Re: [tuning] Re: plausibility measure

>> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, February 02, 2002 12:02 AM
> Subject: [tuning] Re: plausibility measure
> [was, ironically:] The one that doesn't talk about numbers

yup, that sure is ironic

> If we're talking about _pitches_ labeled 31/16 and 125/64, then I'm
> in complete agreement with you and Monz. But if we're talking about
> the _intervals_ 31:16 and 125:64, then I'm not. I thought we were
> talking about the _intervals_ 31:16 and 125:64, since Monz invoked
> harmonic entropy to justify his point of view, and harmonic entropy
> concerns _intervals_, not _pitches_.
>
> > [klaus]
> > Examples: 5/4 consists of the primes 2, 2, and 5. The second
> > 2 is taken to the power of 0, and I add 5+2+1, 8. 25/24
> > consists of the prime 5 twice (-> 5+1), 3 once, and 2 three
> > times (-> 2+1+0.5).
>
> > The plausibility value is 5+3+3.5, 11.5.
> > 125/64 consists of 5 three times (-> 5+1+0.2) and 2 six
> > times (-> 2+1+0.5+0.25+0.125+0.0625) and is thus 10.1375.
> >
> > 31/16 already has prime 31. With the reduced factors of 2
> > added, this is 34.875. Very different.
> >
> > What to make of it? I'm supposing two things: a) listeners
> > don't readily jump to interpretations implying a much higher
> > value, and b) ratios with higher values need to be tuned
> > much more accurately.
>
> [paul]
> My point of view is that there are two levels of ratio-
> interpretation: (1) the actual psychophysical character of an
> _interval_, which is what harmonic entropy talks about; (2) the
> action of the music theorist when faced with a piece of music which
> moves around, can follow the movement musically speaking, and hence
> decides to multiply the _interval_ ratios to derive the _pitch_
> ratios. 125/64 is a result of the _latter_ process, and in reckoning
> its complexity (like Barlow, Wilson, and many others) you take into
> account the number (or some function thereof) of degrees of musical
> memory and movement recognition that are involved. Then, based on
> some picture of what the music might be like, you _combine_ the types
> of reckoning in (1) and (2).
>
> What I thought Monz was talking about, since he invoked harmonic
> entropy, was strictly the first part. This part alone merits serious
> study, because if you don't get part (1) right, how can you combine
> it with your model of (2) afterwards?
>
> ...
>
> > So if a piece of music (or a collection of pitches :) ) uses
> > an interval of around 1160c "understandably" as 125/64
> > (1159c),
>
> So you mean a _pitch_ labeled 125/64, then?
>
> > and there is another one 10-15c smaller , this will
> > more likely be heard as 35/18 (1151c) than as a 31/16
> > (1145c), unless the context consists of the harmonic series
> > exclusively.
>
> OK . . . but you pretty much have to assume a context that includes
> 5/4 and 25/16 for 125/64 to be "understandable" in the first place,
> don't you?
>

ok, I should have specified what I was really thinking of

not of bare dyads, but rather in a systemic sense, i.e.
listening to a piece of "JI" (whatever that means) music,
likely to have lots of 5-limit harmonies in it, and an
instance of a 31:16

in a case like that, the listener is very likely to
"understand" 31:16 as 125:64

now i believe paul will agree with that, right paul?

but i might have contradicted myself by starting from
two logically incompatible assumptions

since we're making a big deal about context, here's the orignal
statement i made in its original context:

> > > > [JG]
> > > > 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 5 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.

[the prime-factor 5 label was erroneously given as 11 before]

so i suppose that a piece "likely to have lots of 5-limit
harmonies in it" would not actually provide "a context
where these two ratios would be audibly indistinguishable"

hmmm ...

i think maybe what it boils down to is that i must believe
that when listening to actual music that has a texture
generally thicker than dyadic, the additional notes
influence our sense of harmonic perception in ways which do
not equate to the sum of the affect of the dyads involved

this is along the lines of why no-one thinks the 1/(6:5:4)
triad sounds anywhere near as concordant as the 4:5:6, while
they both have exactly the same intervallic composition

again, my theory is that we boil everything down to a
prime-factor lattice and "plot" the pitches as we hear them

and of course, the prime basis of everyone's lattice is
different, and the same person's lattice varies at any
given time too

-monz

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

> ok, I should have specified what I was really thinking of
>
> not of bare dyads, but rather in a systemic sense, i.e.
> listening to a piece of "JI" (whatever that means) music,
> likely to have lots of 5-limit harmonies in it, and an
> instance of a 31:16
>
> in a case like that, the listener is very likely to
> "understand" 31:16 as 125:64
>
> now i believe paul will agree with that, right paul?

are you talking about the _intervals_ 31:16 and 125:64, or the
_pitches_ 31/16 and 125/64?

> i think maybe what it boils down to is that i must believe
> that when listening to actual music that has a texture
> generally thicker than dyadic, the additional notes
> influence our sense of harmonic perception in ways which do
> not equate to the sum of the affect of the dyads involved

how about a context for this example

> this is along the lines of why no-one thinks the 1/(6:5:4)
> triad sounds anywhere near as concordant as the 4:5:6, while
> they both have exactly the same intervallic composition

but that's an issue which I think can be settled in realm (1) from my
last essay in this thread, so nothing like these realm (2) issues you
seem to me referring to.