was forgotten.

I think that's a great idea, spelling the specific term with an uppercase C. Thanks.

Re: primary comma for ,'|( as 1/1225k or 1/121k ?

Posted: Fri Jun 19, 2020 3:51 am
by cmloegcmluin
Dave Keenan wrote:
Thu Jun 18, 2020 11:46 pm
It's perfectly reasonable (even desirable) for 121 to beat 1225 in the kleisma size category but vice versa in another size category.
Desirable because it results in a greater variety of intervals able to be notated exactly, I take it?

The 3-exponent of the 121C would be 7, so, slightly worse than the 3-exponent of the 1/121k, which is 5. But more importantly, it's worse than the 3-exponent of the 1225C, which is 3.

Maybe we need to run whatever we settled on for a munged abs3exp from the Magrathean thread against this problem.
Sigh. That's not true. I suggested "komma" for the generic term.
See https://yahootuninggroupsultimatebackup ... 55471.html
Aw, scheisse. If it was the other way around, I can certainly imagine that you'd've hoped I'd intuit it that way. Sorry I failed its test! Or it failed my test.

Yeah, I did think it would be more likely you'd propose it as the specific size category. I did doubt that for a moment — because of its shared initial letter of 'k' with kleisma — but then I realized that wouldn't be a problem, since it could be an uppercase K and kleisma's k lowercase.
And gave up on the idea soon after, due to the extreme hostility it was met with, when I used it (as the generic term) in an exposition on the standardised size categories, in a later thread that I'd prefer was forgotten.
I'm sorry to have resurfaced that event and its associated negative feelings. I knew it was touchy so I could have at least acknowledged that. We'll speak no more of it.
I think that's a great idea, spelling the specific term with an uppercase C. Thanks.
I'm glad it works. One of those things that just cropped up serendipitously as I followed patterns in how I was typing my explanations.

Re: primary comma for ,'|( as 1/1225k or 1/121k ?

Posted: Fri Jun 19, 2020 9:02 pm
by Dave Keenan
cmloegcmluin wrote:
Fri Jun 19, 2020 3:51 am
Desirable because it results in a greater variety of intervals able to be notated exactly, I take it?
Correct.
Maybe we need to run whatever we settled on for a munged abs3exp from the Magrathean thread against this problem.
I don't think we've settled on anything yet.

Re: primary comma for ,'|( as 1/1225k or 1/121k ?

Posted: Sat Jun 20, 2020 1:51 am
by cmloegcmluin
Dave Keenan wrote:
Maybe we need to run whatever we settled on for a munged abs3exp from the Magrathean thread against this problem.
I don't think we've settled on anything yet.
Right. I meant to write "settle".