Which commas matter? (Was: Patent val page)

Michael said:

> > Here's the thing...now I get how/why commas vanish. But given a val...
> > how do I know which commas matter?

Mike replied:

> Do you mean how do you know which commas some val tempers out?

If Michael is like me, then no, that's not what he means; it's not hard to
find out what commas a val tempers out. The question is, what commas matter,
and why?

When a newbie comes to this list and to the wiki for the first time -- okay,
I really mean "when I came [etc]", but let's roll with it -- he sees a
shocking amount of information with little assistance (that he can
understand, anyway) in interpreting it. A huge amount of that information
pertains to commas and EDOs. (That makes sense, of course, given the regular
mapping paradigm. I'm not complaining, I'm just relating my experience.)

A typical example from the 121 EDO page:

"The 121 equal temperament divides the octave into 121 equal steps of 9.917
cents each. It has a distinctly sharp tendency, in that the odd primes from
3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit;
4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and
1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the
13-limit; 256/255, 375/374 and 442/441 in the 17-limit; 190/189 and 361/360
in the 19-limit. It also serves as the optimal patent val for 13-limit
grendel temperament."

I now understand what this is saying, but -- and I think this is where
Michael is coming from along with me -- I don't know why these particular
commas are musically useful.

I understand why tempering out 81/80 is useful: I'm able to play ii-V7-I
without a wolf interval.

I understand why tempering out 243/242 is useful: two approximate 11/9
neutral thirds make up an approximate 3/2.

I don't really understand why tempering out 15625/15552, the Kleisma, is
useful. I went to the Kleismic family page, which says, "...the generator is
a minor third, and that to get to the interval class of major thirds will
require five of these, and so to get to fifths will require six." In 12-tET,
stacks of minor thirds would bring me back to the octave: C-Eb-Gb-A-C, so my
melodic and chordal relationships will be different in Kleismic / Hanson /
Whatever temperament than they would in 12-tET. 19-tET *does* temper out the
Kleisma, so I'd get 315-632-947-63-378 cents instead of a cycle. But how
often do I repeatedly stack minor thirds on each other? The occasional
diminished chord, maybe, but not often. So what does this *really* mean to
me? (No, I don't expect you to answer that. It no doubt takes a lot of
experimentation.)

By and large, a comma list doesn't mean anything to me yet. It's like
Arabic: I know a few words and can read the letters, but I only rarely
derive meaning from it. For instance, I don't have a clue why I'd want to
temper out 4000/3969, but I stumbled across a reason to temper 6144/6125
once (though I didn't write anything using that scale).

I'm not complaining or anything, just saying that even when I get the
mathematical parts of what's going on, I think I rarely get the musical
parts.

On the other hand, sometimes I see explanations I can understand. One of the
reasons I'm probably going to play with Breedsmic is that it has some great
descriptions on its page:

-----
Breedsmic temperaments are rank two temperaments tempering out the breedsma,
2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and
by which two 60/49 intervals fall short. Either of these represent a neutral
third interval which is highly characteristic of breedsmic tempering...

It is also the amount by which four stacked 10/7 intervals exceed 25/6:
10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the
chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and
49/40 * (10/7)^2 = 5/2, relationships which will be significant in any
breedsmic temperament. As a consequence of these facts, the 49/40+60/49
neutral third and the 7/5 and 10/7 intervals tend to have relatively low
complexity in a breedsmic system.
-----

That's great information, and I think I understand how this temperament can
help me make music.

Regards,
Jake

Hi Jake.
Not sure if you use Scala but if you do and if you have a recent version, try locating the files "pump_kleisma.seq" and "shansx.scl" and then type something like "example pump_kleisma.seq pump_kleisma.mid".
Petr

Jake Freivald <jdfreivald@...> wrote:

> A typical example from the 121 EDO page:
>
> "The 121 equal temperament divides the octave into 121
> equal steps of 9.917 cents each. It has a distinctly
> sharp tendency, in that the odd primes from 3 to 19 all
> have sharp tunings. It tempers out 15625/15552 in the
> 5-limit; 4000/3969, 6144/6125, 10976/10935 in the
> 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit;
> 325/324, 352/351, 364/363 and 625/624 in the 13-limit;
> 256/255, 375/374 and 442/441 in the 17-limit; 190/189 and
> 361/360 in the 19-limit. It also serves as the optimal
> patent val for 13-limit grendel temperament."
>
> I now understand what this is saying, but -- and I think
> this is where Michael is coming from along with me -- I
> don't know why these particular commas are musically
> useful.

There probably the simplest commas that get tempered out.
Finding them is a short vectors in a lattice problem.

Giving meaning to commas is a different problem. It means
somebody has to look at each one in turn, and write
something about it.

I've uploaded my rough and ready list of 13-limit commas:

http://x31eq.com/commas.of.13

Some temperament names may have changed. Some comments are
out of date. I left out 11-limit commas because that's
what I decided to do.

Graham

Hello, Jake and all.

Certainly the regular mapping paradigm is one approach to
alternative tunings, but I wouldn't assume that it would be the
ideal approach for all experienced people in this area, let alone
all beginners.

The regular mapping paradigm does seem like an especially
interesting approach for people focusing on styles of harmony
involving different kinds of comma pumps, or seeking new types of
systems analogous to but different from some kind of familiar
diatonic set.

However, for beginners and others, there are a range of possible
directions, just or tempered, which invite other approaches and
paradigms.

First, a range of world musics tend to focus, to the degree they
involve fixed-pitch instruments, on customized irregular
tunings. The tuning of the Persian tar and setar, often with 17
notes per octave, is a good example (e.g. persian.scl or
persian-far.scl in the Scala archive). We find that Nelly Caron
and Dariouche Safvate describe one style of tuning, Hormoz Farhat
another, and Dariush Tala`i yet another. An instrument with fewer
notes per octave like the Persian santur also illustrates the
technique of customizing a tuning to fit a given family of
modes used in a performance -- with retuning to fit a different
modal family.

Secondly, irregular temperaments can be very attractive for
beginners as well as more experienced students of alternative
tunings, with George Secor's 17-note well-temperament (17-WT) as
a fine example.

<http://anaphoria.com/Secor17puzzle.pdf>

Third, focusing on the types or "families" of intervals a tuning
supports may often be more easily and quickly comprehensible than
starting with a list of the commas tempered out or observed ==
although both, naturally, may be of some interest.

For example, "at this location we have a small neutral seventh
around 1037 cents, somewhere between 20/11 at 1035 cents and
51/28 at 1038 cents" might be more immediately understandable to
many people than "this tuning effectively disregards the
561/560." Of course, the comparison of 20/11 and 51/28 might
engage one's curiosity as to the distinction between these
ratios, leading one to the 561/560 -- but starting with the
interval of 1037 cents actually there (nicely suited for a
Turkish Maqam Segah, for example) might be a "user-friendly"
approach, although simply listing the commas might be
"expert-friendly" in certain circles.

Also, both beginners and more experienced people in our area may
want to weigh the advantages and disadvantages of tempering out a
given comma. For maqam music, I would far rather observe the
243/242 or equivalent (27/22 vs. 11/9, or 12/11 vs. 88/81) than
temper it out, because traditional musicians relish the subtle
distinctions between larger and smaller neutral intervals, even
if some instruments tend toward 24-ED2.

Similarly, while one opinion holds that it's a disadvantage to
require 21/16 in a chain of fourths to get a septimal ratio like
7/3, I find that 9:12:16:21 is quite delicious! Someone sharing
my tastes might still, of course, want enough notes per octave so
as to have 4/3 as well as 21/16 available at a good number of
locations.

In short, I would consider the regular mapping paradigm as one
interesting approach, together with various types of irregular
temperaments, not to mention just intonation systems. For
beginners and others, which direction you are moving in may
influence your chosen tools for description and analysis.

With many thanks,

Margo Schulter
mschulter@...