Re: names for 5-limit commas

This is a followup to Monz's query on the main tuning list. I posted it there too, which I probably shouldn't have done.

In the following, I will write "komma" for the more general term and "comma" for the restricted size range.

During the development of the sagittal notation system, George and I looked at a _lot_ of kommas, and eventually came up with a descriptive naming system that worked for us, in the sense that it
(a) is consistent with most existing names,
(b) automatically gives _unique_ names to hundreds of the most commonly encountered notational kommas,
(c) gives _short_ names to hundreds of the most commonly encountered notational kommas,
(d) gives names that make it possible to unpack the komma ratio from the name, if the naming-system is known, (and maybe even if it isn't).

The latter property is very valuable and is something that naming after people, or pieces of music, can never give.

This system won't necessarily suit the temperament-cataloging project, but it can probably be adapted.

The kommas we examined were generated by starting with a list of all the ratios that appear in the Scala archive, along with counts of their number of ocurrences (courtesy of Manuel Op de Coul).

The factors of 2 and 3 were removed from these ratios, then like ratios had their ocurrences combined and they were sorted by number of ocurrences (popularity). Then for each 2,3-reduced ratio we found all the kommas having an absolute 3-exponent not greater than 12 and a size not greater than 70 cents. I imagine these ranges would need to be increased for the temperament project. There was another constraint on komma "slope" which I won't go into here as it probably isn't relevant to your purposes.

The first part of the komma name is simply the two sides of the 2,3-reduced ratio, with a colon between. As a convention, we put the smallest number first. If one side of the reduced ratio is unity, we omit it, and the colon. When speaking, the colon is not pronounced. And you are welcome to pronounce 1 (which is equivalent to 3 in this system) as "Pythagorean", 5 as "classic", 7 as "septimal", 11 as "undecimal", etc., but I find it's easier just to say them in English.

The second part of the name is one of

schismina
schisma
kleisma
comma
small diesis
(medium) diesis
large diesis

This part of the name is made to do double duty. It not only gives the size category, but it distinguishes multiple kommas for the same 2,3-reduced ratio. To do this, the boundaries between categories have to be chosen very carefully. An intial set of category names and approximate boundaries were obtained by looking at the existing komma names in Scala's intnam.par.

Then we worked our way down the ratio popularity list, and whenever we found two kommas for the same ratio in the same category, we moved an existing boundary if we could do so without upsetting any previous ratios, or otherwise added a new boundary and category.

We didn't bother moving a boundary to separate two commas for the same ratio when they were less than about 0.7 c apart, but I imagine you would need to do this for your purposes. We also made boundary decisions based on the boundaries between sagittal symbols which would not be relevant. So you might repeat this process with larger 3-exponents etc., and come up with slightly different boundaries and more size categories. Some of the boundaries we obtained are very similar to some obtained by Monz in
http://sonic-arts.org/td/monzo/o483-26new5limitnames.htm

Here's what we are using (cents)
0
schismina
0.98
schisma
4.5
kleisma
13.47
comma
36.93
small diesis
45.11
(medium) diesis
56.84
large diesis
68.57

Some of these are very precisely defined as they are exact square-roots of 3-limit kommas. The boundaries seem to naturally want to fall on these. 56.84 is actually a half apotome. 45.11 is actually a half Pythagorean limma. The kleisma/comma boundary really wants to be half the Pythagorean comma and the comma/small-diesis boundary seems to want to be about 33.38 cents, which looks like half the [27,-17] comma. The medium/large diesis boundary is at half of [-30,19].

Maybe the next higher boundary (large diesis/small semitone?) should be at 78.49 c, half of [35,-22]. The schismina/schisma and schisma/kleisma boundaries were not actually needed (by us) to distinguish same-ratio kommas, and so are somewhat arbitrary, but maybe they can usefully be assigned roots of other 3-limit commas.

And by the way, I think that, in general, new or obscure temperaments are more usefully named after their generators, than their vanishing kommas.

Here are possible systematic descriptive names for all the 5-limit commas in the tuning list database, in order of decreasing size. By the way, why is this database at tuning and not tuning-math?

I pronounce "^" below as "to the".
Also note that "1" or "3" can be pronounced "Pythagorean", and "5" can be pronounce "classic" or not pronounced at all.

Current name (Scala) Possible systematic name
-------------------------------------------
large limma 25-semitone
diatonic semitone 5-semitone
major limma 5-limma
limma 1-limma or 3-limma
classic
chromatic semitone 25-small-semitone
major diesis 625-large-diesis
<unnamed> 625-medium-diesis
maximal diesis 125-medium-diesis
minor diesis 125-small-diesis
small diesis 5^5-comma
minimal diesis 625-comma
Pythagorean comma 1-comma or 3-comma
syntonic comma 5-comma
diaschisma 25-comma
<unnamed> 125-comma
<unnamed> 5^7-comma
Wuerschmidt's comma 5^8-kleisma
semicomma 5^7-kleisma
<unnamed> 5^9-kleisma
kleisma 5^6-kleisma
<unnamed> 5^5-kleisma
parakleisma 5^13-kleisma
<unnamed> 625-schisma
<unnamed> 5^14-schisma
19-tone comma 5^19-schisma
<unnamed> 5^17-schisma
<unnamed> 125-schisma
schisma 5-schisma
<unnamed> 5-schismina
<unnamed> 5^15-schismina
<unnamed> 5^17-schismina
ennealimma 5^18-schismina
<unnamed> 5^31-schismina
<unnamed> 5^16-schismina
<unnamed> 5^33-schismina
<unnamed> 5^51-schismina
monzisma 25-schismina
<unnamed> 5^14-schismina
<unnamed> 5^47-schismina
<unnamed> 5^35-schismina
<unnamed> 5^37-schismina
<unnamed> 5^49-schismina
atom of Kirnberger 5^12-schismina

So it looks like the schismina/schisma boundary needs to be at 1.807523 c which is half the [-84,53] 3-schisma.

--- In tuning-math@yahoogroups.com, David C Keenan <d.keenan@b...>
wrote:
> Here are possible systematic descriptive names for all the 5-limit
commas
> in the tuning list database, in order of decreasing size.

You've given the names, but not the commas.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, David C Keenan <d.keenan@b...>
> wrote:
> > Here are possible systematic descriptive names for all the 5-limit
> commas
> > in the tuning list database, in order of decreasing size.
>
> You've given the names, but not the commas.

And I suppose you want me to fetch your pipe and slippers too. ;-)

Please save me some typing (and typing errors) and just go to the
5-limit-commas database associated with the Yahoo Group "tuning", and
click the appropriate heading to sort it by size in cents, then click
the same again to make it _decreasing_ order. My list is intended to
exactly match that one, and if it doesn't, the current name I gave
should help.

Maybe we need a new column in that database for these systematic names.

Here's a better idea. Instead of looking up the database, test the naming system and see if you can unambiguously extract the commas from the names.

In case anyone missed it, the neat thing about this naming system is that the last part of the name not only gives the size category, but it also gives the powers of 2 and 3, indirectly.

I've shown the boundaries between the size categories below.

To extract the comma from the name, try combining the number in the first part of the name with successive powers of 3, in the series 0, 1, -1, 2, -2, 3, -3, 4, -4, ... and whatever power of 2 is required to bring its size back into the range -600 c to +600 c. The first one to have its absolute value fall into the correct size category is it.

Possible systematic name
------------------------
sqrt(2^2/3^1) ~249.0 c ?
1-tone or 3-tone
sqrt(2^5/3^3) ~147.1 c ?
25-semitone
5-semitone
sqrt(3^2/2^3) ~102.0 c
5-limma
1-limma or 3-limma
sqrt(3^31/2^49) ~80.3 c
25-small-semitone
sqrt(3^19/2^30) ~68.6 c
625-large-diesis
sqrt(3^7/2^11) ~56.8 c
625-medium-diesis
125-medium-diesis
sqrt(2^8/3^5) ~45.1 c
125-small-diesis
sqrt(2^27/3^17) ~33.4 c
5^5-comma
625-comma
1-comma or 3-comma
5-comma
25-comma
125-comma
5^7-comma
sqrt(3^12/2^19) ~11.7 c
5^8-kleisma
5^7-kleisma
5^9-kleisma
5^6-kleisma
5^5-kleisma
5^13-kleisma
sqrt(2^317/3^200) ~4.5 c ?
625-schisma
5^14-schisma
5^19-schisma
5^17-schisma
125-schisma
5-schisma
sqrt(3^53/2^84) ~1.8 c
5-schismina
5^15-schismina
5^17-schismina
5^18-schismina
5^31-schismina
5^16-schismina
5^33-schismina
5^51-schismina
25-schismina
5^14-schismina
5^47-schismina
5^35-schismina
5^37-schismina
5^49-schismina
5^12-schismina
0

I wrote:
"The first one to have its absolute value fall into the correct size
category is it."

I meant "absolute value of its size in cents". And should then have
said, "If it's a negative number of cents then we flip the signs of
all the prime exponents, and that's it."

And the last boundary in the list should not have been simply "0", but
"sqrt(3^0/2^0) 0 c"

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> > You've given the names, but not the commas.
>
> And I suppose you want me to fetch your pipe and slippers too. ;-)

No, but if you list names for commas, it hardly makes sense not to
list the commas.

This scheme is of course _very_ similar to the one by Joe Monzo in
http://sonic-arts.org/td/monzo/o483-26new5limitnames.htm

The differences are:
(a) it uses 5, 25, 125, 625, 5^5, 5^6, etc. instead of pental,
bipental, tripental, tetrapental, pentapental, hexapental, etc. By the
way, it seems reasonable to me, to expand the power when the result is
3 digits or less.
Hellenising the numbers as Monz did, gets a bit unwieldy when we go
beyond the 5-limit and have 5:7 kommas and 35 kommas and 7:25 kommas
and 11:35 kommas and so on.
(b) our choice of size-category boundaries allows us to ignore the
sign of the power of 5 relative to the powers of 2 and 3, and so we
don't need the "super" or "sub" prefixes.
(c) in addition to small movements of the boundaries, we find no need
for a boundary near 25 cents, and so Monz's small-dieses and commas
all become our commas. Also, his great dieses become our small and
medium dieses; his small-semitones become our large dieses and small
semitones, and his limmas become our limmas and semitones.

By the way, I should have shown the small-semitone/limma boundary as
tentative too (with a question-mark). It's still unclear to me whether
it is best set at
sqrt(3^31/2^49) ~80.3 c
or
sqrt(2^35/3^22) ~78.5 c