Re: definition of scale

Gene said :

> I proposed a definition which everyone hated:
>
> Scale
>
> A discrete set of real numbers, containing 0 and regarded as
representing intervals logarithmically (e.g., in terms of cents), and
such that the distance between sucessive elements of the scale is
bounded both below and above by positive real numbers. The least
upper bound of the intervals between successive elements of the
scale is the maximum scale step, and the greatest lower bound is
the minimum scale step. The element of the scale obtained by
counting up n scale steps is the nth degree, by counting down is
the -nth degree; 0 is the 0th degree.
>

I don't hate it... but I would like to see something that expands
upon this to differentiate between Scale and Mode (or "parent-thing"
and Scale/Mode if you wish).

I will clarify. I tend to think of a Scale as a set of pitches which
are ordered by size (frequency). A Mode is a specific mapping of the
Scale as described above, with a zero-th element (the root). Since
most musicians tend to think of scales and modes as being the same
thing (a rooted and ordered collection of pitches), I have no problem
with some term being given to this "parent-thing" and Scale/Mode being
interchangeable names for a specific rooted orderring.

The other piece here is that MOST people use scales with some notion of
an interval of equivalence where the scale is said to repeat at. Here
the intervals in your definition should repeat, as well as the numberring
(though we can go to subscripts to define which "octave" this "0" is
in). Once this interval of equivalence is created, then I would say that
the "parent-thing" is the set of all ordered rotations in a
single "octave" (noting that those ordered rotations are occurring in all
octaves simultaneously) and a scale/mode is a specific ordered rotation.

Now if we really want to say that scale, parent-thing and mode are all
different, then I would say that a scale is a mode which supports a
tonality (which needs a definition as from Pauls paper for instance).

Bob Valentine

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:

> The other piece here is that MOST people use scales with some notion of
> an interval of equivalence where the scale is said to repeat at.

Does it follow from this that a scale necessarily has an infinite number of degrees? You can certainly make "scale" more restrictive than I did, but we want to cover everything we would like included.

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:

/tuning/topicId_36050.html#36050

>
> Gene said :
>
> > I proposed a definition which everyone hated:
> >
> > Scale
> >
> > A discrete set of real numbers, containing 0 and regarded as
> representing intervals logarithmically (e.g., in terms of cents),
and
> such that the distance between sucessive elements of the scale is
> bounded both below and above by positive real numbers. The least
> upper bound of the intervals between successive elements of the
> scale is the maximum scale step, and the greatest lower bound is
> the minimum scale step. The element of the scale obtained by
> counting up n scale steps is the nth degree, by counting down is
> the -nth degree; 0 is the 0th degree.
> >
>
> I don't hate it... but I would like to see something that expands
> upon this to differentiate between Scale and Mode (or "parent-thing"
> and Scale/Mode if you wish).
>
> I will clarify. I tend to think of a Scale as a set of pitches which
> are ordered by size (frequency). A Mode is a specific mapping of
the
> Scale as described above, with a zero-th element (the root). Since
> most musicians tend to think of scales and modes as being the same
> thing (a rooted and ordered collection of pitches), I have no
problem
> with some term being given to this "parent-thing" and Scale/Mode
being
> interchangeable names for a specific rooted orderring.
>
> The other piece here is that MOST people use scales with some
notion of
> an interval of equivalence where the scale is said to repeat at.
Here
> the intervals in your definition should repeat, as well as the
numberring
> (though we can go to subscripts to define which "octave" this "0"
is
> in). Once this interval of equivalence is created, then I would say
that
> the "parent-thing" is the set of all ordered rotations in a
> single "octave" (noting that those ordered rotations are occurring
in all
> octaves simultaneously) and a scale/mode is a specific ordered
rotation.
>
> Now if we really want to say that scale, parent-thing and mode are
all
> different, then I would say that a scale is a mode which supports a
> tonality (which needs a definition as from Pauls paper for
instance).
>
> Bob Valentine

***Hi Bob and Gene!

Why isn't a "scale" just a collection of pitches ordered by
frequency??

jp

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> Why isn't a "scale" just a collection of pitches ordered by
> frequency??

I wouldn't want to call the pitch continuum a scale. I wouldn't even want to call {1/n cents | n is a non-zero integer} a scale.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_36050.html#36060

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > Why isn't a "scale" just a collection of pitches ordered by
> > frequency??
>
> I wouldn't want to call the pitch continuum a scale. I wouldn't
even want to call {1/n cents | n is a non-zero integer} a scale.

Hi Gene!

Well, that's a good point. Is there any serious consideration that
people would want to make scales that consisted of pitches less than
a cent? It seems people can't even really decide if people can
actually hear and reproduce one cent... ??

jp

>***Hi Bob and Gene!
>
>Why isn't a "scale" just a collection of pitches ordered by
>frequency??

You might not want to order it by frequency. But I say it
should be ordered in some way.

-Carl