plausibility measure [was, ironically:] The one that doesn't talk about numbers

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

> > 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_.

I tend to go with Monz, and I have in mind real life
conditions where ratios aren't exact and where generating
intervals tend to be used more than once; not too different
from the way intervals work in a carefully figured out
temperament, I think they tend to be taken for the things
they mean, not necessarily for the things they are.

Comments about the following welcome. I am not really
interested in defending my opinions as the one truth, but of
course I'd like to hear who's had the same ideas before and
why they didn't catch on.

My(?) measure of interval plausibility simply (I think) adds
the prime factors as they occur for the first time, but
weights them less logarithmically if they repeat: p^1 for
the first time, p^0 the second, p^(2-n) for the nth time. (I
could give higher primes more weight initially by taking
their complement to 1 when their exponents are negative, but
eventually this penalizes longer generator chains.)

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.

So if a piece of music (or a collection of pitches :) ) uses
an interval of around 1160c "understandably" as 125/64
(1159c), 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. 125/64 has a (well, really negative)
plausibility of 10.1375; 35/18 has 18; 31/16 34.975.
On the other hand, if you use an interval like 31/16
melodically (freely, i.e. outside of the harmonic series,
which is a special, primarily harmonic context this
plausibility doesn't account for), you'd be advised to use
e.g. 45/23 (1162c) as a higher variation of this pitch,
because it has an equally high plausibility value of 32. A
dip to the lower values of 3-limit intervals might spoil
this high-prime intervals for you.

While I have a hunch this supposed principle is at work in
Jackie Ligon's unabashed no-limit music, my own experience
stays firmly within 5-limit: 5/4 and 81/64 outside their
context make a huge difference for me, whereas I don't care
much about jumps of one minor 6th or the other. 5/4 and
81/64 have 8 and 8.3819444.., respectively; 8/5 and 128/81
have 8.5 and 8.41319444...

klaus schmirler

--- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
>
>
>
> paulerlich schrieb:
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > > 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_.
>
>
> I tend to go with Monz,

Can you provide an example, as requested above?

> and I have in mind real life
> conditions where ratios aren't exact and where generating
> intervals tend to be used more than once;

Me too!

> not too different
> from the way intervals work in a carefully figured out
> temperament, I think they tend to be taken for the things
> they mean, not necessarily for the things they are.

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_.

> Comments about the following welcome. I am not really
> interested in defending my opinions as the one truth, but of
> course I'd like to hear who's had the same ideas before and
> why they didn't catch on.
>
> My(?) measure of interval plausibility simply (I think) adds
> the prime factors as they occur for the first time, but
> weights them less logarithmically if they repeat: p^1 for
> the first time, p^0 the second, p^(2-n) for the nth time.

OK, this is Klaus complexity, then.

> (I
> could give higher primes more weight initially by taking
> their complement to 1 when their exponents are negative, but
> eventually this penalizes longer generator chains.)
>
> 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.

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?

> 125/64 has a (well, really negative)
> plausibility of 10.1375; 35/18 has 18; 31/16 34.975.
> On the other hand, if you use an interval like 31/16
> melodically (freely, i.e. outside of the harmonic series,
> which is a special, primarily harmonic context this
> plausibility doesn't account for), you'd be advised to use
> e.g. 45/23 (1162c) as a higher variation of this pitch,
> because it has an equally high plausibility value of 32.

I'm not following . . . I thought you said plausibility doesn't
account for such a case, and yet you proceed to use it . . . I'm
confused. Can you help me understand you better?