Re: minority report, huh?

Message text written by INTERNET:tuning@onelist.com
>If we want to talk about the realtive dissonance of intervals (or chords
for that metter) without considering frequency, then the only sensible
thing to do is to compare them all with the same average frequency, hence
n+d or (n+d)/2 is the right approximation for dyads, not d.

Regards,
-- Dave Keenan<

I'm trying to follow this discussion closely, so excuse my delayed response
here.

You really can't consider the relative dissonance of intervals without
considering frequency, and an average frequency is going to eventually have
the same problem.

Dissonance has to be evaluated in terms of frequency and of ratio. The test
I performed used demonstrated to my satisfaction that an increase of
dissonance followed an increase in d, when all ratios were in lowest terms.
Thus, with composite harmonic wave forms, the set with the ratio (0/0) was
the most consonant, the set (2/1, 3/1, 4/1,...)next most consonant and so
on.

However, as the intervals get smaller when one moves from one series to the
next, the frequency-dependent aspect of roughness overrides the ratio
aspect, and roughness itself is eventually overcome by beating and then
that by the limit of pitch difference discrimination.

Imagine, then, if you will, the list of ratios ordered by d to be something
like the map of the Ponderosa Ranch, and the flame slowly burning up the
map is frequency which renders rational intervals into roughness and
eventually perceptual unisons...

Anyone got a neat formula describing the relationship between f, roughness
and d?

Daniel Wolf wrote,

>Thus, with composite harmonic wave forms, the set with the ratio (0/0)
was
>the most consonant

0/0 can be any number, since any number times 0 equals 0. So what is
that, white noise?

>Imagine, then, if you will, the list of ratios ordered by d to be
something
>like the map of the Ponderosa Ranch, and the flame slowly burning up
the
>map is frequency which renders rational intervals into roughness and
>eventually perceptual unisons...

Or approximations of other simple ratios.

Message text written by INTERNET:tuning@onelist.com
>
0/0 can be any number, since any number times 0 equals 0. So what is
that, white noise?
<

Zero divided by zero is one. It's also a terribly convenient way for
setting unisons apart from all other intervals.

On Wed, 24 Mar 1999, Daniel Wolf wrote:

> From: Daniel Wolf <DJWOLF_MATERIAL@compuserve.com>
>
> Message text written by INTERNET:tuning@onelist.com
> >
> 0/0 can be any number, since any number times 0 equals 0. So what is
> that, white noise?
> <
>
> Zero divided by zero is one. It's also a terribly convenient way for
> setting unisons apart from all other intervals.
>
> ------------------------------------------------------------------------

From a mathematical perspecive 0/0 can be any number (indeterminate form)
as the first author in this n-ologue expressed. A simple analogy could be
the 'time' at the north pole. I never thought of it as white noise though.
I have to think about it. It is an interesting thought!

Can Akkoc
Alabama School of Mathematics and Science
Mobile, Alabama 36604-2519
Phone: (334) 441-2126
Fax : (334) 441-3290

Message text written by INTERNET:tuning@onelist.com
>Or approximations of other simple ratios.<

No. The quality of the approximation is not a function of f.

And, in contrast to the discrimination of near unisons, the tolerance of
the approximations of simple ratios depends on the sample period.

I wrote,

>>Or approximations of other simple ratios.<

Daniel Wolf wrote

>No. The quality of the approximation is not a function of f.

Did I say it was or wasn't?

>And, in contrast to the discrimination of near unisons, the tolerance
of
>the approximations of simple ratios depends on the sample period.

I would argue that there is only a difference of degree -- especially in
the case of harmonic timbres, where simple ratios between the
fundamentals entail unisons among the partials.