Gene's PB formula, generalized (was: a notation for Schoenberg's...)

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 8:36 PM
> Subject: [tuning-math] Re: a notation for Schoenberg's rational
implications
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > And the JI periodicity-block scale derived from this
>
> again, i'm baffled. how do you get a PB when you have one too many
> UVs?

i'm using the formula Gene posted, which i generalized to:

> for a set of i rational unison-vectors {u1/v1, ... ui/vi},
> for any non-zero I can define a scale by calculating for 0<=n<d
>
> step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d)
> (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d)

but oops! ... i realize now that this is still not entirely
generalized.

all those numbers (7,12,7,-2,5) are from a particular set of
homomorphisms (the first Schoenberg PB Gene calculated, back
around Christmas), and need to be replaced by variables.

i don't really know what to call them, so i'll just make this
do: {hv, hw, hx, hy, hz}. it's the top row of numbers in the
adjoint (or is it a unimodular inverse?) of the kernel.

so the generalized formula really is:

for a set of i rational unison-vectors {u1/v1, u2/v2,... ui/vi},
where {hx, hy, ...hq} is the top row of the unimodular adjoint
of the kernel matrix of the unison-vectors, for any non-zero
I can define a scale by calculating for 0 <= n < d :

step[n] = (u1/v1)^round(hx*n/d) * (u2/v2)^round(hy*n/d)
* ... (ui/vi)^round(hq*n/d) .

-monz

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> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 9:03 PM
> Subject: [tuning-math] Gene's PB formula, generalized (was: a notation for
Schoenberg's...)
>
>
>
> > From: paulerlich <paul@stretch-music.com>
> > To: <tuning-math@yahoogroups.com>
> > Sent: Sunday, February 03, 2002 8:36 PM
> > Subject: [tuning-math] Re: a notation for Schoenberg's rational
> implications
> >
> >
>
> so the generalized formula really is:
>
>
> for a set of i rational unison-vectors {u1/v1, u2/v2,... ui/vi},
> where {hx, hy, ...hq} is the top row of the unimodular adjoint
> of the kernel matrix of the unison-vectors, for any non-zero
> I can define a scale by calculating for 0 <= n < d :
>
> step[n] = (u1/v1)^round(hx*n/d) * (u2/v2)^round(hy*n/d)
> * ... (ui/vi)^round(hq*n/d) .

oops ... that last bit should read * ... (ui/vi)^round(hi*n/d) .

-monz

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

> i don't really know what to call them, so i'll just make this
> do: {hv, hw, hx, hy, hz}. it's the top row of numbers in the
> adjoint (or is it a unimodular inverse?) of the kernel.

I would call those hv(2), hw(2), hx(2), hy(2) and hz(2), where the h's are vals, which you can equate to column vectors.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > i don't really know what to call them, so i'll just make this
> > do: {hv, hw, hx, hy, hz}. it's the top row of numbers in the
> > adjoint (or is it a unimodular inverse?) of the kernel.
>
> I would call those hv(2), hw(2), hx(2), hy(2) and hz(2), where the
h's are vals, which you can equate to column vectors.

So why is there an "extra vector" used in constructing the PB? Is
this just an arbitrary vector independent of the others, or is it a
step vector, or what?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> So why is there an "extra vector" used in constructing the PB? Is
> this just an arbitrary vector independent of the others, or is it a
> step vector, or what?

It's the step vector, the rest are the commas.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > So why is there an "extra vector" used in constructing the PB? Is
> > this just an arbitrary vector independent of the others, or is it
a
> > step vector, or what?
>
> It's the step vector, the rest are the commas.

aha! monz please take note.