Fwd: Re: interval of equivalence, unison-vector, period

--- In metatuning@y..., "paulerlich" <paul@s...> wrote:
--- 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.
--- End forwarded message ---

--- In metatuning@y..., "paulerlich" <paul@s...> wrote:
And recall that the interval of equivalence is usually a _large_
interval, usually an octave, so not really much like a _unison_ at
all!
--- End forwarded message ---

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, January 30, 2002 9:52 PM
> Subject: [tuning-math] Fwd: Re: interval of equivalence, unison-vector,
period
>
>
> >
> > 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".

But what *is* the period? I mean, not what interval or size,
but what is it? What significance does it have?

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, January 30, 2002 9:52 PM
> Subject: [tuning-math] Fwd: Re: interval of equivalence, unison-vector,
period
>
>
> And recall that the interval of equivalence is usually a _large_
> interval, usually an octave, so not really much like a _unison_ at
> all!

But ... but ...

Say the syntonic comma is a unison-vector. So pick a reference note;
the note a comma away is considered equivalent. But on an
"8ve"-equivalent lattice, the note *an "8ve" and a comma away
(~1222 cents) is also considered equivalent*!! So then why
not the "8ve" itself? I don't get it.

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Wednesday, January 30, 2002 9:52 PM
> > Subject: [tuning-math] Fwd: Re: interval of equivalence, unison-
vector,
> period
> >
> >
> > >
> > > 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".
>
>
> But what *is* the period? I mean, not what interval or size,
> but what is it? What significance does it have?

It's the smallest interval at which a scale can be transposed without
changing the scale at all. For example, the diminished (octatonic)
scale in 12-tET has a period of 1/4-octave. For another, my
symmetrical decatonic scale in 22-tET has a period of 1/2-octave.

>> > And recall that the interval of equivalence is usually a _large_
>> > interval, usually an octave, so not really much like a _unison_
>at
>> > all!
>
>
>> But ... but ...
>
>> Say the syntonic comma is a unison-vector. So pick a reference
note;
>> the note a comma away is considered equivalent. But on an
>> "8ve"-equivalent lattice, the note *an "8ve" and a comma away
>> (~1222 cents) is also considered equivalent*!! So then why
>> not the "8ve" itself?

On an octave-equivalent lattice, yes, it would be considered
equivalent. That's not enough to make it a unison vector, though. The
way Gene does things, unison vectors are all _small intervals_
defined with specific ratios, for example 81:80 but not 81:40, and
then Gene can construct temperaments or whatever, and then octave-
equivalence can be stuck back in at the end, if desired. If you don't
do it this way, you won't be able to deal with torsion properly.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, January 31, 2002 1:02 PM
> Re: interval of equivalence, unison-vector, period
>
>
> > But what *is* the period? I mean, not what interval
> > or size, but what is it? What significance does it have?
>
> It's the smallest interval at which a scale can be
> transposed without changing the scale at all. For example,
> the diminished (octatonic) scale in 12-tET has a period
> of 1/4-octave. For another, my symmetrical decatonic scale
> in 22-tET has a period of 1/2-octave.

Ah ... OK, I can grasp that.

But then what makes the "interval of equivalence" different
from that?

> The way Gene does things, unison vectors are all
> _small intervals_ defined with specific ratios, for
> example 81:80 but not 81:40, and then Gene can construct
> temperaments or whatever, and then octave-equivalence
> can be stuck back in at the end, if desired. If you don't
> do it this way, you won't be able to deal with torsion
> properly.

Can you explain why not?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Thursday, January 31, 2002 1:02 PM
> > Re: interval of equivalence, unison-vector, period
> >
> >
> > > But what *is* the period? I mean, not what interval
> > > or size, but what is it? What significance does it have?
> >
> > It's the smallest interval at which a scale can be
> > transposed without changing the scale at all. For example,
> > the diminished (octatonic) scale in 12-tET has a period
> > of 1/4-octave. For another, my symmetrical decatonic scale
> > in 22-tET has a period of 1/2-octave.
>
>
> Ah ... OK, I can grasp that.
>
> But then what makes the "interval of equivalence" different
> from that?

Well, the interval of equivalence is what you explicitly decide you
want to treat as a kind of "generator" in your scale, in that the
scale will automatically repeat every interval of equivalence because
we'll just be hearing the "same" scale again, only higher or lower in
pitch. The period, however, comes in at 1/N octaves, where N is an
integer (usually 1, but not always), just because of the way the
unison vectors work out.

> > The way Gene does things, unison vectors are all
> > _small intervals_ defined with specific ratios, for
> > example 81:80 but not 81:40, and then Gene can construct
> > temperaments or whatever, and then octave-equivalence
> > can be stuck back in at the end, if desired. If you don't
> > do it this way, you won't be able to deal with torsion
> > properly.
>
>
> Can you explain why not?

Recall that a particular unison vector (or product of unison vectors,
etc.) is candidate for torsion if it's a power (square, cube, etc.)
of some other interval. Let's say you don't keep track of the factors
of 2 making up the unison vectors. Now let's say you notice that a
particular unison vector (or product of unison vectors, etc.) has all
its prime-factorization exponents as multiples of N. Then it appears
to be an Nth power of some unison vector, right? Well, not
necessarily. If the power of 2 that you threw away was also a
multiple of N, then you're fine -- the Nth root of the small interval
is some even smaller interval. But if it wasn't, then you're really
taking the Nth root of something close to an octave, or to two
octaves, etc. . . . which may not be a small interval at all!

For example:

6561:6400 = 2^-8 * 3^8 * 5^-2

This is the square of

81:80 = 2^-4 * 3^4 * 5^-4

So any periodicity block where is 6561:6400 is a unison vector, or
the product of the unison vectors, etc., will be torsional.

HOWEVER:

50:49 = 2^1 * 5^2 * 7^-2

What if we ignore the factors of 2?

50:49 "=" 5^2 * 7^-2

This is the square of

5^1 * 7^-1

which is a tritone, or tritone plus octave, or . . . etc., depending
on how many factors of two you put back in.

But a tritone is no kind of unison vector! Instead, it (as 1/2
octave) becomes the _period_ for any system involving the 50:49
unison vector.

Catchin' on?

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, February 01, 2002 1:01 AM
> Subject: [tuning-math] Fwd: Re: interval of equivalence, unison-vector,
period
>
>
> Recall that a particular unison vector (or product of unison vectors,
> etc.) is candidate for torsion if it's a power (square, cube, etc.)
> of some other interval. Let's say you don't keep track of the factors
> of 2 making up the unison vectors. ...
> <etc.>
>
> But a tritone is no kind of unison vector! Instead, it (as 1/2
> octave) becomes the _period_ for any system involving the 50:49
> unison vector.
>
> Catchin' on?

YUP! Thanks, Paul!

This is the kind of "come on Joey, give me your hand" explanation
that I really need sometimes when I'm buried in this math stuff.

-monz

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