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

Posted:

**Thu Jun 18, 2020 5:32 am**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Recently @Dave Keenan pointed out to me that Scala's configuration files for Sagittal were out of date, especially with respect to the ET notations, due to the developments on bad-fifth ETs that culminated in the Periodic Table of EDOs.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:

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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:

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?

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 ,| ?

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.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).

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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?