Naming commas and other intervals: a compilation of various sources

[Dave Keenan]

So you can see that every n-edo has a vanishing 3-comma whose 3-exponent is n, but you think there may be some 3-commas with 3-exponent of n that do not vanish in n-edo. So find me one.

[Dave Keenan]

Only click the button below after you've had a go at finding a counterexample to my claim that a 3-comma with 3-exponent n always vanishes in n-edo.

It's easy to find a 3-"comma" with 3-exponent n that does not vanish in n-edo. Just take the obvious mapping for n-edo (have I mentioned how much I hate the term "val", and particularly "patent val") and add or subtract 1 from its mapping of 3.

But what does that mean for the comma? It means adding or subtracting 1 from its 2-exponent. That means either making the comma an octave larger or taking the octave-complement of the comma. Either way, it's not really a comma any more is it? So I only have to specify that, to be considered a comma, a ratio must be smaller than a half-octave, in order for there to be no counterexamples. That's something most people would assume without question. Hence the scare-quotes on "comma" above.

[cmloegcmluin]

I promise I didn't open your spoiler until I gave it a solid go. More or less I arrived at that conclusion, though some of my specifics were slightly off. I wrote a huge wall of disorganized mess as I had nested insight after nested insight and continuously tried to update my response. But it's just not worth cleaning up. Thanks for the challenge.

[cmloegcmluin]

I just noticed in the Comma Namer shared here that the preferred name for the size category with abbreviation "MS" — for "medium semitone" — is "limma". I had known that "limma" was an option since our exchange here, but I hadn't realized until now that it was in fact the preferred name, not an alternative (and that "medium semitone" is the alternative). I've updated the code base accordingly.

Similarly, I see that the preferred name for the size category with abbreviation "M" — for "medium (diesis)" — is "diesis", and it is "medium diesis" which is the alternative name. I began updating the code base to reflect this, but then I noticed that in the Sagittal-SMuFL-Map the commatic intervals in this size category are labelled with "medium diesis", not "diesis". Is this a mistake? Or is the Comma Namer out of date? Or is this an intended inconsistency?

[cmloegcmluin]

In the Comma Namer, some alternatives are given for the diesis names: 1/5-tone, 1/4-tone, and 1/3-tone. I suggest instead spelling those fractions out: fifth-tone, quarter-tone, and third-tone, for two reasons:

• it's a pain in the butt for my code to parse the comma names if there're any numeric characters in the size category part of their names, and
• it's also weird for humans to parse a name like e.g. the 1/5-1/5-tone, [ -23 16 -1 ⟩, ~44.966¢; I think the 1/5-fifth-tone is better.

[cmloegcmluin]

Our size-category boundaries are at the square-roots of certain 3-prime-limit ratios...

schisma (s)
[317 -200>/2 ~= 4.4999 cents (half a complex Pythagorean kleisma)

I think "Pythagorean-supercomplex-kleisma" would be the correct name for this interval according the rules described on this topic. [ -168 106 ⟩ is the simplest 3-limit kleisma, and [ -252 159 ⟩ is the complex one.

[Dave Keenan]

I just noticed in the Comma Namer shared here that the preferred name for the size category with abbreviation "MS" — for "medium semitone" — is "limma". I had known that "limma" was an option since our exchange here, but I hadn't realized until now that it was in fact the preferred name, not an alternative (and that "medium semitone" is the alternative). I've updated the code base accordingly.

Unlike the category names given in footnote 7 on page 8 of the XH article, not a lot of thought was given to the names for the categories beyond large dieses. I don't remember ever discussing them with George. I think I came up with them when developing that CommaNamer spreadsheet. Although George didn't object to them.

I now think that the terms apotome and limma are so strongly associated with pure-fifths/Pythagorean tuning that I feel something needs to be pretty close to the simple 3-limit versions of them to be considered one of them. This condition is satisfied for the apotome size category (because it's not very wide), but it's not satisfied for the medium semitone category. So perhaps "limma" should remain an alternative, accepted on input, but not given as output. Instead we give "medium semitone" or "(medium-)semitone" on output (you choose). Of course if you choose to use "(medium-)semitone" on output, you have to allow "semitone" on input, for this size category. Maybe you want to allow "semitone" on input anyway.

Similarly, I see that the preferred name for the size category with abbreviation "M" — for "medium (diesis)" — is "diesis", and it is "medium diesis" which is the alternative name. I began updating the code base to reflect this, but then I noticed that in the Sagittal-SMuFL-Map the commatic intervals in this size category are labelled with "medium diesis", not "diesis". Is this a mistake? Or is the Comma Namer out of date? Or is this an intended inconsistency?

The difference/inconsistency was not intended. Maybe we should always use "medium-diesis" or "(medium-)diesis" on output, but accept "diesis" on input, for this size category.

[Dave Keenan]

In the Comma Namer, some alternatives are given for the diesis names: 1/5-tone, 1/4-tone, and 1/3-tone. I suggest instead spelling those fractions out: fifth-tone, quarter-tone, and third-tone, for two reasons:

• it's a pain in the butt for my code to parse the comma names if there're any numeric characters in the size category part of their names, and
• it's also weird for humans to parse a name like e.g. the 1/5-1/5-tone, [ -23 16 -1 ⟩, ~44.966¢; I think the 1/5-fifth-tone is better.

I totally agree. They should only be accepted as "fifth-tone", "quarter-tone", and "third-tone".

[Dave Keenan]

Our size-category boundaries are at the square-roots of certain 3-prime-limit ratios...

schisma (s)
[317 -200>/2 ~= 4.4999 cents (half a complex Pythagorean kleisma)

I think "Pythagorean-supercomplex-kleisma" would be the correct name for this interval according the rules described on this topic. [ -168 106 ⟩ is the simplest 3-limit kleisma, and [ -252 159 ⟩ is the complex one.

Thanks for checking that. But I feel it should be written in the order "supercomplex-Pythagorean-kleisma", "sc3k", as "Pythagorean" is a synonym for "3" in naming commas. And of course now we'd prefer to call it the "200edo-3-kleisma" or the "200edo Pythagorean kleisma", "200e3k". Feel free to write it as "200-edo-..." or "200-EDO-...".

I have updated it to:

schisma (s)
[317 -200>/2 ~= 4.4999 cents (half a supercomplex Pythagorean kleisma = half a 200edo Pythagorean kleisma)

[cmloegcmluin]

Okay, time for more questions. Well, let me phrase things in the form of a minimal table of examples with my assumptions:

comma name input	resulting comma	reasoning
1/5C	[-4 4,-1 ⟩
5C	[-4 4 -1 ⟩	directed form is preferred, but undirected form is valid (and historical)
c1/5C	(error)	c1/5C cannot be interpreted as "short for" the c5C
c5C	[-34 20 1⟩
1/5k	(error)	do not assume you mean the c1/5k if you don't ask for complexity
5k	[-99 61 1⟩
c1/5k	[-153 98 -1⟩
c5k	[-153 98 -1 ⟩

Please let me know if you think otherwise for any of these.