Triprime commas

Here's a definition:

Triprime comma

Suppose W is a bival in some prime limit q. It has a basis we may
write as b_[pi, pj] where pi, pj are primes <= q; the coefficient of W
with basis b_[pi, pj] we may call w_[pi, pj]. If p = ithprime(n), we
define e(p) as +1 if n is even, and -1 if n is odd; hence e(2) = -1,
e(3) = +1, e(5) = -1 and so forth. Suppose p1 < p2 < p3 are three
primes <= q; we let

C = p1^(e(p2)e(p3)w_[p2, p3]) p2^(e(p1)e(p3)w_[p1, p3])
p3^(e(p2)e(p3)w_[p1, p2])

If C<1 we take the inverse, and if it is a power we reduce it by
taking the gcd of the exponents and dividing it out; in other words if
it is an nth power we take the nth root of it. The result is the
{p1, p2, p3}-triprime comma of the wedgie W; taken together these are
the triprime commas of W. It should be noted that sometimes the comma
is simply a 1; however any comma of W is a product of its triprime commas.

For trivals you would similarly get quadprime commas, and so forth;
and obviously for vals we can easily define the biprime commas.

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

> C = p1^(e(p2)e(p3)w_[p2, p3]) p2^(e(p1)e(p3)w_[p1, p3])
> p3^(e(p2)e(p3)w_[p1, p2])

Should be

C = p1^(e(p2)e(p3)w_[p2, p3]) p2^(e(p1)e(p3)w_[p1, p3])
p3^(e(p1)e(p2)w_[p1, p2])

of course.

hi Gene,

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

> Suppose W is a bival in some prime limit q.

please write me a defintion of "bival".

while you're at it, take a look at "val" and see if
it needs help.

> It has a basis

should i get a definition of this for my lattice-basis page?
or do i need a separate page? in any case, please define.

> we may write as b_[pi, pj] where pi, pj are primes <= q;
> the coefficient of W with basis b_[pi, pj] we may call
> w_[pi, pj]. If p = ithprime(n), we define e(p) as +1 if
> n is even, and -1 if n is odd; hence e(2) = -1, e(3) = +1,
> e(5) = -1 and so forth. Suppose p1 < p2 < p3 are three
> primes <= q; we let
>
> C = p1^(e(p2)e(p3)w_[p2, p3]) p2^(e(p1)e(p3)w_[p1, p3])
> p3^(e(p2)e(p3)w_[p1, p2])
>
> If C<1 we take the inverse, and if it is a power we reduce
> it by taking the gcd of the exponents and dividing it out;
> in other words if it is an nth power we take the nth root
> of it. The result is the {p1, p2, p3}-triprime comma of
> the wedgie W; taken together these are the triprime commas
> of W. It should be noted that sometimes the comma is simply
> a 1; however any comma of W is a product of its triprime
> commas.
>
> For trivals you would similarly get quadprime commas, and
> so forth; and obviously for vals we can easily define the
> biprime commas.

and now we can expect myriad posts from you with long dry
lists of numbers, as illustrations of this ... right?

i sure hope so! :)

when you post those illustrations, give me some ideas on
how i can make cool diagrams showing ordinary people what
you're going on about. thanks.

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > C = p1^(e(p2)e(p3)w_[p2, p3]) p2^(e(p1)e(p3)w_[p1, p3])
> > p3^(e(p2)e(p3)w_[p1, p2])
>
> Should be
>
> C = p1^(e(p2)e(p3)w_[p2, p3]) p2^(e(p1)e(p3)w_[p1, p3])
> p3^(e(p1)e(p2)w_[p1, p2])

Still screwed up. Let sign(n) be the signum function, +1 if n is
postive, and -1 if n is negative; in other words |n|/n, and let
e(a,b,c) = sign((a-b)*(b-c)*(c-a)). Then

C = p1^(e(p2,p3,p1)*w_[p2, p3]) *
p2^(e(p1,p3,p2)*w_[p1, p3]) *
p3^(e(p1,p2,p3)*w_[p1, p2])

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

> please write me a defintion of "bival".

A bival is the wedge product of two vals; in particular, a linear
temperament wedgie.

> should i get a definition of this for my lattice-basis page?
> or do i need a separate page? in any case, please define.

Define what?

> and now we can expect myriad posts from you with long dry
> lists of numbers, as illustrations of this ... right?
>
> i sure hope so! :)

From the 7-limit meantone wedgie, <<1 4 10 4 13 12||, we get the four
triprime comms |0 12 -13 4>, |6 0 -5 2>, |-13 10 0 -1>, |-4 4 -1 0>
which are the {3,5,7}, {2,5,7}, {2,3,7} and {2,3,5} commas
respectively. Going up to the 11-limit would add six more such commas;
for {2,3,11}, {2,5,11}, {2,7,11}, {3,5,11}, {3,7,11} and {5,7,11},
which of course would be different for the two different versions of
11-limit meantone.

> when you post those illustrations, give me some ideas on
> how i can make cool diagrams showing ordinary people what
> you're going on about. thanks.

I duuno. I find these commas useful for various purposes, such as
obtaining a comma basis, but no pictures spring to my mind.

http://tonalsoft.com/enc/triprime-comma.htm

i took the license to rearrange the text for what
i hope is a clearer presentation.

now, just waiting on the example data, so i can
make some pretty pictures ...

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> http://tonalsoft.com/enc/triprime-comma.htm
>
>
> i took the license to rearrange the text for what
> i hope is a clearer presentation.

Thanks, Monz.

> now, just waiting on the example data, so i can
> make some pretty pictures ...

Maybe this will help: we think of 7-limit temperaments sharing a
5-limit comma as belonging to the same extended family, and the name
family if they have corresponding periods and generators. This is a
stronger relationship than merely sharing a comma, since it means that
if you reduce down to the {2,3,5} temperament, they are the same.

However, sharing any tricomma will do something similar. For instance,
meantone and hemiwuerschmidt share the same {2,5,7} comma, 3136/3125.
They define the same {2,5,7} linear temperament, whose generator is a
whole tone which is half of a major third, and where 5/2 of a major
third, or five tones, is a 7/4. If you took a piece in meantone and
tossed out all of the notes which came out to an odd number of fifths
in terms of the circle of fifths, you'd get something which could be
retuned to hemiwuerschmidt (99&130), a much more accurate temperament
; whereupon you could go stick notes back in again, trying to come
close to what you started with. I haven't tried it but I wonder what
would happen if I did! Hemithirds is another temperament you could
play this game with, boosting your meantone piece to 118-equal.

hi Gene,

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

> > should i get a definition of this for my lattice-basis page?
> > or do i need a separate page? in any case, please define.
>
> Define what?

basis, as you used it here

> From the 7-limit meantone wedgie, <<1 4 10 4 13 12||,
> we get the four triprime comms |0 12 -13 4>, |6 0 -5 2>,
> |-13 10 0 -1>, |-4 4 -1 0> which are the {3,5,7}, {2,5,7},
> {2,3,7} and {2,3,5} commas respectively. Going up to the
> 11-limit would add six more such commas; for {2,3,11},
> {2,5,11}, {2,7,11}, {3,5,11}, {3,7,11} and {5,7,11},
> which of course would be different for the two different
> versions of 11-limit meantone.

i think you'll like the nice neat version of this which
i've added to the webpage.

> > when you post those illustrations, give me some ideas on
> > how i can make cool diagrams showing ordinary people what
> > you're going on about. thanks.
>
> I duuno. I find these commas useful for various purposes,
> such as obtaining a comma basis, but no pictures spring
> to my mind.

please explain to me how you find them useful for
"obtaining a comma basis".

also, what other "various purposes"?

i want to show with lattices why triprime commas are useful.

-monz

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

> i want to show with lattices why triprime commas are useful.

i'm thinking of an applet that will let the reader
switch back and forth between lattices of different
temperaments which share the same comma.

sort of like what's at the bottom of the "bingo-card" page.

http://tonalsoft.com/enc/bingo.htm

-monz