an unnamed (?) 7-limit interval of interest

65536/64827 (~ 18.8314 cents) is |16 -3 -4>. I found this while studying lattices. It is derived in similar fashion to the kleisma/semicomma majeur (15625/15552), defined as |-6 -5 6>.

What should it be named, if it's not already? It's too big to be a kleisma, so I'm thinking "septimal semicomma majeur". Or is it one of the "Beta" ratios?

~Danny~

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> 65536/64827 (~ 18.8314 cents) is |16 -3 -4>. I found this while
studying
> lattices. It is derived in similar fashion to the kleisma/semicomma
majeur
> (15625/15552), defined as |-6 -5 6>.

Should be |16 -3 0 -4>, of course. That makes it one of those cool
{2,3,7} commas, like 64/63, 256/243, 1029/1024 or 118098/117649.

> What should it be named, if it's not already?

I dunno. It is supported by 53, 58 and 68, and the 53&58 temperament
starts off with a lot of no-fives stuff which reflects this comma.
This has got a generator of 21/16, which is the no-fives generator. So
3/(21/16)^4 = 65536/64827, and (7/16)(21/16)^3 ~ 64827/65536. Scala
thinks 21/16 is a narrow fourth if you want to work that into a name.

> 65536/64827 (~ 18.8314 cents) is |16 -3 -4>. I found this while studying
> lattices. It is derived in similar fashion to the kleisma/semicomma majeur
> (15625/15552), defined as |-6 -5 6>.
>
> What should it be named, if it's not already? It's too big to be a > kleisma,
> so I'm thinking "septimal semicomma majeur"...

That name won't work either, since it's also too big to be a semicomma. Maybe "septimal diaschisma".

From: "Gene Ward Smith":

>> What should it be named, if it's not already?
>
> I dunno. It is supported by 53, 58 and 68, and the 53&58 temperament
> starts off with a lot of no-fives stuff which reflects this comma.
> This has got a generator of 21/16, which is the no-fives generator. So
> 3/(21/16)^4 = 65536/64827, and (7/16)(21/16)^3 ~ 64827/65536. Scala
> thinks 21/16 is a narrow fourth if you want to work that into a name.

"Narrow septimal comma" it is then. And don't forget 41-equal, which is itself septimal-friendly. And undecimal.

I also see a need for 41-tone temperament that uses a cubic curve instead of a sine curve. And I think I like 665-tone better as a measurement for 7-limit.

~Danny~

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> 65536/64827 (~ 18.8314 cents) is |16 -3 -4>. I found this while
studying
> lattices. It is derived in similar fashion to the kleisma/semicomma
majeur
> (15625/15552), defined as |-6 -5 6>.
>
> What should it be named, if it's not already? It's too big to be a
kleisma,
> so I'm thinking "septimal semicomma majeur". Or is it one of
the "Beta"
> ratios?
>
> ~Danny~

Dave Keenan would call this the 2401-comma.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> Dave Keenan would call this the 2401-comma.

Where's the 2401?

Gene Ward Smith wrote (about my

> "George D. Secor" wrote:
>
>> Dave Keenan would call this the 2401-comma.
>
> Where's the 2401?

2401 = 7^4, and 64827 = 3^3 * 7^4. The comma is derived from four septimal minor sevenths and three perfect fifths.

~Danny~

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> Gene Ward Smith wrote (about my
>
> > "George D. Secor" wrote:
> >
> >> Dave Keenan would call this the 2401-comma.
> >
> > Where's the 2401?
>
> 2401 = 7^4, and 64827 = 3^3 * 7^4. The comma is derived from four
septimal
> minor sevenths and three perfect fifths.

That's not a 2401, it's a 64827. Why isn't this the 64827 comma?
If you want a 2401 in there, I'd suggest (2048/2025)/(2401/2400),
but then 2401/2400 still gets to be the 2401 comma, which was my real
point.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
>
> > Dave Keenan would call this the 2401-comma.

I've never seen it before, but I plugged it into my comma-namer
spreadsheet and "7^4-comma" (pronounced "seven-to-the-fourth comma")
is what it actually said. But it could certainly also be called the
"2401-comma" since 7^4 = 2401.

> Where's the 2401?

To me, the most significant thing about a comma, once you know its
approximate size, is its powers of primes greater than 3. By knowing
these and that it is in the range of about 12 to 33 cents (because it
is called a "comma" and not a kleisma" or "diesis" etc.) then you can
work out the relevant powers of 2 and 3 (by trial and error if
necessary) as the smallest that will bring 2401 into that range.

If they were not the smallest such powers, but the next smallest, it
would be prefixed with the word "complex". In practice the only such
comma I've seen is the complex Pythagorean comma [65 -41>.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> > Gene Ward Smith wrote (about my
> >
> > > "George D. Secor" wrote:
> > >
> > >> Dave Keenan would call this the 2401-comma.
> > >
> > > Where's the 2401?
> >
> > 2401 = 7^4, and 64827 = 3^3 * 7^4. The comma is derived from four
> septimal
> > minor sevenths and three perfect fifths.
>
> That's not a 2401, it's a 64827. Why isn't this the 64827 comma?
> If you want a 2401 in there, I'd suggest (2048/2025)/(2401/2400),
> but then 2401/2400 still gets to be the 2401 comma, which was my real
> point.

In my scheme, 2401/2400 is a schismina (i.e. smaller than 1.8 cents),
not a comma (except in the generic sense of being a small interval)
and it gets to be the 5^2:7^4-schismina or 25:2401-schismina.

I admit these names get rather ugly when you get into larger numbers
like these. But they are at least guaranteed unique while giving an
idea of the size. They usually have fewer syllables than the full
ratios, but not in this case.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> "Narrow septimal comma" it is then.

So that's for
[-16 3, 0 4> 18.8 c 65536:64827

Are you sure? :-)

Here are a few other interesting ratios that could reasonably be
called the "narrow septimal comma".

[ 0 -5, 1 2> 14.2 c 243:245
[ 6 3, -1 -3> 13.1 c 1715:1728
[ 1 5, 1 -4> 20.8 c 2401:2430
[-15 3, 2 2> 16.1 c 32768:33075
[-5 -3, 3 1> 21.9 c 864:875
[ 1 2, -3 1> 13.8 c 125:126
[ 5 -4, 3 -2> 13.5 c 3969:4000
[-9 3, -3 4> 22.3 c 64000:64827

-- Dave Keenan

And here are a couple with a zero power of 5 that could also
reasonably be called the "narrow septimal comma", but admittedly not
as interesting as
[-16 3, 0 4> 18.8 c 65536:64827

[-9 11, 0 -3> 15.0 c 175616:177147
[-20 2, 0 6> 16.9 c 1048576:117649

I know, I'm just a pedant and a spoil-sport. ;-)

-- Dave Keenan