Naming commas and other intervals: a compilation of various sources

[Dave Keenan]

I imagine the reader would think that too. Which is largely why I recommended directed comma names in the first place.

I don't have a problem with using the simplified form 5C in situations where it doesn't confuse the issues at hand. But in this particular context, maybe the answer is to just use the full name, the 1/5C? Why in particular do you "want" to write "the 5-comma" here?

It seems bizarre to me, to consider "5C" a simplified form of "1/5C". I could however consider it a simplified form of "5/1C", "1:5C" or "5:1C".

I want to write "the 5 comma" so that the reader gets the correspondence between + = and 5 × 11 = 55.

It seems to me, that using "1/5C" here has the same problem as using "5C down", namely that the reader might reasonably expect 1/5 × 11 = 11/5.

Please spell out your preferred version of that sentence about + =  .

In what sense are your preferred names "directed" if you still need to put "down" after them to give them a direction other than up? That was the case with the original undirected names with the colons.

I see I already asked that, and you responded, and we went on and on, and I thought it was resolved, beginning when you wrote:

= 7/5k (5103/5120) and = 5/7k (5120/5103) is good.

and ending here when you said you preferred:

"The right barb is therefore defined as the 55 comma (55/54), i.e., the 11M diesis (33/32) plus the 5 comma (80/81), where the upward and downward left barbs cancel."

over:

"The right barb is therefore defined as the 55 comma (55/54), i.e., the 11M diesis (33/32) less the 1/5 comma (81/80)."

[Dave Keenan]

I've posted a relevant new response here:
viewtopic.php?p=3057#p3057

[cmloegcmluin]

Okay. I had a sleep on it. And I think I get it now.

It could be said that a primary goal of yours is that you really really really want there to be something out there whose preferred name is "the 5C". It's not enough for that to be an accepted alternative. You worked really hard to convince people to start calling the syntonic comma the 5C and that was hard enough. You have strong doubts that we'd ever convince people to use "1/5C" and "1/5C down", and while I'm often lapse into not being mindful of this consideration, when I am mindful of it, I must admit I wouldn't expect the general public, burdened by convention, to go along with it.

In what sense are your preferred names "directed" if you still need to put "down" after them to give them a direction other than up? That was the case with the original undirected names with the colons.

I see I already asked that, and you responded,

Right. And what I responded was, in bold text:

The purpose of encoding the direction into the name is to disambigutate the orientation of the comma's prime content with respect to whether you are moving up or down by it.

Which I can see I got grumpy about you not responding directly to... but it looks like implicitly I came to realize that you didn't so much disagree with that idea as much as wish to deprioritize it. Which I eventually went along with. But I see that I've naturally circled back to this personal fixation of mine, because though we agreed your way was better before, it didn't fully sink in for me.

So there are two aspects of a comma involving direction: the direction of its prime content (which 2,3-free primes are in the numerator, and which are in the denominator), and its direction itself (is it positive or negative when written as cents), and I clearly have a deep-seated desire to separate the information about these two types of direction, encoding the former in the comma name, and the latter by accompanying it with the word "(up)" or "down". I suppose it's my engineering instincts which draw me to this clean implementation. But I recognize now that there are more important considerations than cleanliness. The users of this system will not be amenable to this approach, and our mandate is to devise a system which people (read: actual composers, not technico-aesthetic nerd hobbyists like myself) will actually take to.

And so I've come around, again, to the approach of using the comma name to represent both the direction of the prime content as well as the direction of the comma. We could say, then, that each size category (except the unison) has a pair of bounds mirrored by the unison, e.g. that a kleisma is any comma in the range -11.73¢ to -4.5¢ or 4.5¢ to 11.73¢.

So then what of that one massive worm, the case of the ambiguous 5-limit MS+A counterexample to the naming scheme?



(get it, prim^:-)ordial?)

Mostly I say: eff it. We shouldn't make decisions that affect how practical Sagittal's accompanying comma naming system is on such ludicrous counterexamples.

I almost set out to write a script this morning that would see if there were any simpler counterexamples to worry about, but then I had the thought while grinding my coffee... why can't we just use the 2-exponent as a tie-breaker?

In other words, why not name them like this:

[336 2 -146> = 1/112...625MS+A
[-342 2 146> = c112...625MS+A

Then we could also say:

[-336 -2 146> = 112...625MS+A
[342 -2 -146> = c1/112...625MS+A

I think there might still be ties possible. You'd have to find a comma where when you flip the sign on both the 2-exponent and the 3-exponent (while not flipping the signs for the 2,3-free prime content; otherwise you'd just have flipped the whole thing) you end up in the same size category. That'd mean that the combination of 2-exponent and 3-exponent on its own would have to be really small, e.g. the 3s, [-84 53⟩, ~3.6¢. But keep in mind that since in this counterexample scenario we don't have recourse to octave-reduction, we still have to find some 2,3-free prime content to accompany such a thing which is also itself comma-sized.

I found many, many examples. Here's just one:

[0 0 0 -1 2 0 -1⟩

That's 121/119, ~28.855¢, what we'd call the 121/(7⋅17)C. So this:

[-84 53 0 -1 2 0 -1⟩

at ~32.470¢, is the hc121/(7⋅17)C (that's right, it's the hypercomplex one... there're two other simpler 121/(7⋅17) commas before we reach this one which has the property we're after). So then, according to my code now, this:

[84 -53 0 -1 2 0 -1⟩

at ~25.240¢, is also the hc121/(7⋅17)C. And the point here is, that even we used the 2-exponent as a tie-breaker, we'd still have a problem.

I'm open to suggestions on how to break the tie at this point. Maybe we just say that the one whose 3-smooth content is super is the simpler one, i.e. the one which adds the 3s is simpler and the one which subtracts it is the more complex?

It seems bizarre to me, to consider "5C" a simplified form of "1/5C".

It now seems bizarre to me, too, in the face of the unmovable object which is the community's refusal to accept "1/5C down" over "5C".

Please spell out your preferred version of that sentence about + =  .

The same one I came to prefer before: "The right barb is therefore defined as the 55 comma (55/54), i.e., the 11M diesis (33/32) plus the 5 comma (80/81), where the upward and downward left barbs cancel."

[cmloegcmluin]

Alright, so how's about this table then. I'm now regretting the fact that for my previous table at the last minute I collapsed my "up" sections into the sections with no up or down as a parenthetical "(up)", since now I need to break them out, and it would have been nice to have better parallelism between the two tables. Oh well, no big deal. I removed the 5-limit MS+A counterexample, and added the 7C and 11M, anyway.

comma name input	resulting comma	cents	preferred
5C	[4 -4 1⟩	-	*
5/1C	[4 -4 1⟩	-
1/5C	[-4 4 -1⟩	+	*
5:1C	[4 -4 1⟩	-
1:5C	[4 -4 1⟩	-
5C up	[-4 4 -1⟩	+
5/1C up	[-4 4 -1⟩	+
1/5C up	[-4 4 -1⟩	+
5:1C up	[-4 4 -1⟩	+
1:5C up	[-4 4 -1⟩	+
5C down	[4 -4 1⟩	-
5/1C down	[4 -4 1⟩	-
1/5C down	[4 -4 1⟩	-
5:1C down	[4 -4 1⟩	-
1:5C down	[4 -4 1⟩	-
c5C	[-34 20 1⟩	+	*
c5/1C	[-34 20 1⟩	+
c1/5C	[34 -20 -1⟩	-	*
c5:1C	[-34 20 1⟩	+
c1:5C	[-34 20 1⟩	+
c5C up	[-34 20 1⟩	+
c5/1C up	[-34 20 1⟩	+
c1/5C up	[-34 20 1⟩	+
c5:1C up	[-34 20 1⟩	+
c1:5C up	[-34 20 1⟩	+
c5C down	[34 -20 -1⟩	-
c5/1C down	[34 -20 -1⟩	-
c1/5C down	[34 -20 -1⟩	-
c5:1C down	[34 -20 -1⟩	-
c1:5C down	[34 -20 -1⟩	-
5k	[-99 61 1⟩	+	*
5/1k	[-99 61 1⟩	+
1/5k	[99 -61 -1⟩	-	*
5:1k	[-99 61 1⟩	+
1:5k	[-99 61 1⟩	+
5k up	[-99 61 1⟩	+
5/1k up	[-99 61 1⟩	+
1/5k up	[-99 61 1⟩	+
5:1k up	[-99 61 1⟩	+
1:5k up	[-99 61 1⟩	+
5k down	[99 -61 -1⟩	-
5/1k down	[99 -61 -1⟩	-
1/5k down	[99 -61 -1⟩	-
5:1k down	[99 -61 -1⟩	-
1:5k down	[99 -61 -1⟩	-
c5k	[-153 98 -1⟩	+	*
c5/1k	[-153 98 -1⟩	+
c1/5k	[153 -98 1⟩	-	*
c5:1k	[-153 98 -1⟩	+
c1:5k	[-153 98 -1⟩	+
c5k up	[-153 98 -1⟩	+
c5/1k up	[-153 98 -1⟩	+
c1/5k up	[-153 98 -1⟩	+
c5:1k up	[-153 98 -1⟩	+
c1:5k up	[-153 98 -1⟩	+
c5k down	[153 -98 1⟩	-
c5/1k down	[153 -98 1⟩	-
c1/5k down	[153 -98 1⟩	-
c5:1k down	[153 -98 1⟩	-
c1:5k down	[153 -98 1⟩	-
5/7k	[10 -6 1 -1⟩	+	*
7/5k	[-10 6 -1 1⟩	-	*
5:7k	[-10 6 -1 1⟩	-
7:5k	[-10 6 -1 1⟩	-
5/7k up	[10 -6 1 -1⟩	+
7/5k up	[10 -6 1 -1⟩	+
5:7k up	[10 -6 1 -1⟩	+
7:5k up	[10 -6 1 -1⟩	+
5/7k down	[-10 6 -1 1⟩	-
7/5k down	[-10 6 -1 1⟩	-
5:7k down	[-10 6 -1 1⟩	-
7:5k down	[-10 6 -1 1⟩	-
7C	[-6 2 0 1⟩	-	*
7/1C	[-6 2 0 1⟩	-
1/7C	[6 -2 0 -1⟩	+	*
7:1C	[-6 2 0 1⟩	-
1:7C	[-6 2 0 1⟩	-
7C up	[6 -2 0 -1⟩	+
7/1C up	[6 -2 0 -1⟩	+
1/7C up	[6 -2 0 -1⟩	+
7:1C up	[6 -2 0 -1⟩	+
1:7C up	[6 -2 0 -1⟩	+
7C down	[-6 2 0 1⟩	-
7/1C down	[-6 2 0 1⟩	-
1/7C down	[-6 2 0 1⟩	-
7:1C down	[-6 2 0 1⟩	-
1:7C down	[-6 2 0 1⟩	-
11M	[-5 1 0 0 1⟩	+	*
11/1M	[-5 1 0 0 1⟩	+
1/11M	[5 -1 0 0 -1⟩	-	*
11:1M	[-5 1 0 0 1⟩	+
1:11M	[-5 1 0 0 1⟩	+
11M up	[-5 1 0 0 1⟩	+
11/1M up	[-5 1 0 0 1⟩	+
1/11M up	[-5 1 0 0 1⟩	+
11:1M up	[-5 1 0 0 1⟩	+
1:11M up	[-5 1 0 0 1⟩	+
11M down	[5 -1 0 0 -1⟩	-
11/1M down	[5 -1 0 0 -1⟩	-
1/11M down	[5 -1 0 0 -1⟩	-
11:1M down	[5 -1 0 0 -1⟩	-
1:11M down	[5 -1 0 0 -1⟩	-

So we have the 5C. It is down, inherently. So saying "the 5C down" is redundant, but fine. We could say "the 5C up" to flip it, or "the 1/5C". So when we're considering the commas in the contexts where the comma is undirected, like, choosing tina commas or whatnot, we call this comma "the 1/5C", because in these same contexts its important for the 2,3-free prime content to be directed. But in the context of notating a JI piece, it's generally more helpful to think of it as "the 5C", because with respect to the harmonic series, its the comma which notates the otonal 5.

So I still get it "my way" insofar as direction of the comma is separated from direction of the 2,3-free prime content, because anything with an explicit "up" is definitely positive cents, and anything with an explicit "down" is definitely negative cents; the up/down is not required, but it's allowed. And you get it "your way" so that the comma which notates the harmonic is the nC and the one which notates the subharmonic is the 1/nC (which is similar to my insistence that the name matches the sign of the 2,3-free prime content, but without the fixation on the undirected comma forcing whichever one is down to be written the same way but with "down" appended). And I still get it "my way" in the sense that when dealing with undirected commas the comma names are directed, i.e. the name of the comma class I need to worry about when writing Sagittal code is the 1/5C.

[volleo6144]

@volleo6144, when you found that counterexample, did you mean to imply that it was the simplest possible counterexample? Or is it still an open question whether there are many much simpler counterexamples?

I certainly didn't mean it as the simplest possible (there's certainly another prime combination out there that would let you do this with an N2D3P9 below 7×10⁵⁷), but I'm still not sure if there are any that are simple enough to ever be musically relevant.

...wait, we already have sub-tina diacritics and therefore sub-tina distinctions, who am I kidding that "musical relevance" is relevant? (Did I ever mention that I felt that 8539edo—and other edos—is a better choice for comma boundaries—and for tinas in general—than 809eda—and other edas—or did I delete that post before posting it? Either way, it's nearly irrelevant on the scale of even L-dieses, where the distinction changes the L/SS boundary from 487.95 to 487.97.)

[Dave Keenan]

Phew! I think we're in agreement. But ...

did you really mean to write:
5:7k [10 -6 1 -1⟩ +
7:5k [10 -6 1 -1⟩ +
?

Because our convention is that 5:7 = 7:5 = 7/5 (≠ 5/7), and you (correctly) have:
7/5k [-10 6 -1 1⟩ -

You were just testing to see if I was paying attention. Right?

[cmloegcmluin]

Phew indeed!

I've corrected 5:7k and 7:5k. I figured I couldn't make a table that big and repetitive without at least one error.

[cmloegcmluin]

I certainly didn't mean it as the simplest possible (there's certainly another prime combination out there that would let you do this with an N2D3P9 below 7×10⁵⁷), but I'm still not sure if there are any that are simple enough to ever be musically relevant.

I found a much simpler example of a pair of commas which break the naming scheme's complexity aspect. Feast your eyes upon the 5²⋅7/11C:

[15 -12 2 1 -1⟩

or is this the 5²⋅7/11C:

[-23 12 2 1 -1⟩

they are -33.325¢ and 13.595¢, respectively, the former falling just underneath the upper bound of the Comma size category, downward, and the latter falling just above the lower bound of the Comma size category, upward. As you can see, they are separated by two Pythagorean commas, [-19 12⟩. They have N2D3P9 of 98.032, so theoretically this 2,3-free class even just barely makes the top 100 most popular ones Sagittal would seek to notate.

So according to my 2-exponent tie-breaker, the latter comma would be demoted to the c5²⋅7/11C, because abs(-23) > abs(15).

[cmloegcmluin]

Just to be sure, I added a new column to my most recent table of comma name inputs to monzos, indicating with asterisks which of the name inputs are preferred, i.e. would be the ones given for those monzos on output. Please let me know if there is any disagreement there.

I have one additional question regarding comma names: I've seen the unabbreviated names given with hyphens, e.g. the complex-5-Comma. That's nice because there are no spaces in it, which is sometimes good for some humans, and its definitely good for code being able to recognize the start and end of a comma name. But it's certainly possible for the code to work with comma names with spaces in them, say, if you just put them in quotes, like "complex 5 Comma". Should I support that, or is there any reason (I can't imagine any) why that would be bad? And if so, should that be the preferred way of outputting them?

[cmloegcmluin]

Additional question re: comma naming.

So I've found I have a need to regularly refer to the half apotome. In code, I have a precise representation of its value. However, when I want to give it as an input, currently the only way for me to provide it is in the form of cents. I don't have its cents amount memorized, and I really don't want to. Mightn't we prescribe an official way to indicate half commas?

sq3A
half-3A
√3A

Whatever it is, it'd have to contend with the existing complexity prefixes. None of these suggestions conflict or create ambiguities.

Alternatively, we could say that these aren't even JI values themselves anymore and are outside the purview of a comma naming scheme. Perhaps I should instead just use the "scaled monzo" syntax Dave and I were discussing somewhat recently:

[-11 7⟩(1/2)

Which may be preferable because it allows for easy specification of any fractional amount, e.g. a step of 809-EDA is
[-11 7⟩(1/809).

Okay, bonus question: can we assume 3 if it's not stated? i.e. should my code parse just "A" to [-11 7⟩ (and accordingly "C" to [-19 12⟩, etc.?)