Naming commas and other intervals: a compilation of various sources

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volleo6144
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Re: Naming commas and other intervals: a compilation of various sources

Post by volleo6144 »

Dave Keenan wrote: Mon Jun 15, 2020 10:21 am Therefore, we only need to ensure that our comma size categories are no wider than 23.46 cents (upper bound minus lower bound) to ensure that any "complex"-named comma must differ from its base comma by at least 41 in its 3-exponent. Our widest comma size category is the specific "comma" category, which ranges approximately from 11.73 cents to 33.38 cents, a width of approximately 21.65 cents, which is indeed less than the above-mentioned 23.46 cents.
Yeah.
 Category HalfSizeMonzo    Cents
 -------- -------------    -----
schismina [ - 84   53 > 1.807523 - close to 5s :'::|:
  schisma [  401 -253 > 2.692391 - between 91s :`::'::|: and 19:4375s :,::)|:
  kleisma [ -336  212 > 7.230092 - 3.74E+50:3.76E+50 = 106edo-kleisma exactly
    comma [   46 - 29 > 21.65249 - close to 5C :/|: and slightly less than 12edo-comma
 S-diesis [ - 19   12 > 11.73001 - close to 11:35k :'::)|(:
 M-diesis [ - 19   12 > 11.73001
 L-diesis [ - 19   12 > 11.73001
Ssemitone [ - 19   12 > 11.73001
    limma [   46 - 29 > 21.65249
Lsemitone [   65 - 41 > 9.922482 - between 7:11k :)|(: and 275k :`::)|(:
  apotome [ -168  106 > 3.615046 - 1.93E+25:1.94E+25 = 53edo-schisma exactly
Note that, while 29edo has a slightly better fifth than 12edo in terms of absolute error, its comma is actually larger (the 29edo-S-diesis of 43.30497¢) than the 12edo comma (of 23.46001¢), so we don't have to worry about it being too big.

Unless the commas are non-directionally named, in which case we get a similar issue to the one with the two 253-commas, with, for example, [336 2 -146> (1.1E+109:1.3E+109 = 202.1078¢) and [-342 2 146> (9.0E+102:1.0E+103 = 205.7122¢) both being 5^146MS+A's with the same 3-exponent and not just the same absolute value of the 3-exponent. Again, these examples probably don't have any musical relevance, but still...
I'm in college (a CS major), but apparently there's still a decent amount of time to check this out. I wonder if the main page will ever have 59edo changed to green...
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Re: Naming commas and other intervals: a compilation of various sources

Post by Dave Keenan »

volleo6144 wrote: Tue Jun 16, 2020 12:54 am Unless the commas are non-directionally named, in which case we get a similar issue to the one with the two 253-commas, with, for example, [336 2 -146> (1.1E+109:1.3E+109 = 202.1078¢) and [-342 2 146> (9.0E+102:1.0E+103 = 205.7122¢) both being 5^146MS+A's with the same 3-exponent and not just the same absolute value of the 3-exponent. Again, these examples probably don't have any musical relevance, but still...
Thanks for checking the proof. That's a wicked "counterexample". Any time I want someone to look for counterexamples I'm gonna call you. :)

But yeah. That's another reason to use directed ratios instead of undirected, in the comma names, as @cmloegcmluin suggests. I note that we use a slash instead of a colon, to indicate a directed ratio. But for something as complex as that, we'd just use the monzos (for the positive cents). And of course, such monzos are inherently directed.

I find it annoying to have to write 1/5-comma and 1/7-comma instead of 5-comma and 7-comma, but I'll get used to it. It does give the user useful extra information about how to use the symbols.
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Re: Naming commas and other intervals: a compilation of various sources

Post by cmloegcmluin »

Dave Keenan wrote: Mon Jun 15, 2020 4:57 pm I just noticed that in http://sagittal.org/SagittalJI.gif, George has 1/1 = [0⟩ as "1n" (as you originally had it, @volleo6144), and [-19 12⟩ as 1C, and since I want 1 and 3 to be interchangeable in these comma names, this contradicts what I said earlier, when attempting to be musician-friendly, about 1/1 not being in any comma size category, and so 1n = 3n = [485 -306⟩ (1.77¢).

Any suggestions for how this should be resolved?
George's original JI calculator spreadsheet also gives 1/1 as the 1n.

I am okay with 1n; I don't foresee major mathematical or coding-related concerns with the schismina size category extending all the way down to and including 0¢. But I agree that the average musician might be put off by this. Any opportunity Sagittal has to conform to entrenched terminological expectations, where its intent is not to push the community in a better direction, it should take. And I highly doubt that the unison being a comma is one of these battles Sagittal is trying to win for great microtonal glory.

Have we ever considered calling it the 1u, for unison? What I like about this is that it's fairly obvious when you see it on the page. And I also appreciate how when you call it the 3u, it subtly underlines how with respect to this system of commas, moving in pitch by a factor of 3 is a non-move. Also u is an upside-down n so it just fits real nicely down in there underneath all the n's. And I don't foresee us needing the letter u for any other size category abbreviations.
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Re: Naming commas and other intervals: a compilation of various sources

Post by Dave Keenan »

cmloegcmluin wrote: Tue Jun 16, 2020 2:20 pm Have we ever considered calling it the 1u, for unison? What I like about this is that it's fairly obvious when you see it on the page. And I also appreciate how when you call it the 3u, it subtly underlines how with respect to this system of commas, moving in pitch by a factor of 3 is a non-move. Also u is an upside-down n so it just fits real nicely down in there underneath all the n's. And I don't foresee us needing the letter u for any other size category abbreviations.
That's a great solution. :)

I hereby declare that the lowerbound of the schismina category is sqrt( [-16785921 10590737 ) = 4.69387×10⁻⁵ ¢. :twisted: :o :)

Everything smaller than that is a "unison". And so 1/1 = 1u, while [485 -306 = 3n.

Actually, we don't need to define such a boundary. Such a definition would itself be musician-unfriendly. We only need to say that all comma size category boundaries are exclusive, including the lower bound of schisminas at 1/1. So, just as sqrt( [-84 53 ) is not a schismina (or a schisma), sqrt( [0 0 ) is not a schismina either. And 1/1 is the single example of a unison, which is not (really) a comma size category.
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Re: Naming commas and other intervals: a compilation of various sources

Post by cmloegcmluin »

Great, glad you like it!

That’s an astoundingly unnoticeable comma you’re found there! But I agree having exclusive bounds is the better solution.

I’ll change things to use 1u in the various Sagittal resources.
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Re: Naming commas and other intervals: a compilation of various sources

Post by Dave Keenan »

cmloegcmluin wrote: Wed Jun 17, 2020 6:00 am That’s an astoundingly unnoticeable comma you’re found there! But I agree having exclusive bounds is the better solution.
I cheated. I just chose a 3-exponent from this sequence (which I already linked above),
https://oeis.org/A005664
and found the corresponding 2-exponent as = -ROUND(3_exp*LOG(3,2),0).

I actually wish I'd chosen sqrt( [301994 -190537⟩ ) = 5.58095×10⁻⁵ cents for my "joke", because it's only 20% larger, but it "only" requires 6 digit exponents instead of 8 digit.

I’ll change things to use 1u in the various Sagittal resources.
Thanks.
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Re: Naming commas and other intervals: a compilation of various sources

Post by cmloegcmluin »

Dave Keenan wrote: Wed Jun 17, 2020 12:41 pm I cheated. I just chose a 3-exponent from this sequence (which I already linked above),
https://oeis.org/A005664
and found the corresponding 2-exponent as = -ROUND(3_exp*LOG(3,2),0).
Not sure this game has any rules you could cheat!

My intent is not to hijack @volleo6144's original topic, but I am curious if you know anything about David Ryan's paper linked from that OIES entry, "An algorithm to assign musical prime commas to every prime number and construct a universal and compact free Just Intonation musical notation": https://arxiv.org/abs/1612.01860
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Re: Naming commas and other intervals: a compilation of various sources

Post by cmloegcmluin »

I briefly had a notification on the forum that you, @Dave Keenan, had quoted me, but on an older post: viewtopic.php?p=943#p943

As soon as I saw it, I recognized the name Dave Ryan as the guy who helped a bunch with Prime Factor notation over here.

The notification is gone and I don't know if anything was maybe changed on those posts. Just letting you know.
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Re: Naming commas and other intervals: a compilation of various sources

Post by Dave Keenan »

cmloegcmluin wrote: Wed Jun 17, 2020 2:49 pm I briefly had a notification on the forum that you, @Dave Keenan, had quoted me, but on an older post: viewtopic.php?p=943#p943

The notification is gone and I don't know if anything was maybe changed on those posts. Just letting you know.
I just fixed a typo where I had 8:11 instead of 8:13.
As soon as I saw it, I recognized the name Dave Ryan as the guy who helped a bunch with Prime Factor notation over here.
Yes, indeed.
cmloegcmluin wrote: Wed Jun 17, 2020 1:34 pm My intent is not to hijack @volleo6144's original topic, but I am curious if you know anything about David Ryan's paper linked from that OIES entry, "An algorithm to assign musical prime commas to every prime number and construct a universal and compact free Just Intonation musical notation": https://arxiv.org/abs/1612.01860
Oh, yes! I reviewed the first version for him in 2016. You'll find that the acknowledgements section at the end contains: "Dave Keenan who offered constructive criticism on a previous algorithm which chose some suboptimal commas, in particular for primes 11 and 13."

After that, he and I (and others) had extensive discussions about his algorithm in one of the facebook groups. I think he's kind of nutty to worry about how one should notate primes with 4 or more digits. But a lovable and very intelligent kind of nutty. :)

The first prime we disagree about is 17, but I'm happy to have his choice for that as an option, just not the standard. The next prime we disagree about is 59. No option required for that one. :)

His notation is not a staff notation, but is designed for text entry, and for that purpose makes perfect sense. Revo Sagittal couldn't use his algorithm for choosing higher prime commas if it wanted to, because it would require an unlimited series of symbols corresponding to triple sharps, quadruple sharps etc.
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Re: Naming commas and other intervals: a compilation of various sources

Post by volleo6144 »

The algorithm shown in the paper only disagrees with the "Sagittal" algorithm it describes for five primes below 100:
Pr DRComma SagComma
-- ------- --------
17 1 /17 k 17 / 1 C (DR = :~|: / SAG = :~|(:; sum = 12edo3C)
59 1 /59 M 59 / 1 L (DR ~ :'::/|): / SAG ~ :.::(|\:; sum = 7edo3A)
67 1 /67 S 1 / 67 k (DR ~ :,::'::(|: / SAG ~ :``::)|(:; sum = 1/4489M ~ :`::'::/ /|:; diff = 12edo3C)
83 83/ 1 S 1 / 83 M (DR ~ :,,::/ /|: / SAG ~ :,,::/|):; sum = 5edo3MS)
89 1 /89 S 1 / 89 C (DR ~ :.::/ /|: / SAG ~ :~~|:; sum = 1/7921L ~ :,::.::(|):; diff = 12edo3C)
The DR algorithm uses some sort of badness measure equal to the comma's Tenney height times its size, while the "Sagittal" algorithm just chooses the first 3-exponent within some L-diesis (or, in other words, within half of the 19edo3C+A) of the target prime. In the case of that exponent landing on the side of the prime, this usually results in more complex-looking commas.

The "KG2" algorithm described disagrees in a lot more ways, because it focuses more on assigning intervals to primes instead of assigning commas to primes, which leads to the "more complex" 11- and 13-dieses (11L and 13M) being the commas for their respective primes (with 8:11 being an A4 and not a P4, and 8:13 being a m6 and not a M6), as well as a number of other ... questionable choices in the context of 3-limit-plus-prime commas. (For the record, the 53edo3s is a negative sevenfold-diminished sixth.)
I'm in college (a CS major), but apparently there's still a decent amount of time to check this out. I wonder if the main page will ever have 59edo changed to green...
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