Short commas

If the "length" of a comma is the length of its minimal comma pump, not counting any factors of 2 or 1/2 which are irrelevant, then if we look at superparticular commas under the 17-limit with length less than 4, we get the Archipelago suspects: 325/324, 364/363, 676/675 and 1001/1000. By this I mean that these other short 13-limit commas, 325/324, 364/363 and 1001/1000 seemed to work well with 676/675 and turned up when temperaments were being considered. Putting them all together yields tritikleismic, and that might be something to look at more deeply just because of all those short comma pumps.

Another gang of short and accurate commas in the 19-limit is 969/968, 1331/1330, 1445/1444, 1729/1728. Adding the short 17-limit comma 833/832 and the short 13-limit comma 1001/1000 (the lummaism?) yields a rank two temperament for which 198edo could be used as a tuning, in which case the generator would be 43\198. This might seem absurdly complex, which is certainly what you'd think looking at it in terms of MOS scales. However, all of those short sequences might come to our rescue in a non-MOS scale such as a MODMOS or hobbit, etc. Looking at the complexity of the various pumps involved in this temperament would be a place to start.

On Thu, Apr 14, 2011 at 11:20 AM, genewardsmith
<genewardsmith@...t> wrote:
>
> If the "length" of a comma is the length of its minimal comma pump, not counting any factors of 2 or 1/2 which are irrelevant, then if we look at superparticular commas under the 17-limit with length less than 4, we get the Archipelago suspects: 325/324, 364/363, 676/675 and 1001/1000. By this I mean that these other short 13-limit commas, 325/324, 364/363 and 1001/1000 seemed to work well with 676/675 and turned up when temperaments were being considered. Putting them all together yields tritikleismic, and that might be something to look at more deeply just because of all those short comma pumps.

Why superparticular?

> Another gang of short and accurate commas in the 19-limit is 969/968, 1331/1330, 1445/1444, 1729/1728. Adding the short 17-limit comma 833/832 and the short 13-limit comma 1001/1000 (the lummaism?)

The lummasma? The lummisma? The lumasma?

> yields a rank two temperament for which 198edo could be used as a tuning, in which case the generator would be 43\198. This might seem absurdly complex, which is certainly what you'd think looking at it in terms of MOS scales. However, all of those short sequences might come to our rescue in a non-MOS scale such as a MODMOS or hobbit, etc. Looking at the complexity of the various pumps involved in this temperament would be a place to start.

I should add that developing further this concept of there being a
"moment of omnitetrachordality," which is probably better described as
a moment of local quasi-periodicity, might lead to some of the most
practical and/or "tonal" scales for complex temperaments that we've
seen yet. Maybe it makes sense to define the local sub-period in such
a sense that the resulting scale has a lot of these comma pumps in it.
As these scale will generalize tetrachordality, it may also be
relevant that tetrachordal thinking for temperaments like porcupine
seems to generate MODMOS's more often than not.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Why superparticular?

Because I was lazy. I didn't have a comma list around and didn't want to make one.

> > Another gang of short and accurate commas in the 19-limit is 969/968, 1331/1330, 1445/1444, 1729/1728. Adding the short 17-limit comma 833/832 and the short 13-limit comma 1001/1000 (the lummaism?)
>
> The lummasma? The lummisma? The lumasma?

Lummaisma is what I was trying to type.

> I should add that developing further this concept of there being a
> "moment of omnitetrachordality," which is probably better described as
> a moment of local quasi-periodicity, might lead to some of the most
> practical and/or "tonal" scales for complex temperaments that we've
> seen yet.

And if I could figure out how to construct one, I might think about it. But I'm not clear what you are saying with these subperiods.

Gene & Mike wrote:

> > Another gang of short and accurate commas in the 19-limit
> > is 969/968, 1331/1330, 1445/1444, 1729/1728. Adding the short
> > 17-limit comma 833/832 and the short 13-limit comma 1001/1000
> > (the lummaism?)
>
> The lummasma? The lummisma? The lumasma?

I'd vote for lummisma. But why did my name come to mind
regarding 1001/1000?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Gene & Mike wrote:
>
> > > Another gang of short and accurate commas in the 19-limit
> > > is 969/968, 1331/1330, 1445/1444, 1729/1728. Adding the short
> > > 17-limit comma 833/832 and the short 13-limit comma 1001/1000
> > > (the lummaism?)
> >
> > The lummasma? The lummisma? The lumasma?
>
> I'd vote for lummisma. But why did my name come to mind
> regarding 1001/1000?

So far as I know, you were the first to mention it.

Gene wrote:

> > > The lummasma? The lummisma? The lumasma?
> >
> > I'd vote for lummisma. But why did my name come to mind
> > regarding 1001/1000?
>
> So far as I know, you were the first to mention it.

I just searched the archives. It appeared in Paul's "list
of superparticular unison vectors (within 13-limit)" in 2001.
You mentioned it as one of the "12 smallest 13-limit
superparticular commas" in 2003. It showed up in a
denominator-limited comma search of mine in 2005 (1716/1715
was better). There may be other references I missed.

-Carl

On Thu, Apr 14, 2011 at 1:37 PM, genewardsmith
<genewardsmith@...> wrote:
>
> And if I could figure out how to construct one, I might think about it. But I'm not clear what you are saying with these subperiods.

I guess a lot of what I'm trying to get at is this:

/tuning-math/message/19085

As you can see, 81/80 still vanishes in this scale, as evidenced by
the fact that 9/8 and 10/9 are still the same size. However, it's
constructed from a different paradigm than just stacking generators
one after the other. Since sometimes the MOS's of various temperaments
don't really do the trick, then perhaps constructing scales in this
way can be useful. These will all be MODMOS's of the temperament.

One useful property that I think that they'll have is that sometimes,
if you can pin down a theorem that eludes me, I think they'll be
"locally periodic" with respect to some subperiod. In the case of
Paul's pentachordal scales, this is the case with the 4/3. It seems
like Paul's scales are basically constructed this way, e.g. instead of
using the true period of the scale, which is 1/2 oct, he went with the
tempered 3/2 as the pseudo-period of the scale, and in so doing
arrived at an omnitetrachordal scale.

Lastly, I'm trying to figure out exactly what causes
omnitetrachordality to occur at all. It seems like
4/3-omnitetrachordal scales, like Paul's SPM scales and meantone,
occur because the generator is a fourth. I can't get the same results
with some MODMOS of augmented[9], though, no matter how hard I seem to
try.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Apr 14, 2011 at 1:37 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > And if I could figure out how to construct one, I might think about it. But I'm not clear what you are saying with these subperiods.
>
> I guess a lot of what I'm trying to get at is this:
>
> /tuning-math/message/19085

I don't see a definition of periods within periods.