Name that comma!

So far no one has given a name to 2 (25/27)^9, which is a comma of
considerable significance. If that keeps up, it means it's time to
play

***** NAME THAT COMMA! *****

The winner, I hope, gets to have his or her name placed on Manuel's
list. It belongs there--for instance, here is the line on Paul's
chart we were talking of: 27, 72, 99, 171, 198, 270, 342, 414, 441,
612, 684, 1224, 1395. All of these are divisible by 9 (since nine
27/25 make up the octave) and all are worthy of notice as scale
divisions.

--- In tuning@y..., genewardsmith@j... wrote:

> So far no one has given a name to 2 (25/27)^9, which is a comma of
> considerable significance.

Another power-of-nine, 5-limit comma to go with it tells us that nine
Pythagorean thirds is almost exactly three octaves and a chromatic
semitone, that is (81/64)^9 ~ 8 25/24. It doesn't seem to figure as
importantly, turning up in 53, 612, and the 665 (which owns an even
more impressive 3-limit comma than does 53.)

--- In tuning@y..., genewardsmith@j... wrote:
> So far no one has given a name to 2 (25/27)^9, which is a comma of
> considerable significance. If that keeps up, it means it's time to
> play
>
> ***** NAME THAT COMMA! *****
>
> The winner, I hope, gets to have his or her name placed on Manuel's
> list. It belongs there--for instance, here is the line on Paul's
> chart we were talking of: 27, 72, 99, 171, 198, 270, 342, 414, 441,
> 612, 684, 1224, 1395. All of these are divisible by 9 (since nine
> 27/25 make up the octave) and all are worthy of notice as scale
> divisions.

I certainly don't want my name on it, but it could be called the "nonolimmal" comma or something
like that . . .

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:

> I certainly don't want my name on it, but it could be called
the "nonolimmal" comma or something
> like that . . .

What *would* you want your name on? I like 22^3/(22^3-1) myself,
because it has a perfect cube as numerator, but 2^12 and 3^6 are kind
of nice also. As for "nonolimmal", it sounds like it is being named
in honor of Luigi Nono, and I say "no no" to that. :(

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., genewardsmith@j... wrote:
>
> > I certainly don't want my name on it, but it could be called
> the "nonolimmal" comma or something
> > like that . . .
>
> What *would* you want your name on?

Something I use in my music -- like 50:49.

> I like 22^3/(22^3-1) myself,
> because it has a perfect cube as numerator, but 2^12 and 3^6 are kind
> of nice also.

Does 22^3-1 factor into 2^12 and 3^6??

> As for "nonolimmal", it sounds like it is being named
> in honor of Luigi Nono, and I say "no no" to that. :(

Ennealimmal?

[Paul E wrote:]
>Does 22^3-1 factor into 2^12 and 3^6??

10647: 3^2 7 13^2

Would you like a prime-factorization program?

JdL

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Ennealimmal?

That has a nice sound--better than enneagram, for starters--but
shouldn't it be "ennealimma"? What does "limmal" mean?

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Paul E wrote:]
> >Does 22^3-1 factor into 2^12 and 3^6??
>
> 10647: 3^2 7 13^2
>
> Would you like a prime-factorization program?
>
> JdL

Matlab does it but I'm not in the office on weekends . . .

So I wonder what Gene meant?

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Ennealimmal?
>
> That has a nice sound--better than enneagram, for starters--but
> shouldn't it be "ennealimma"? What does "limmal" mean?

It's the adjective form of limma. Think of "tritonic comma", for
example. But since the interval in question literally _is_ nine
limmae (reduced by an octave), I think "ennealimma" is fine!

[Paul E wrote:]
>>>Does 22^3-1 factor into 2^12 and 3^6??

[I wrote:]
>>10647: 3^2 7 13^2

>>Would you like a prime-factorization program?

[Paul:]
>Matlab does it but I'm not in the office on weekends . . .

>So I wonder what Gene meant?

He's identified a genuine comma, a "surprising" one, perhaps, since
the numbers are so large yet differ only by 1. Henceforth, this must
be know as the genewardensmith comma.

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Paul E wrote:]
> >>>Does 22^3-1 factor into 2^12 and 3^6??
>
> [I wrote:]
> >>10647: 3^2 7 13^2
>
> >>Would you like a prime-factorization program?
>
> [Paul:]
> >Matlab does it but I'm not in the office on weekends . . .
>
> >So I wonder what Gene meant?
>
> He's identified a genuine comma, a "surprising" one, perhaps, since
> the numbers are so large yet differ only by 1. Henceforth, this
must
> be know as the genewardensmith comma.

But what did 2^12 and 3^6 come from?

P.S. Gene -- did this discovery have anything to do with the ABC
stuff you were talking about on tuning-math@yahoogroups.com?

P.P.S. What ETs swallow this "comma"?

P.P.P.S. Most of the time, the term "comma" is used for larger
intervals than "kleisma", etc. . . . there seems to be a set of
general terms for really small intervals.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> > He's identified a genuine comma, a "surprising" one, perhaps,
since
> > the numbers are so large yet differ only by 1. Henceforth, this
> must
> > be know as the genewardensmith comma.

Such a great name. :)

> But what did 2^12 and 3^6 come from?

They are commas also--2^12/(2^12-1) = 2^12 / 3^2 5 7 13 and
3^6/(3^6-1) = 3^6 / 2^3 7 13. It's too late for anyone to get their
name on 3^4 or 7^4, but these seem to be free. There are also things
like squares of triangles of triangles, etc.

> P.S. Gene -- did this discovery have anything to do with the ABC
> stuff you were talking about on tuning-math@y...?

I was just looking at superparticular commas which have things like
powers higher than squares as numerator; possibly ABC was an indirect
inspiration.

> P.P.S. What ETs swallow this "comma"?

It rounds up many of the usual suspects: 16,17,22,24,29,41,46,
58,80,87,121,224,270,311,494,684,1178,1506,2190,2684.

> P.P.P.S. Most of the time, the term "comma" is used for larger
> intervals than "kleisma", etc. . . . there seems to be a set of
> general terms for really small intervals.

I don't have the impression it is that organized; in any case, some
dictionary definitions for "comma" make it sound as if it is a
generic term, even if that is not what was intended.

--- In tuning@y..., genewardsmith@j... wrote:

> They are commas also--2^12/(2^12-1) = 2^12 / 3^2 5 7 13 and
> 3^6/(3^6-1) = 3^6 / 2^3 7 13.

Oh, I get it. As superparticulars, these should all be in John
Chalmers's list in Xenharmonikon 17, under the 13-limit.

> > P.P.S. What ETs swallow this "comma"?
>
> It rounds up many of the usual suspects: 16,17,22,24,29,41,46,
> 58,80,87,121,224,270,311,494,684,1178,1506,2190,2684.

One of the reasons I like to stick with ETs consistent with respect
to the basic consonant intervals making up the comma in question when
such a question is posed, is that one may have reasons to use a
different mapping from the consonant intervals to the ET than any
given assumed mapping. I hope you'd agree. So presumably, all these
ETs are consistent with respect to the set {1,3,7,11,13}?
>
> > P.P.P.S. Most of the time, the term "comma" is used for larger
> > intervals than "kleisma", etc. . . . there seems to be a set of
> > general terms for really small intervals.
>
> I don't have the impression it is that organized;

It is, roughly -- "schisma" refers to intervals that are even
smaller . . .

> in any case, some
> dictionary definitions for "comma" make it sound as if it is a
> generic term, even if that is not what was intended.

How generic? No lower limit, and no upper limit either?

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, October 29, 2001 1:10 PM
> Subject: [tuning] Re: Name that comma!
>
>
> > > P.P.P.S. Most of the time, the term "comma" is used for larger
> > > intervals than "kleisma", etc. . . . there seems to be a set of
> > > general terms for really small intervals.
> >
> > I don't have the impression it is that organized;
>
> It is, roughly -- "schisma" refers to intervals that are even
> smaller . . .
>
> > in any case, some
> > dictionary definitions for "comma" make it sound as if it is a
> > generic term, even if that is not what was intended.
>
> How generic? No lower limit, and no upper limit either?

I was on a quest to straighten out this terminology
myself, a while back. Here's my attempt:
http://www.ixpres.com/interval/td/monzo/o483-26new5limitnames.htm

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> I was on a quest to straighten out this terminology
> myself, a while back. Here's my attempt:
> http://www.ixpres.com/interval/td/monzo/o483-26new5limitnames.htm

"Anomaly" suggests something anomalous, which makes no sense. If a
new term needs to be coined, I'll suggest "jot". If a precise
definition needs to be made, perhaps for p/q>1 reduced to lowest
terms we could bound the value of ln(p/q)ln(p-q+1) to some agreed on
figure.

--- In tuning@y..., genewardsmith@j... wrote:

> "Anomaly" suggests something anomalous, which makes no sense.

It makes plenty of sense, and has good precedent in the literature.
The interval is an anomaly of JI with respect to any system in which
it serves as a unison vector.

> If a
> new term needs to be coined, I'll suggest "jot".

A new term for what?

> If a precise
> definition needs to be made, perhaps for p/q>1 reduced to lowest
> terms we could bound the value of ln(p/q)ln(p-q+1) to some agreed
on
> figure.

I don't understand what this is about. I thought Monz and I were just
showing that "limma", "diesis", "comma", "kleisma", and "schisma" (or
anything ending with a consonant +isma) seem to refer to successively
smaller classes of intervals.

What are you trying to do with the math here?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> It makes plenty of sense, and has good precedent in the literature.
> The interval is an anomaly of JI with respect to any system in
which
> it serves as a unison vector.

That means it only begans to make sense if there is a val hanging
about, but we were talking about small intervals _per se_. In any
case, I think "anomaly" is a bad word, given that it could be applied
to the fact that a small interval may be anomalously large (as 81/80
is in the 3-et, for example) or an interval which is larger than 1
may anomalously be represeted by something less than the unison. That
something is in the kernel of an et can't make it an anomaly, or else
pretty well everything is an anomaly.

> > If a
> > new term needs to be coined, I'll suggest "jot".

> A new term for what?

Small interval arising in music theory.

> > If a precise
> > definition needs to be made, perhaps for p/q>1 reduced to lowest
> > terms we could bound the value of ln(p/q)ln(p-q+1) to some agreed
> on
> > figure.

> What are you trying to do with the math here?

Give a precise, relativized meaning to "small", but I need to stick
in something about the p-limit as well.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > It makes plenty of sense, and has good precedent in the
literature.
> > The interval is an anomaly of JI with respect to any system in
> which
> > it serves as a unison vector.
>
> That means it only begans to make sense if there is a val hanging
> about, but we were talking about small intervals _per se_. In any
> case, I think "anomaly" is a bad word, given that it could be
applied
> to the fact that a small interval may be anomalously large (as
81/80
> is in the 3-et, for example) or an interval which is larger than 1
> may anomalously be represeted by something less than the unison.
That
> something is in the kernel of an et can't make it an anomaly, or
else
> pretty well everything is an anomaly.

Well, the term anomaly should at least be allowed its meaning in the
literature, as an interval that vanishes in 12-tET. So maybe 250:243
and 2401:2400 are not anomalies, but 81:80 and
>
> > > If a
> > > new term needs to be coined, I'll suggest "jot".
>
> > A new term for what?
>
> Small interval arising in music theory.

Too late -- jot already has a meaning in microtonal theory:

http://www.xs4all.nl/~huygensf/doc/measures.html
>
> > > If a precise
> > > definition needs to be made, perhaps for p/q>1 reduced to
lowest
> > > terms we could bound the value of ln(p/q)ln(p-q+1) to some
agreed
> > on
> > > figure.
>
> > What are you trying to do with the math here?
>
> Give a precise, relativized meaning to "small", but I need to stick
> in something about the p-limit as well.

Well anyway, what about the original point, which was that
a "schisma" or "x-ma" is usually smaller than a "diesis" which is
usually smaller than a "comma" which is usually smaller than
a "diesis" which is usually smaller than a "limma"?

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, October 29, 2001 5:55 PM
> Subject: [tuning] Re: Name that comma!
>
> I don't understand what this is about. I thought Monz and I were just
> showing that "limma", "diesis", "comma", "kleisma", and "schisma" (or
> anything ending with a consonant +isma) seem to refer to successively
> smaller classes of intervals.

Yes, Paul, you speak well for me in this case... that
was exactly my intention.

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, October 29, 2001 6:39 PM
> Subject: [tuning] Re: Name that comma!
>
>
> Too late -- jot already has a meaning in microtonal theory:
>
> http://www.xs4all.nl/~huygensf/doc/measures.html

BTW, I also have a "jot" entry
in the dictionary which gives some additional info:
http://www.ixpres.com/interval/dict/jot.htm

> Well anyway, what about the original point, which was that
> a "schisma" or "x-ma" is usually smaller than a "diesis" which is
> usually smaller than a "comma" which is usually smaller than
> a "diesis" which is usually smaller than a "limma"?

Be careful, Paul!... here, you've reversed the positions
of "diesis" and "comma" in the pantheon, whose order you
stated correctly before!

For intervals smaller than the 12-tET "semitone" 2^(1/12),
it's: limma > diesis > comma > schisma.

Diaschisma also figures in there somewhere... approximately
between diesis and comma.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Well anyway, what about the original point, which was that
> a "schisma" or "x-ma" is usually smaller than a "diesis" which is
> usually smaller than a "comma" which is usually smaller than
> a "diesis" which is usually smaller than a "limma"?

It's not generic. I think I'll stick with "comma".

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> > Well anyway, what about the original point, which was that
> > a "schisma" or "x-ma" is usually smaller than a "diesis" which is
> > usually smaller than a "comma" which is usually smaller than
> > a "diesis" which is usually smaller than a "limma"?
>
>
> Be careful, Paul!... here, you've reversed the positions
> of "diesis" and "comma" in the pantheon, whose order you
> stated correctly before!

Actually, I put diesis in two places above . . . oops!
>
> Diaschisma also figures in there somewhere... approximately
> between diesis and comma.
>
Trouble is that "the" diaschisma is only 19.5 cents.