Taking the inner product of a val and a monzo

Hi all,

If you take the inner product of a val and a monzo, you get a scalar.
If for some v and m <v|m> = 0, then v tempers m. OK.

What do other integer values of <v|m> end up denoting, however? Does
sign(<v|m>) specify if an interval is reversed? What does abs(<v|m>)
mean?

This seems relevant here, from the xenwiki page on hobbits:

> Denoting the OE seminorm for any element x of interval space by T(x), we first define the hobbit of an odd-numbered scale; that is, a scale for which v[1] is an odd number. If v[1] is odd then for each integer j, 0 < j < v[1], we choose a corresponding monzo m such that <v|m> = j, 0 < <J|m> < 1 where J is the JI mapping <log2(2) log2(3) ... log2(p)|, and T(m) is minimal.

Does j somehow end up meaning the number of iterations of the
generator that m maps to in v? Is that it?

-Mike

Mike Battaglia <battaglia01@gmail.com> wrote:

> If you take the inner product of a val and a monzo, you
> get a scalar. If for some v and m <v|m> = 0, then v
> tempers m. OK.
>
> What do other integer values of <v|m> end up denoting,
> however? Does sign(<v|m>) specify if an interval is
> reversed? What does abs(<v|m>) mean?

They count generator steps. For equal temperaments, that
means scale steps. The sign specifies the interval is
descending instead of ascending. The absolute value gives
you the undirected interval -- "a fourth" instead of "an
ascending fourth" or "a descending fourth".

> This seems relevant here, from the xenwiki page on
> hobbits:
>
> > Denoting the OE seminorm for any element x of interval
> > space by T(x), we first define the hobbit of an
> > odd-numbered scale; that is, a scale for which v[1] is
> > an odd number. If v[1] is odd then for each integer j,
> > 0 < j < v[1], we choose a corresponding monzo m such
> > that <v|m> = j, 0 < <J|m> < 1 where J is the JI mapping
> > <log2(2) log2(3) ... log2(p)|, and T(m) is minimal.
>
> Does j somehow end up meaning the number of iterations of
> the generator that m maps to in v? Is that it?

Isn't v the val that determines the scale steps? In that
case, j is the index of the pitch in the scale. 0 for a
unison, 1 for a second, 2 for a third, and so on, for the
diatonic scale.

The 0 < <J|m> < 1 condition is to make sure the interval is
between a unison and an octave. It looks redundant. Gene
will have to clarify.

Graham