commas from wedgies (was: Proposal: a high-order septimal schisma)

(sorry ... the previous version of this post got away from me
too soon; ignore it.)

hi Gene,

> From: "genewardsmith" <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, September 04, 2002 11:27 AM
> Subject: [tuning] Re: Proposal: a high-order septimal schisma
>
>
> <snip>
>
> Let p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4, then
>
> p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2,
> u1*v3-v1*u3,u1*v4-v1*u4]
>
> Let r be the mapping to primes of an equal temperament given
> by r = [u1, u2, u3, u4], and s be given by [v1, v2, v3, v4]. This
> means r has u1 notes to the octave, u2 notes in the approximation
> of 3, and so forth; hence [12, 19, 28, 24] would be the usual
> 12-equal, and [31, 49, 72, 87] the usual 31-et. The wedge now is
>
> r ^ s = [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4,
> u4*v2-u2*v4,u2*v3-v2*u3]
>
> Whether we've computed in terms of commas or ets, the wedge product
> of the linear temperament is exactly the same, up to sign.
>
> If the wedgie is [u1,u2,u3,u4,u5,u6] then we have commas given by
>
> 2^u6 3^(-u2) 5^u1
> 2^u5 3^u3 7^(-u1)
> 2^u4 5^(-u3) 7^u2
> 3^u4 5^u5 7^u6

at last, i finally understand how you're calculating wedgies!

but that last bit has me a little confused.

from the example meantone wedgie [1,4,10,12,-13,4] which you gave here:

> From: "Gene Ward Smith" <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, September 09, 2002 10:50 PM
> Subject: [tuning-math] [tuning] Re: Proposal: a high-order septimal
schisma
>
> <snip>
>
> Wedge of two intervals:
>
> > > p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2,
> > > u1*v3-v1*u3,u1*v4-v1*u4]
>
> For example p = 126/125 and q=81/80, then p = 2^1 3^2 5^(-3) 7^1,
> so in vector form it is [1,2,-3,1]. Similarly,
> q=2^(-4) 3^4 5^(-1) 7^0, which in vector form is [-4,4,-1,0].
> Wedging the two gives the wedgie for meantone, but 126/125 ^ 225/224,
> for example, will work also.

i calculated these commas

[ 4 -4 1] = 80 / 81
[-13 10 -1] = 59049 / 57344
[ 12 -10 4] = 9834496 / 9765625
[ 12 -13 4] = 1275989841 / 1220703125

OK, so the syntonic comma (81/80) is there ...
but what happened to 126/125 and 225/224? why are
they not in this list, and why are the other ones there?

-monz
"all roads lead to n^0"

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

> at last, i finally understand how you're calculating wedgies!

For the 7-limit, at least. :)

> OK, so the syntonic comma (81/80) is there ...
> but what happened to 126/125 and 225/224? why are
> they not in this list, and why are the other ones there?

Those four commas are special, in that they all leave out a prime; you should have written them as 4-vectors, not 3-vectors:

[-4,4,-1,0] -- the 5-limit comma
[-13,10,0,-1] -- the {2,3,7} comma
[12,0,-10,4] -- the {2,5,7} comma
[0,12,-13,4] -- the odd comma

The "kernel" is generated by these commas but also by 81/80 and
126/125, etc; so they are all in the kernel, but 126/125 and 225/224
have all four 7-limit primes in their factorization.