superparticulars

Recently there was a discussion on this list of why superparticular
ratios "work..." Surely, they include some of our most "common"
ratios.

Somehow my brain has lost this information. Could someone please
refresh my memory on it. I believe Paul Erlich was involved in this
discussion, but I forget. Otherwise, I'm OK...

J. Pehrson

----- Original Message -----
From: "jpehrson2" <jpehrson@rcn.com>

> Recently there was a discussion on this list of why superparticular
> ratios "work..." Surely, they include some of our most "common"
> ratios.
>
> Somehow my brain has lost this information. Could someone please
> refresh my memory on it. I believe Paul Erlich was involved in this
> discussion, but I forget. Otherwise, I'm OK...
>
> J. Pehrson

The difference tone is always 1, giving fundimental support
to the interval.

* David Beardsley
* http://biink.com
* http://mp3.com/davidbeardsley

In a message dated 7/18/02 4:17:17 PM Eastern Daylight Time, jpehrson@rcn.com
writes:

> Recently there was a discussion on this list of why superparticular
> ratios "work..." Surely, they include some of our most "common"
> ratios.
>
>

Hi Joseph,

Try this explanation on for size. When a singer sings a rich musical tone,
or even a bassoonist plays a juicy and rich note, it is really the overtone
series at work.

See, if the harmonics of a string, or of a bassoon, are lined up to bring out
their sound, you will have all the superparticular ratios represented in the
note itself.

Now, if all these superparticular resonance which is the apex of musical
timbre, are liberated to form intervals of actual musical usage, they take on
peculiar characteristics.

They are definitive consonance in the sense that they are so in synch with
each other that they lose some of their individuality in favor of blending.
And all those superparticular intervals are the straight row of numbers: 1,
2, 3, 4, 5, 6, 7, etc.

Joseph, it is an essential relationship between things.

best, Johnny Reinhard

--- In tuning@y..., Afmmjr@a... wrote:

/tuning/topicId_38697.html#38699

> In a message dated 7/18/02 4:17:17 PM Eastern Daylight Time,
jpehrson@r...
> writes:
>
>
> > Recently there was a discussion on this list of why
superparticular
> > ratios "work..." Surely, they include some of our most "common"
> > ratios.
> >
> >
>
> Hi Joseph,
>
> Try this explanation on for size. When a singer sings a rich
musical tone,
> or even a bassoonist plays a juicy and rich note, it is really the
overtone
> series at work.
>
> See, if the harmonics of a string, or of a bassoon, are lined up to
bring out
> their sound, you will have all the superparticular ratios
represented in the
> note itself.
>
> Now, if all these superparticular resonance which is the apex of
musical
> timbre, are liberated to form intervals of actual musical usage,
they take on
> peculiar characteristics.
>
> They are definitive consonance in the sense that they are so in
synch with
> each other that they lose some of their individuality in favor of
blending.
> And all those superparticular intervals are the straight row of
numbers: 1,
> 2, 3, 4, 5, 6, 7, etc.
>
> Joseph, it is an essential relationship between things.
>
> best, Johnny Reinhard

***Thanks, Johnny. That's a great explanation. I guess when a
string, for example, vibrates, all the other divisions vibrate as
well...

That's interesting, but I wonder why all the divisions that vibrate
are derived from *integers*...(or rather the *inversions* of
integers, and the *sounds* are integers...)

I wonder why, scientifically it works out that way. Somebody here
knows the answer to that question, I am sure...

best,

Joseph

> Joseph:
> Recently there was a discussion on this list of why superparticular
> ratios "work..." Surely, they include some of our most "common"
> ratios.

> Johnny
> Try this explanation on for size. When a singer sings a rich
musical
> tone, or even a bassoonist plays a juicy and rich note, it is really
> the overtone series at work.
>
> See, if the harmonics of a string, or of a bassoon, are lined up to
> bring out their sound, you will have all the superparticular ratios
> represented in the note itself.
[snip]

There is another explanation that is, in a way, just another wording
of Johnny's explanation, using harmonic entropy. First, we should note
that the octave is "stricto sensu" the simplest superparticular ratio.

Then we may ask the question; how is it possible to divide a
superparticular ratio in order to introduce two new ratio with the
smallest harmonic entropy wrt the two existing degrees? I think we can
rephrase this by replacing "harmonic entropy", by "highest
consonnance" or "maximum number of common partials" but I am a little
affraid to trigger another holy war on the NG ;-).

The answer is starting from N:N+1 we get 2N:2N+1:2N+2; i.e. stacking
two superparticular ratio to interleave a new degree between initial
ratio. Try to find simpler, it is not possible (as far as I can
conjecture, even tough I cannot provide a formal proof). So we have a
process of "superparticular split" that may had a role in derivation
of scale in various tradition.

1:2 -> 2:3:4 fifth and fourth from octave
2:3 -> 4:5:6 major third and minor third from fifth
3:4 -> 6:7:8 "pygmea" thirds from fourth
4:5 -> 8:9:10 major and minor second from major third

That raises a few question like:

Why the division of the fourth 6:7:8 is is not used in western
tradition?
why the minor third is divided 40:45:48 and not simply 10:11:12?

The answer is probably something like: because of the need to build a
scale with a great number of usable simple ratio from a limited (7)
number of degrees. Certainly other people are more historicaly
informed to provide better answers to those questions (Paul? Margo?).

Peculiarly, the diatonic semitone 15:16 seems to arise from the
superparticular split of the interval 7:8 that is not used in western
tradition.

I read somewhere that 6:7:8 is one of the basis of the magnificent
pygmea Aka polyphonies. Is it used in other musical cultures?

Yours truly

François Laferrière