interval of equivalence, unison-vector, period

Hi guys,

I've been diligently studying the tuning-math archives, and
am really confused about one thing.

(OK, many things ... but let's start here...)

> tuning-math message 823
> From: graham@m...
> Date: Thu Aug 23, 2001 7:22 am
> Subject: Re: Interpreting Graham's matrix
/tuning-math/message/823?expand=1
>
> The things that make this system different to the one
> before is that it isn't unitary, and only one column of
> the inverse depends on the first generator. It's the second
> criterion that allows us to draw the non-arbitrary
> distinction between "interval of equivalence" and
> "unison vector", and so throw away the former.

I'm having a really hard time understanding the differences
between "interval of equivalence", "period", and "unison-vector".

Why aren't they *all* unison-vectors?

-monz

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Oops ... my bad. Sent that one to the wrong list by mistake.
Sorry.

-monz

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--- In metatuning@y..., "monz" <joemonz@y...> wrote:
> Hi guys,
>
>
> I've been diligently studying the tuning-math archives, and
> am really confused about one thing.
>
> (OK, many things ... but let's start here...)
>
>
> > tuning-math message 823
> > From: graham@m...
> > Date: Thu Aug 23, 2001 7:22 am
> > Subject: Re: Interpreting Graham's matrix
> /tuning-math/message/823?expand=1
> >
> > The things that make this system different to the one
> > before is that it isn't unitary, and only one column of
> > the inverse depends on the first generator. It's the second
> > criterion that allows us to draw the non-arbitrary
> > distinction between "interval of equivalence" and
> > "unison vector", and so throw away the former.
>
>
> I'm having a really hard time understanding the differences
> between "interval of equivalence", "period", and "unison-vector".
>
> Why aren't they *all* unison-vectors?

The period is often 1/2-octave, 1/3-octave, 1/4-octave, 1/9-
octave, . . . so that's clearly not a "unison-vector".

The "interval of equivalence" is a unison vector in Graham's system,
but Graham's system seems more limited than Gene's. Gene treats it as
only one of the "constructing" consonant intervals, and then
somehow "sticks it back in at the end" with some LLL reduction of
something.

And recall that the interval of equivalence is usually a _large_
interval, usually an octave, so not really much like a _unison_ at
all!