Capstone temperament

Before we get to the range which defines the supertemperament (where
the nullspace of the matrix of monzos has dimension one, defining a
val, and thus an equal temperament) we have a range where the
nullspace is dimension two, and defines a linear temperament--which
I've mentioned before, under the name of the capstone temperament.
Here are mappings for capstones up to the 19-limit, and generators in
terms of the corresponding supertemperament. (This last is a little
crude, but does get us into the ballpark.)

5 limit: 81/80 meantone

7 limit: [[9, 15, 22, 26], [0, -2, -3, -2]] ennealimmal
generators: [1/9, 1/24]

11 limit: [[18, 28, 41, 50, 62], [0, 2, 3, 2, 1]] hemiennealimmal
generators: [1/18, 1/72]

13 limit: [[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]]
generators: [1/2, 47/270]

17 limit: [[1, 35, 221, 161, -5, 197, 367],
[0, -79, -517, -374, 20, -457, -858]]
generators: [1, 637/1506]

19 limit: [[1, 250, 324, 62, -178, -481, 1579, 258],
[0, -512, -663, -122, 374, 999, -3246, -523]]
generators: [1, 4143/8539]

The 13-limit capstone has reduced basis

[1716/1715, 2080/2079, 3025/3024, 4096/4095]

and wedgie

[22, 48, -38, -34, -54, 25, -122, -130, -167, -223,
-245, -303, 36, -11, -61].

It is well-covered by 494-et, and can be thought of as the
270&494 13-limit linear temperament. I don't want anyone to pitch a
fit, but I suppose Cap13 is one possible name.

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Before we get to the range which defines the
> supertemperament (where the nullspace of the matrix
> of monzos has dimension one, defining a val, and thus
> an equal temperament) we have a range where the
> nullspace is dimension two, and defines a linear
> temperament--which I've mentioned before, under
> the name of the capstone temperament.
> Here are mappings for capstones up to the 19-limit, and
> generators in terms of the corresponding supertemperament.
> (This last is a little crude, but does get us into the
> ballpark.)
>
> 5 limit: 81/80 meantone
>
> 7 limit: [[9, 15, 22, 26], [0, -2, -3, -2]] ennealimmal
> generators: [1/9, 1/24]

this looks important to me, and i want to understand it.
can you help, by labeling these lists of numbers?

in this example, i think i recognize [0, -2, -3, -2] as
a monzo, but it doesn't make sense as a [2, 3, 5, 7]-monzo.

and is [1/9, 1/24] meant to represent 2^(1/9) and 2^(1/24)?
if not, then what?

and what about [9, 15, 22, 26]? what's that?

Gene, it would help me a lot if you would *always* put
a brief legend before lists like this. pretty please?

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Gene,
>
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
> > Before we get to the range which defines the
> > supertemperament (where the nullspace of the matrix
> > of monzos has dimension one, defining a val, and thus
> > an equal temperament) we have a range where the
> > nullspace is dimension two, and defines a linear
> > temperament--which I've mentioned before, under
> > the name of the capstone temperament.
> > Here are mappings for capstones up to the 19-limit, and
> > generators in terms of the corresponding supertemperament.
> > (This last is a little crude, but does get us into the
> > ballpark.)
> >
> > 5 limit: 81/80 meantone
> >
> > 7 limit: [[9, 15, 22, 26], [0, -2, -3, -2]] ennealimmal
> > generators: [1/9, 1/24]
>
>
>
> this looks important to me, and i want to understand it.
> can you help, by labeling these lists of numbers?
>
> in this example, i think i recognize [0, -2, -3, -2] as
> a monzo, but it doesn't make sense as a [2, 3, 5, 7]-monzo.

it's the same thing that's shown in the linear temperament table on
your et dictionary page -- the mapping from generators to primes. so
for example, the third entry in each of the vectors, 22 and -3
respectively, refer to the fact that the third prime, 7, is
approximated in the temperament by stacking 22 of the first generator
(otherwise known as the period) and -3 of the second generator
(otherwise known as, simply, the generator).

> and is [1/9, 1/24] meant to represent 2^(1/9) and 2^(1/24)?

yes.

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> it's the same thing that's shown in the linear temperament
> table on your et dictionary page -- the mapping from
> generators to primes. so for example, the third entry in
> each of the vectors, 22 and -3 respectively, refer to the
> fact that the third prime, 7, is approximated in the
> temperament by stacking 22 of the first generator
> (otherwise known as the period) and -3 of the second
> generator (otherwise known as, simply, the generator).
>
> > and is [1/9, 1/24] meant to represent 2^(1/9) and 2^(1/24)?
>
> yes.

oh, OK. thanks.

-monz

spurred on by the work i did on Dave's list of "kommas",
i've made a huge list of all the 11-limit intervals where
the <3, 5, 7, 11>-monzo exponents are
<-15...+15, -10...+10, -4...+4, -2...+2>, arranged in
order by cents.

it takes quite a while to download ... there are over
28,000 intervals in the list. i hope it will be very
useful as a reference.

http://sonic-arts.org/dict/interval-list.htm

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > it's the same thing that's shown in the linear temperament
> > table on your et dictionary page -- the mapping from
> > generators to primes. so for example, the third entry in
> > each of the vectors, 22 and -3 respectively, refer to the
> > fact that the third prime, 7, is approximated in the
> > temperament by stacking 22 of the first generator
> > (otherwise known as the period) and -3 of the second
> > generator (otherwise known as, simply, the generator).
> >
> > > and is [1/9, 1/24] meant to represent 2^(1/9) and 2^(1/24)?
> >
> > yes.
>
>
>
> oh, OK. thanks.
>
>
>
>
> -monz

sorry, the third prime is 5, of course . . .

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> it takes quite a while to download ... there are over
> 28,000 intervals in the list. i hope it will be very
> useful as a reference.

I am, of course, expecting a name for each and every one of them.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > it takes quite a while to download ... there are over
> > 28,000 intervals in the list. i hope it will be very
> > useful as a reference.
>
> I am, of course, expecting a name for each and every one of them.

of course! ... but this time, we use *my* naming convention!

;-)

-monz