definitions of period, equivalence, etc. (was: Re: graham's linear temperament page)

hi Dave,

> From: "dkeenanuqnetau" <d.keenan@uq.net.au>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, May 13, 2002 3:56 PM
> Subject: [tuning-math] Re: graham's linear temperament page
>
>
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
> > (and if anyone would care to explain that "matter of formalism"
> > which differentiates a "generator" from an "equivalence interval",
> > i'd sure like to read it! after much discussion of it a few months
> > ago, i'm still not clear about it.)
>
> Unfortunately I have a slightly different take on this to Graham's
> recent statement. I'm wondering if Graham really meant to say the
> following.
>
> A linear temperament is generated by two "generating intervals" (or
> generators for short), one of which is distinguished as the "interval
> of periodicity" (or "period" for short) and the other is simply
> called "the generator". The "period" is distinguished from the other
> generator by the fact that the "interval of equivalence" is a whole
> number multiple of the period. The interval of equivalence is usually
> the octave. A particular scale within a linear temperament (e.g. a
> diatonic scale within meantone) will have a fixed number of
> (non period) generators but an unspecified number of intervals of
> equivalence.

hmmm... thanks for that. it seems to me that i should put this
into a definition as an addendum, but the "linear temperament"
definition is really only appropriate for the first part of it.

i still don't have a definition for "equivalence interval" or
"interval of equivalence", mainly because of my confusion over
exactly what it is and why it's different from the "interval
of periodicity".

i think it's really best for me to stay out of it until you,
Graham, Paul, and Gene define exactly what all these terms mean.
then let me know, and i'll update the Dictionary.

thanks.

-monz

--- In tuning-math@y..., "monz" <monz@a...> wrote:
> i still don't have a definition for "equivalence interval" or
> "interval of equivalence", mainly because of my confusion over
> exactly what it is and why it's different from the "interval
> of periodicity".
>
> i think it's really best for me to stay out of it until you,
> Graham, Paul, and Gene define exactly what all these terms mean.
> then let me know, and i'll update the Dictionary.

Graham, Paul and Gene, please let Monz know if you agree with the
following.

"interval of equivalence" = "equivalence interval" = "formal octave"
is that interval (much larger than a unison) which, when it occurs
between two pitches, we consider them to be, in some sense, (formally
if not perceptibly) the same note. For most scales this is the octave
1:2, and when it is not the octave it is usually some other highly
consonant interval such as the "tritave" 1:3. But the essential
feature of the interval of equivalence in relation to definitions of
scales and types of scales is that when we describe a scale we
describe only the pitches that fall within a single interval of
equivalence, and we leave it up to the instrument builder to decide
the range of the instrument and therefore how many times (including
fractions) the interval of equivalence should be repeated.

"interval of periodicity" = "periodic interval" = "period"
is that generator of a regular temperament (whether linear, planar, or
n-dimensional) which generates the interval of equivalence all by
itself. This means that the period is either equal to the interval of
equivalence or fits into the interval of equivalence a whole number of
times.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Graham, Paul and Gene, please let Monz know if you agree with the
> following.

It's fine, but I think we should be clear that a linear temperament does not require an interval of equivalence, of an octave or anything else.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Graham, Paul and Gene, please let Monz know if you agree with the
> > following.
>
> It's fine, but I think we should be clear that a linear temperament
does not require an interval of equivalence, of an octave or anything
else.

Agreed. But I'd prefer to see the definition assume an IoE at first,
so as not to lose the musicians entirely, and then add that caveat at
the end. The situation of no IoE is extremely rare, and this fact is
why it makes sense to call them "linear" when mathematically they are
more simply treated as rank 2. Can someone name a popular or
historical scale/tuning that doesn't have an interval of equivalence?
i.e. one where every pitch must be listed, over the entire compass of
the instrument.

Note that those with an IoE _can_ be treated mathematically as rank
one, provided all arithmetic is modulo the IoE.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
> > i still don't have a definition for "equivalence interval" or
> > "interval of equivalence", mainly because of my confusion over
> > exactly what it is and why it's different from the "interval
> > of periodicity".
> >
> > i think it's really best for me to stay out of it until you,
> > Graham, Paul, and Gene define exactly what all these terms mean.
> > then let me know, and i'll update the Dictionary.
>
> Graham, Paul and Gene, please let Monz know if you agree with the
> following.
>
> "interval of equivalence" = "equivalence interval" = "formal octave"
> is that interval (much larger than a unison) which, when it occurs
> between two pitches, we consider them to be, in some sense,
(formally
> if not perceptibly) the same note. For most scales this is the
octave
> 1:2, and when it is not the octave it is usually some other highly
> consonant interval such as the "tritave" 1:3. But the essential
> feature of the interval of equivalence in relation to definitions
of
> scales and types of scales is that when we describe a scale we
> describe only the pitches that fall within a single interval of
> equivalence, and we leave it up to the instrument builder to decide
> the range of the instrument and therefore how many times (including
> fractions) the interval of equivalence should be repeated.
>
> "interval of periodicity" = "periodic interval" = "period"
> is that generator of a regular temperament (whether linear, planar,
or
> n-dimensional) which generates the interval of equivalence all by
> itself. This means that the period is either equal to the interval
of
> equivalence or fits into the interval of equivalence a whole number
of
> times.

i agree, and note:

a linear temperament is described by a generator and a period, *not*
by a generator and an interval of equivalence (which is what graham
wrote).

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > Graham, Paul and Gene, please let Monz know if you agree with
the
> > > following.
> >
> > It's fine, but I think we should be clear that a linear
temperament
> does not require an interval of equivalence, of an octave or
anything
> else.
>
> Agreed. But I'd prefer to see the definition assume an IoE at
first,
> so as not to lose the musicians entirely,

agreed so far

> and then add that caveat at
> the end. The situation of no IoE is extremely rare,

but should it be?

> and this fact is
> why it makes sense to call them "linear" when mathematically they
are
> more simply treated as rank 2.

right -- so in the case of two generators and no IoE, i don't know if
i feel comfortable with the term 'linear' anymore.

> Can someone name a popular or
> historical scale/tuning that doesn't have an interval of
equivalence?
> i.e. one where every pitch must be listed, over the entire compass
of
> the instrument.

i don't think we should discount such an approach as less valid or
inherently interesting, regardless of how few examples have been
brought to our attention so far.

> Note that those with an IoE _can_ be treated mathematically as rank
> one, provided all arithmetic is modulo the IoE.

i used to think so, but it seems gene was able to convince me
otherwise. i think that you can't handle torsion properly unless you
express the unison vectors in IoE-specific, rather than IoE-
equivalent/IoE-invariant terms.