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

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

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Recently @Dave Keenan pointed out to me that Scala's configuration files for Sagittal were out of date, especially with respect to the ET notations, due to the developments on bad-fifth ETs that culminated in the Periodic Table of EDOs.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.

----

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?

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

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.

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

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.

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

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.

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

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".

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

Using the final LPEI badness metric described in this post, I find the badness of the 1/1225k and 1/121k to be 7.35 and 5.82, respectively. Therefore we should go with the 1/121k.

I'm not surprised, since as @Dave Keenan pointed out earlier, the 1/121k has lower ATE and lower SoPF>3, which speaks to low uselessness and unpopularity scores, respectively, both of which factor into the LPEI badness metric.

Again, this is how it is in the Scala configuration files, but not in George's spreadsheet calculator. We were just waiting for the completion of a consolidated badness metric before returning to this topic to settle on one or the other commas using said metric as an objective approach.

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

cmloegcmluin wrote: Thu Nov 05, 2020 3:39 pm Using the final LPEI badness metric described in this post, I find the badness of the 1/1225k and 1/121k to be 7.35 and 5.82, respectively. Therefore we should go with the 1/121k.

I'm not surprised, since as @Dave Keenan pointed out earlier, the 1/121k has lower ATE and lower SoPF>3, which speaks to low uselessness and unpopularity scores, respectively, both of which factor into the LPEI badness metric.

Again, this is how it is in the Scala configuration files, but not in George's spreadsheet calculator. We were just waiting for the completion of a consolidated badness metric before returning to this topic to settle on one or the other commas using said metric as an objective approach.
It would be nice to tidy up this loose end of the 15th mina. Unfortunately it's not so easy as assuming the .par file is correct and the JI Notation spreadsheet has an error, just because the .par file has the lowest badness comma. I spent way too much time researching this in old emails and so didn't get much else Sagittal done today. Sorry.

I note that the sag_ji4.par file, (which has 121k) is dated 2007-Nov-11, while the email that gives 1/1225k is dated earlier, on 2007-Oct-02. And I looked at 6 different JI Notation spreadsheet versions that George sent me, that are dated both before and after the .par file, and all of them give 1225k. George was the author of all these documents.

Early JI Notation spreadsheets also give 121k, but only as the comma for the "super-olympian" symbol .~|'

Super-olympian is a obsolete never-released notation that went beyond Olympian but did not use any additional flags or accents. It only combined the existing ones in additional ways, and had 36 additional split minas. Its purpose was to help decide what element combinations and commas to use for Olympian.

And by the way, the "X" in Herculean-X stands for "maXimum splits". Herculean-X is what we now call simply Herculean (Ultra), and it provided the DAfLL (drop accents for lower level) property when dropping mina accents from Olympian (Extreme). The original Herculean was much closer to 58-EDA.

I found the email to which the sag_ji4.par file was attached, and there was no comment about changing the comma for the 15th mina, or anything else. I can only assume that George made a transcription error, or copy and paste error, when going from the JI Notation spreadsheet (which at the time still contained super-olympian) to the .par file. i.e. I can only assume he intended for the .par file to have 1225k for the 15th mina of Olympian, just as every other document does.

It took you to find his error, 13 years later.

But that means that if we were to change the spreadsheet, and your code, to use 121k instead of 1225k, we would not be correcting an error, we would be changing an intentional comma definition on the basis of our LPEI badness metric. But we know there are 20 symbols in the Olympian (Extreme) notation that do not have the lowest badness comma in their zones. There's nothing special about the 15th mina in that regard. And I don't want to tackle researching the reasons for all their choices at this point in time.

So. I believe the correct thing to do, for now, is to fix the error in sag_ji4.par, by changing its 15th mina commas (up and down) to 1225k.

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

Dave Keenan wrote: Fri Nov 06, 2020 1:38 am It would be nice to tidy up this loose end of the 15th mina. Unfortunately it's not so easy as assuming the .par file is correct and the JI Notation spreadsheet has an error, just because the .par file has the lowest badness comma.
In other words, "Not so fast..."
I spent way too much time researching this in old emails and so didn't get much else Sagittal done today. Sorry.
Well, it's basically impossible to research anything in old emails without spending way too much time on it. No worries.
It took you to find his error, 13 years later.
We'll never fully escape manual error, but I hope the codebase will help reduce it.
I found the email to which the sag_ji4.par file was attached, and there was no comment about changing the comma for the 15th mina, or anything else. I can only assume that George made a transcription error, or copy and paste error, when going from the JI Notation spreadsheet (which at the time still contained super-olympian) to the .par file. i.e. I can only assume he intended for the .par file to have 1225k for the 15th mina of Olympian, just as every other document does.
But that means that if we were to change the spreadsheet, and your code, to use 121k instead of 1225k, we would not be correcting an error, we would be changing an intentional comma definition on the basis of our LPEI badness metric. But we know there are 20 symbols in the Olympian (Extreme) notation that do not have the lowest badness comma in their zones. There's nothing special about the 15th mina in that regard. And I don't want to tackle researching the reasons for all their choices at this point in time.

So. I believe the correct thing to do, for now, is to fix the error in sag_ji4.par, by changing its 15th mina commas (up and down) to 1225k.
Thanks for researching that. Yes, with this new evidence, I totally agree.

And actually, the Scala file was already changed to match the spreadsheet's 1/1225k back in June, as documented in the GitHub discussion thread sampled in the first post here. So we're done for now. We can revisit this issue along with the other 19 at some other time.

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

Here's a spreadsheet from George dated 2007-Oct-09 (a month prior to the creation of sag_ji4.par) that contains the obsolete Super-olympian set. This will probably be useful in understanding others of those 20 not-least-bad comma assignments in Olympian, as well as how easy it would have been to copy and paste the wrong comma to sag_ji4.par for mina 15.
Attachments
JI-Nota.xlsx

cmloegmcluin