huh?

>But then we realise that the interval 2/3 is exactly the same size as the
>interval 3/2 etc. They are the same interval. And yet we have decided that
>2/3 is more dissonant than 3/2. A contradiction? No. Paul Erlich has
>already explained this...

I missed it! Cound somebody quote it? Or point me to digest and post #?

>Those who say dissonance is proportional to d are keeping the lower
>frequency constant.

What exactly do you mean? Here's part of Wolf's experiment...

> --- 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, 11/1, 12/1, 13/1, ...
>
> 3/2, --- 5/2, --- 7/2, --- 9/2, ---- 11/2, ---- 13/2, ...
>
> 4/3, 5/3, --- 7/3, 8/3, --- 10/3, 11/3, ---- 13/3, ...
>
> 5/4, --- 7/4, --- 9/4, ---- 11/4, ---- 13/4, ...
>
> 6/5, 7/5, 8/5, 9/5, ---- 11/5, 12/5, 13/5, ...

You mean that he's kept all of the denominators at the same frequency?

If so, then all of the fractions can (and should) be converted to share the
same denominator. In which case dissonance, if it is proportional to d,
doesn't change across the entire experiment.

>This implies that lower frequencies are inherently more dissonant than
>higher ones.

How so? What does frequency have to do with anything, other than what
Sethares discussed with the sensory dissonance model?

Carl

At 10:55 19/03/99 -0500, you wrote:
>>But then we realise that the interval 2/3 is exactly the same size as the
>>interval 3/2 etc. They are the same interval. And yet we have decided that
>>2/3 is more dissonant than 3/2. A contradiction? No. Paul Erlich has
>>already explained this...
>
>I missed it! Cound somebody quote it? Or point me to digest and post #?

I can't be bothered searching for it, sorry, but the idea is just what I
described, and will again below.

>>Those who say dissonance is proportional to d are keeping the lower
>>frequency constant.
>
>What exactly do you mean? ...
>You mean that he's kept all of the denominators at the same frequency?

Yes. Here's what he (Dan Wolf) said:
>>I made a series of WAV files with sawtooth waveforms and a smooth envelope
>>in the following series of intervals, with the lower tone set to 200, 300
>>and 400 Hz:

So I take it he repeated the whole experiment 3 times.

>If so, then all of the fractions can (and should) be converted to share the
>same denominator. In which case dissonance, if it is proportional to d,
>doesn't change across the entire experiment.

Nah. Why? His whole point was that the dissonance was proportional to d
WHEN THE FRACTION IS IN LOWEST TERMS. This is also true for n+d.

>>This implies that lower frequencies are inherently more dissonant than
>>higher ones.
>
>How so? What does frequency have to do with anything, other than what
>Sethares discussed with the sensory dissonance model?

Maybe it does correspond to Sethares. Notice how we typically play only
octaves and fifths in the bass. The others are too dissonant down there.
What we are suggesting here is that the dissonance is related to the period
(inverse of frequency) of the virtual fundamental (VF). When the ratio for
an interval is in lowest terms, the VF corresponds to a 1.

e.g. If the lower tone is fixed at 400Hz the VF of a 6:4 (= 3:2) is 200Hz
(period 5ms) while the VF of a 5:4 is 100Hz (period 10ms). Periods are
proportional to d.

If instead we fix the average frequency of the two notes at 450Hz. The
notes of a 3:2 will be 360Hz and 540Hz and the VF will be 180Hz (period
5.56ms), while the notes of a 5:4 will be 400Hz and 500Hz and the VF will
be 100Hz (period 10ms). Periods are proportional to n+d or (n+d)/2.

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
http://dkeenan.com

>Notice how we typically play only octaves and fifths in the
bass. The others are too dissonant down there.

#!^%*^&#$%@#&*^@#!!!!... Typically schmipically.

Dan