Fw: The one that doesn't talk about numbers (was: JI definitions...)

Here's a response from a private discussion which I
wanted to post here.

-monz

----- Original Message -----
From: monz <joemonz@yahoo.com>
To: unidala
Sent: Wednesday, January 30, 2002 9:51 PM
Subject: Re: The one that doesn't talk about numbers (was: JI
definitions...)

> > > > [JG]
> > > If so, of what use are all these mathematical constructs
> > > which attempt to conceptually characterize "tunings"?
> >
> > [monz]
> > My own belief is that Paul's harmonic entropy concept is
> > a step in the direction of figuring all this out.
>
> JG: My point was - if we can *only* trust our ears in these
> matters (of purity, tuning "perfection", etc), then what
> are we (you and I and all) doing trying to posit that
> "this or that" mathematical analytical model has validity?

Well, the *hope* is that eventually there will be a
really accurate mathematical model of the whole
auditory-system/psychoacoustics shebang, so that *then*
we can really discuss tuning mathematically in a way
that *really* corresponds to what we hear.

But until then ... I guess there will be those who
make the music, and those who wank off with the math.
But it's my firm conviction that all the wanking off
will eventually lead to something concrete.

>
> > > Questions regarding "Monzo finity":
> > >
> > > (1) How would we manage to isolate - the values of
> > > prime numbers which happen to factor the numerators
> > > and denominators of ratios which describe such ratios
> > > "slightly different" in frequency from a low-numbered
> > > intervallic ratio (such as 3/2) - from the possibility
> > > of the limits of our perception in resolving small
> > > variations in pitch resulting in effects similar to
> > > the "bridging" effects proposed?
> >
> >
> > Again, I refer you to the harmonic entropy concept.
>
> JG: So, if "harmonic entropy" is untrue, then so is
> "bridging" (or, at least, not demonstrable as being
> different from a phenomenon related simply to the
> limits of our ability to discern pitch accurately,
> sort of like the finite number of digits in an
> arithmetical calculation)?

But the "limits of our ability to discern pitch accurately"
is entirely bound up with both the bridging *and* harmonic
entropy concepts. That's exactly what they both explore.

> > Of course, ultimately, my belief is that, whatever the mathematics
> > involved in the actual tuning, our understanding of our *perception*
> > of what we hear is grounded in perception of the lowest few primes.
>
> JG: But I am asking you about *why* you believe what you do
> (about that which is expressible in non-subjective terms).

Well, the vast amount of reading I've done on tuning suggests
to me that there is some type analytical process going on in
the brain when we listen to music.

The business with the "lowest few primes" is simply that,
in calculation, it's my belief that it takes a lot less
"brain power" to comprehend numbers that can be factored
into small primes, like 2, 3, 5, 7, 11 ... you get the idea.

I think that somewhere between 13 and 31 or so, this ability
finally breaks down, and any numbers with higher prime-factors
will simply be understood as audibly-equivalent numbers composed
of lower prime-factors. Of course, these boundaries depend
a lot on the musical context.

> > So we're employing xenharmonic bridges all the time when we hear
> > any tuning that's not alread low-integer RI ("rational intonation").
>
> JG: How could we proceed in proving this to ourselves and others?

One example: nearly everyone who hears a 12-EDO "major" triad
will accept it, at least briefly, as a 4:5:6 proportion (again,
this also depends on context).

A better example: a 16:19:24 JI "minor" triad will almost always
be accepted at first as a 12-EDO "minor" triad.

A still better example: a 12-EDO "perfect 5th" and a Pythagorean
3:2 dyad will almost always be interpreted as the same thing.

Xenharmonic bridges happening in all of these.

> _______________________________________________________
> > >
> > >
> > > JM: In fact, this is exactly what's at the root of the whole
> > > heated debate over how to precisely define "just intonation".
> > >
> > > Intervals and larger sound-conglomerates whose frequencies
> > > can be compared as low-integer ratios reveal that the
> > > lowest primes (as factors in the ratio terms) clearly have
> > > a distinctive audible affect, which is why they are important
> > > to tuning theory.
> > > __________________
> > >
> > > JG: But, since there are *more* "lowest primes" per linear
> > > section of integers which encompass the integer values of
> > > those "lower primes" than in the case of higher primes,
> > > how do we *know* that it is the *small numerical value*
> > > primes which have "more" significance that the *large
> > > numerical value* primes?
> >
> >
> > I think I get the gist of what you're saying here, but it's
> > not too clear. Please elaborate, maybe with some examples.
>
> JG: That was not well put.
>
> I added in the following (amending) post:
>
> << JG: I add to the above statement:
>
> Does the fact that there *are* "more of them" suffice?
>
> Which is not very well put, either.
>
> What I am getting at is along the lines of:
>
> << A second question is expressed by the addition of the
> following statement to the text quoted above:
>
> How, also, do we determine whether it is the relative
> "density" (per section of integers) of the "smaller
> valued primes" which makes them "special", OR the
> "smallness of the integers" (which turn out to be
> largely prime integer values in the case of small
> valued integers)? >>
>
> By the same thinking, how do we "know" that it is
> the small-valued primes (and not, instead the large-
> valued primes) which form a "basis"?
>
>
> > > It seems to me that - in a system considering sinusoidal
> > > sound source "voices" - the distinction of whether the
> > > *lower* valued primes or the *higher* valued primes hold
> > > a more significant place would be a hard thing to establish.
> >
> >
> > Well, it's a foundation of my theory that the brain
> > "understands" the lowest primes far more easily than higher
> > ones. This would be primarily because our perception of
> > pitch is multiplicative rather than additive. The lowest
> > primes, as factors in ratios, form a sort of ultimate
> > reduced basis for our comprehension of harmonic relationships.
>
> JG: But larger valued primes are, themselves, irreducible,
> are they not?

Yes, absolutely. But let's take this example:

prime-factor
2 3 11 31 ratio ~cents

-4 0 0 1 = 31/16 1145.035572
-6 0 3 0 = 125/64 1158.941142

There's a difference between them of only ~13.90556913
(a bit more than 1/2-comma).

In a context where these two ratios would be audibly
indistinguishable, a listener will probably perceive
31/16 *as* 125/64, because of the lower prime-factors.

I have to answer the rest later. Hope that helps.

-monz

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> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, January 31, 2002 1:26 PM
> Subject: [tuning] Re: Fw: The one that doesn't talk about numbers (was: JI
definitions...)
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > Yes, absolutely. But let's take this example:
> >
> > prime-factor
> > 2 3 11 31 ratio ~cents
> >
> > -4 0 0 1 = 31/16 1145.035572
> > -6 0 3 0 = 125/64 1158.941142
> >
> > There's a difference between them of only ~13.90556913
> > (a bit more than 1/2-comma).
> >
> > In a context where these two ratios would be audibly
> > indistinguishable, a listener will probably perceive
> > 31/16 *as* 125/64, because of the lower prime-factors.
>
> I find this absurd. Though 31/16 is already an extremely complex
> ratio, 125/64 is far more complex! It seems like utter nonsense to
> suggest that a listener would perceive an isolated dyad exactly as
> 125/64! Furthermore, 11/6 is even closer to 31/16 than 125/64 is, and
> 11/6 has much smaller numbers, so it seems FAR FAR more likely to me
> that the listener will perceive this as 11/6 than as 31/16 (let alone
> 125/64)!

Well, guess what? ... 11/6 was the comparison I was originally
going to use! ... Take a look at the now-erroneous prime-factor
label above the exponents -- 11 is still there instead of 5!.
So I guess we agree here.

But I wasn't necessarily speaking of an "isolated dyad".
I was thinking more along the lines of hearing the interval
in an actual piece of music, in which harmonic context would
probably strongly persuade the listener to "hear" it as a
5-limit interval.

... Guess I should have stuck with my original idea, 11/6.

-monz

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Thursday, January 31, 2002 1:26 PM
> > Subject: [tuning] Re: Fw: The one that doesn't talk about numbers
(was: JI
> definitions...)
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > > Yes, absolutely. But let's take this example:
> > >
> > > prime-factor
> > > 2 3 11 31 ratio ~cents
> > >
> > > -4 0 0 1 = 31/16 1145.035572
> > > -6 0 3 0 = 125/64 1158.941142
> > >
> > > There's a difference between them of only ~13.90556913
> > > (a bit more than 1/2-comma).
> > >
> > > In a context where these two ratios would be audibly
> > > indistinguishable, a listener will probably perceive
> > > 31/16 *as* 125/64, because of the lower prime-factors.
> >
> > I find this absurd. Though 31/16 is already an extremely complex
> > ratio, 125/64 is far more complex! It seems like utter nonsense to
> > suggest that a listener would perceive an isolated dyad exactly as
> > 125/64! Furthermore, 11/6 is even closer to 31/16 than 125/64 is,
and
> > 11/6 has much smaller numbers, so it seems FAR FAR more likely to
me
> > that the listener will perceive this as 11/6 than as 31/16 (let
alone
> > 125/64)!
>
>
> Well, guess what? ... 11/6 was the comparison I was originally
> going to use! ... Take a look at the now-erroneous prime-factor
> label above the exponents -- 11 is still there instead of 5!.
> So I guess we agree here.
>
> But I wasn't necessarily speaking of an "isolated dyad".
> I was thinking more along the lines of hearing the interval
> in an actual piece of music, in which harmonic context would
> probably strongly persuade the listener to "hear" it as a
> 5-limit interval.
>
> ... Guess I should have stuck with my original idea, 11/6.

It's a good thing you didn't, because it's about 100 cents away from
the ratios above!

Sorry I goofed here . . . I'm deleting my message and trying again.

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> prime-factor
> 2 3 11 31 ratio ~cents
>
> -4 0 0 1 = 31/16 1145.035572
> -6 0 3 0 = 125/64 1158.941142
>
> There's a difference between them of only ~13.90556913
> (a bit more than 1/2-comma).
>
> In a context where these two ratios would be audibly
> indistinguishable, a listener will probably perceive
> 31/16 *as* 125/64, because of the lower prime-factors.

> In a context where these two ratios would be audibly
> indistinguishable, a listener will probably perceive
> 31/16 *as* 125/64, because of the lower prime-factors.

I find this absurd. Though 31/16 is already an extremely complex
ratio, 125/64 is far more complex! It seems like utter nonsense to
suggest that a listener would perceive an isolated dyad exactly as
125/64!

Now, Monz will argue that in a non-isolated context, with harmonic
motion, one could perceive 31/16 as 125/64. I'd like him to provide
an example, and then to offer his own intepretation, from his
example, as to exactly _whay_ is being perceived as _what_.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, January 31, 2002 1:44 PM
> Subject: [tuning] Re: Fw: The one that doesn't talk about numbers (was: JI
definitions...)
>
>
> > > I find this absurd. Though 31/16 is already an extremely complex
> > > ratio, 125/64 is far more complex! It seems like utter nonsense to
> > > suggest that a listener would perceive an isolated dyad exactly as
> > > 125/64! Furthermore, 11/6 is even closer to 31/16 than 125/64 is,
> and
> > > 11/6 has much smaller numbers, so it seems FAR FAR more likely to
> me
> > > that the listener will perceive this as 11/6 than as 31/16 (let
> alone
> > > 125/64)!
> >
> >
> > Well, guess what? ... 11/6 was the comparison I was originally
> > going to use! ... Take a look at the now-erroneous prime-factor
> > label above the exponents -- 11 is still there instead of 5!.
> > So I guess we agree here.
> >
> > But I wasn't necessarily speaking of an "isolated dyad".
> > I was thinking more along the lines of hearing the interval
> > in an actual piece of music, in which harmonic context would
> > probably strongly persuade the listener to "hear" it as a
> > 5-limit interval.
> >
> > ... Guess I should have stuck with my original idea, 11/6.
>
> It's a good thing you didn't, because it's about 100 cents away from
> the ratios above!
>
> Sorry I goofed here . . . I'm deleting my message and trying again.

Whew, we must both be in the twilight zone today! Now I remember,
*that's* precisely the reason why *I* did away with the 11/6 too!
At first I thought it was close to 31/16, but then I realized it's
a whole semitone lower.

-monz

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