## Naming commas and other intervals: a compilation of various sources

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

This ... made me think of writing this all out:
Dave Keenan wrote: Thu May 28, 2020 10:45 am I agree with your argument. But there's no need to check. That property was guaranteed by generating candidates with steadily increasing 3-exponents, starting from zero and working in both positive and negative directions.

Here's the Comma Namer spreadsheet. [attached: CommaNamer.xlsx]
A comma is named as the ratio it notates with any factors of 2 and 3 removed (5103:5120 becomes 7:5, which becomes 5:7 because the lower number comes first), together with a size category (twice 5103:5120 is in between [317 -200> and [-19 12>, so 5103:5120 is a kleisma). These size categories are delineated by halves (or square roots?) of various 3-limit commas, which are mostly relevant to specific EDOs.

These upper bounds are, in increasing order:

schismina (n): [-84 53> / 2, or about 1.81¢. Half of Mercator's comma (1s or 3s), or 26.5 times the interval by which each fifth is flattened in 53-EDO.
schisma (s): [317 -200> / 2, or about 4.50¢.
kleisma (k): [-19 12> / 2, or about 11.73¢. Half of the Pythagorean comma (1C or 3C), or six times the interval by which each fifth is flattened in 12-EDO.
comma (C): [27 -17> / 2, or about 33.38¢.
S-diesis (S): [8 -5> / 2, or about 45.11¢. Half of the Pythagorean limma, or m2.
M-diesis (M): [-11 7> / 2, or about 56.84¢. Half of the Pythagorean apotome, or sharp.
L-diesis (L): [-30 19> / 2, or about 68.57¢.
S-semitone (SS): [-49 31> / 2, or about 80.30¢.
M-semitone or limma (MS): [-3 2> / 2, or about 101.96¢. The 2:3 perfect fifth minus half an octave, or half an 8:9 "major" tone.
L-semitone (LS): [62 -39> / 2, or about 111.88¢.
Apotome (A): [-106 67> / 2, or about 115.49¢. "Apotome" is pronounced similarly to "epitome", and the epitome of mispronunciation is how long I thought it was pronounced with emphasis on the "a".
After these, the existing comma boundaries here are added to [-22 14> / 2 = [-11 7>, the Pythagorean apotome, to yield [295 -186> / 2 for s+A, etc. up to double-apotome (A+A) with an upper bound of [-128 81> / 2 ≈ 229.18¢. Limma-plus-apotome can be named as "whole-tone" instead, because a minor second (a limma) plus a sharp (an apotome) is a major second (a whole tone).

When there is a simpler ratio (by smaller 3-exponent) that fits the primes and the size category, the prefixes "complex", "supercomplex", "hypercomplex", "ultracomplex", "5-complex", "6-complex", etc. may be added to signify how many smaller 3-exponents "work" for it. (Strangely, the spreadsheet in its current form has a glitch where any comma that's supercomplex or beyond will still just be listed as "complex", and I'm not sure which of [50516 -31867, 0 0 -1, 0 0 0, -1> and [-50500 31867, 0 0 -1, 0 0 0, -1> is counted as less complex.)
AP20 in the spreadsheet wrote: [The] "plus-apotome" names are only intended to be used if the interval is really being considered as a comma, otherwise a different naming system applies. See http://users.bigpond.net.au/d.keenan/Mu ... Naming.txt or http://users.bigpond.net.au/d.keenan/Mu ... Naming.htm
users.bigpond.net.au isn't responding (it isn't showing me any kind of error; it just isn't responding), but archive.org exists, so I checked the other sources out. The first one refers to a particular way of assigning "wide" or "narrow" labels to the existing labels used in the second one (based on Miracle temperament, which is 72-EDO but with 7\72 instead of 1\72 as the generator), and the second one lays out a systematic naming system for 11-limit intervals based on 31-EDO.

P1 = 0\31, M2 = 5\31, M3 = 10\31, P4 = 13\31, P5 = 18\31, M6 = 23\31, M7 = 28\31, P8 = 31\31.

For seconds, thirds, sixths, and sevenths, the modifiers for successive 31-EDO steps are: subdiminished, diminished, subminor, minor, neutral, major, supermajor, augmented, superaugmented.
For unisons, fourths, fifths, and octaves, the modifiers are: doubly diminished, subdiminished, diminished, sub, perfect (0), super, augmented, superaugmented, double augmented.

For both the comma system and the larger-interval system, the names "Pythagorean", "classic" (???), "septimal", "undecimal", and "tridecimal" can be used instead of "1" (or "3", which can replace "1" for 3-limit intervals), "5", "7", "11", and "13" when no other prime is present ("classic-to-septimal kleisma" is certainly not a correct full name for the 5:7k) in full names.

Obviously, these should usually accompany the interval's ratio itself, not replace it, but I didn't make the names; the other people here did.

Even with a post as extensive as this, I'm sure I missed something, and I'm sure I'd have missed something no matter how many times I revised this, so this should probably do for now.
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...

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

volleo6144 wrote: Tue Jun 02, 2020 12:47 pm 7:5, which becomes 5:7 because the lower number comes first
Recently Dave and I have been trying to write them in directed form, so the 5:7k would be the 5/7k. Not because the 5 is smaller, but because it is in the numerator while the 7 is in the denominator. So the comma whose monzo is [-16 9 -1 0 0 0 1⟩ would be named the 17/5S. Mainly we think this helps clarify some of the element arithmetic, such as I put in this message to Dave recently:
cmloegcmluin wrote:For example, I think it would be excellent if the fact that and combined into was encoded in their comma names. When they are called the 5C, 55C, and 11M, this relationship is obscured. When we call them the 1/5C, 55C, and 11M, then it is clear.
Let me know what you think of this approach, though!
volleo6144 wrote: These size categories are delineated by halves (or square roots?) of various 3-limit commas
They are the square roots of the commas. But when you're expressing them one logarithmic step down, such as with monzos and cents, taking the square root looks like dividing by two:
• in ratio form: 256/243 → √256/√243 = 16/~15.588
• in monzo form: [8 -5⟩ → [8 -5⟩ / 2 = [4 -2.5⟩
• in cents form: 90.225 -> 90.225 / 2 = 45.1632821917
I think you took your list of size category bounds from this post. Linking it here in case it is of use to anyone.

And this post I made recently may also be of interest, where I explain why I think all existing Sagittal commas are the simplest in their size category (not "complex"): viewtopic.php?p=1666#p1666
volleo6144 wrote: [The] "plus-apotome" names are only intended to be used if the interval is really being considered as a comma, otherwise a different naming system applies. See http://users.bigpond.net.au/d.keenan/Mu ... Naming.txt or http://users.bigpond.net.au/d.keenan/Mu ... Naming.htm
Many links to http://users.bigpond.net.au/d.keenan/ have been simply moved to http://dkeenan.com/. Try:

http://dkeenan.com/Music/IntervalNaming.htm
http://dkeenan.com/Music/Miracle/Miracl ... Naming.txt
http://dkeenan.com/Music/EdoIntervalNames.pdf
Last edited by Dave Keenan on Wed Jun 03, 2020 4:10 pm, edited 1 time in total.
Reason: Added URL at end, for my most recent interval naming scheme http://dkeenan.com/Music/EdoIntervalNames.pdf

Dave Keenan
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### Re: Naming commas and other intervals: a compilation of various sources

volleo6144 wrote: Tue Jun 02, 2020 12:47 pm (Strangely, the spreadsheet in its current form has a glitch where any comma that's supercomplex or beyond will still just be listed as "complex", and I'm not sure which of [50516 -31867, 0 0 -1, 0 0 0, -1> and [-50500 31867, 0 0 -1, 0 0 0, -1> is counted as less complex.)
Thanks for that, @volleo6144, I was surprised to learn that it didn't work for supercomplex or beyond. When I investigated this just now, I noticed that I had headed the relevant column "Complexity level (incomplete heuristic)". Apparently I only needed something that would work for all the commas that I could find on various lists. The only algorithm I know to truly determine this comma complexity level, is to iterate over the 3-exponents in the sequence 0, 1, -1, 2, -2, 3, -3, ... with the same combination of primes above 3, calculating whether there exists an integer 2-exponent that creates a comma whose absolute value of cents is in the given size range.

I applaud your persistence in finding that extraordinary pair of 11.23-commas. Of course, with 3-exponents in the tens of thousands, they are unlikely to have any musical relevance., but we can (somewhat arbitrarily, but memorably) decide on a convention that a negative exponent is more complex than a positive exponent of the same magnitude (because, mnemonically, it requires an extra character to represent it). The signs would be those that give the comma as a positive number of cents.

If we found there were a lot of musically-relevant commas for the same combination of primes above 3 in the same size category, then we would need to consider whether we needed another size boundary. After all, that's how we got the boundaries we have. But I can't see it happening.

I note that, for the pairs of close commas with the same 2,3-reduced ratio that led to the existing boundaries, the boundary was simply the geometric mean of the two comma ratios, in which the primes-above-3 cancelled out, leaving the irrational square root of a 3-limit ratio. That is not the case for these two 11.23-commas. Their geometric mean is the ratio [8 0, 0 0 -1, 0 0 0, -1⟩ or 256/253 (≈ 20.40771037 ¢).

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

Dave Keenan wrote: Thu Jun 04, 2020 2:15 pm
volleo6144 wrote: Tue Jun 02, 2020 12:47 pm [50516 -31867, 0 0 -1, 0 0 0, -1> and [-50500 31867, 0 0 -1, 0 0 0, -1>
I applaud your persistence in finding that extraordinary pair of 11.23-commas.
Actually, I was ... just using whatever comma I thought of first (which turned out to be 1/253C) that met certain criteria that I don't remember had a 3-exponent of zero. The 96c1n* part came from a week ago:
cmloegcmluin wrote: Thu May 28, 2020 9:15 am Trivially, we could make a comma more complex by adding, say, [ -50508 31867 >, which is only 0.012577¢, and therefore small enough in size that it's unlikely to change its size category. But of course we'd notice right away if any of our commas had astronomically large prime exponents like that.
*counting 1:1 as 1n and [485 -306> as c1n
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...

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

The unnoticeable commas page is strangely one of my most frequented pages on the wiki.

Dave Keenan
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### Re: Naming commas and other intervals: a compilation of various sources

cmloegcmluin wrote: Sun Jun 14, 2020 12:31 pm The unnoticeable commas page is strangely one of my most frequented pages on the wiki.
Since Kite's Color names are allowed in the Wiki, it would be good to also have a column for "Secor-Keenan systematic name" or "Sagittal name", with a subcolumn giving the abbreviated form. And of course the heading would need to link to an explainer page.

I just had the thought: I wonder if some kind of histogram of commas would show notches near our size category boundaries?

volleo6144 wrote: Sat Jun 06, 2020 3:19 am *counting 1:1 as 1n and [485 -306> as c1n
I'm glad you wrote that. It embodies two examples where something that makes sense to a mathematician, may not make sense to a musician.

I predict that, to most musicians, the term "1-comma" will make no sense, and 1:1 isn't any kind of comma. So [485 -306⟩ should be listed as the 3-schismina (3n) and [-569 359⟩ as the 3-schisma (3s), even though that may require some special-cases in the code or spreadsheet formula.

I'm guessing that [-1054 665⟩ is the complex 3-schismina (c3n), [24727 -15601⟩ is the supercomplex 3-schismina (sc3n or 2c3n) and [-50508 31867⟩ is the hypercomplex 3-schismina (hc3n or 3c3n).

Are those correct @volleo6144?

Surely those 3-limit comma names like "359-comma", "15601-comma" and "31867-comma" should be "359edo-comma", "15601edo-comma" and "31867edo-comma".

But I don't understand why [24727 -15601⟩ and [-50508 31867⟩ are even on that page. I see no other commas on the wiki with 5 digit exponents. What musical significance do they have?

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

Dave Keenan wrote: Sun Jun 14, 2020 5:09 pm [-569 359> as the 3-schisma (3s)
...hey, wait, are you forgetting the 53edo-comma used in defining the n/s boundary?
Dave Keenan wrote: Sun Jun 14, 2020 5:09 pm I predict that, to most musicians, the term "1-comma" will make no sense, and 1:1 isn't any kind of comma. So [485 -306⟩ should be listed as the 3-schismina (3n) and [-569 359⟩ as the 3-schisma (3s), even though that may require some special-cases in the code or spreadsheet formula.

I'm guessing that [-1054 665⟩ is the complex 3-schismina (c3n), [24727 -15601⟩ is the supercomplex 3-schismina (sc3n or 2c3n) and [-50508 31867⟩ is the hypercomplex 3-schismina (hc3n or 3c3n).

Are those correct @volleo6144?
There are actually a lot more in between. [-1054 665> (0.0756¢) is the c3n if [485 -306> (1.77¢) is the 3n (and not itself the c3n, with 3n being 1:1), but then there's a lot of others, starting with [1539 -971> (1.69¢) as sc3n and [-2108 1330> (0.151¢) as the hc3n. [24727 -15601> (0.0315¢) is the 46c3n, and [-50508 31867> (0.0126¢) is the 95c3n. The ones listed on the wiki are just really small compared to the rest. (94c3n, [-50023 31561>, is just barely below the n/s boundary, at 1.78¢, and 93c3n, [49454 -31202>, is pretty small, at 0.0630¢, but still larger than 46c3n.)
Dave Keenan wrote: Sun Jun 14, 2020 5:09 pm Surely those 3-limit comma names like "359-comma", "15601-comma" and "31867-comma" should be "359edo-comma", "15601edo-comma" and "31867edo-comma".

But I don't understand why [24727 -15601⟩ and [-50508 31867⟩ are even on that page. I see no other commas on the wiki with 5 digit exponents. What musical significance do they have?
I'm not sure they really have any musical significance, really, outside of being particularly small Pythagorean schisminas. The 31867edo-schismina in particular is smaller than the smallest 13-limit superparticular, the 169:1925n of 123200:123201.

On an only-partially-related note, I don't understand what's going on with the names like "171&1547&3125 comma" (this is actually listed in the "Unnoticeable comma" page on the wiki) for the laleruyo [-1 4 11 -11>, also listed here as a potential way to offset a 2 with a schismina (in this case tempering the octave to 1977326743:3955078125).
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...

Dave Keenan
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### Re: Naming commas and other intervals: a compilation of various sources

volleo6144 wrote: Sun Jun 14, 2020 10:47 pm ...hey, wait, are you forgetting the 53edo-comma used in defining the n/s boundary?
Indeed I did. Thanks for correcting me. It was (rightly) not on that page as that page only shows commas smaller than 3.5 cents.

I'm thinking that the whole "n-complex" thing is only convenient if you rarely have to use it. But 3-commas seem to use it too much. And the "n-edo" thing is convenient for 3-commas.

So we have:
3s = [-84 53⟩ = 53edo 3-schisma = 53e3s
c3s = [-569 359⟩ = 359edo 3-schisma = 359e3s
sc3s = [970 -612⟩ = 612edo 3-schisma = 612e3s

There are actually a lot more in between. [-1054 665> (0.0756¢) is the c3n if [485 -306> (1.77¢) is the 3n
Agreed. 3n = 306e3n and c3n = 665e3n.
(and not itself the c3n, with 3n being 1:1), but then there's a lot of others, starting with [1539 -971> (1.69¢) as sc3n and [-2108 1330> (0.151¢) as the hc3n. [24727 -15601> (0.0315¢) is the 46c3n, and [-50508 31867> (0.0126¢) is the 95c3n. The ones listed on the wiki are just really small compared to the rest. (94c3n, [-50023 31561>, is just barely below the n/s boundary, at 1.78¢, and 93c3n, [49454 -31202>, is pretty small, at 0.0630¢, but still larger than 46c3n.)
Thanks for that.
On an only-partially-related note, I don't understand what's going on with the names like "171&1547&3125 comma" (this is actually listed in the "Unnoticeable comma" page on the wiki) for the laleruyo [-1 4 11 -11>, also listed here as a potential way to offset a 2 with a schismina (in this case tempering the octave to 1977326743:3955078125).
I can explain the "&" notation. It combines EDO prime-maps as a way of specifying a temperament of higher rank. In this case rank-3. If you go to this page of Graham Breed's temperament finder. http://x31eq.com/temper/net.html and paste "171&1547&3125" into the first field and "11" into the second field and hit "submit", you will see [-1 4 11 -11> as one of the commas that vanishes in that temperament. But how one would make musical use of such a complex comma or temperament, I have no idea.

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

I want to restate and extend a proof that @cmloegcmluin gave elsewhere.

The list with increasing absolute 3-exponent, of progressively-smaller 3-commas, begins:

3-exp	Size in cents	Description
------	-------------	---------------------
-1	498.0449991	perfect fourth
2	203.9100017	whole tone
-5	90.22499567	limma
12	23.46001038	3-comma
-41	19.84496452	complex 3-comma
53	3.615045866	3-schisma
-306	1.769735191	3-schismina
665	0.075575483	complex 3-schismina
-15601	0.031499087	46-complex 3-schismina (Thanks volleo6144)
31867	0.012577308	95-complex 3-schismina (   "        "    )
... See https://oeis.org/A005664


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.

For notation purposes, we are not interested in commas having an absolute 3-exponent greater than 20 (because of the need for more than 2 sharps or flats). Therefore we are also not interested in commas whose 3-exponent differs from any interesting comma by 41, in either direction, since 20-41 = -21. Therefore we are not interested in any "complex"-named comma for notation purposes.

I personally don't think that "complex"-named commas are of any interest as vanishing commas for temperaments either, except for the 3-commas of equal temperaments.

Dave Keenan