Hearing the same interval as different ratios

Paul Erlich wrote (See http://www-math.cudenver.edu/~jstarret/22ALL.pdf):

"For example, the tritone b-f in 12-equal is ambiguous when heard in isolation, but can be heard as 7:5, 10:7, 17:12 in the chords g-b-d-f, c#-g#-b-f, e-g#-d-b-f respectively."

These imply the approximations 4:5:6:7, 4:6:7:10, 4:5:7:12:17.

Paul, I'm rather dubious about what it could mean to hear something *as* something else in this context. I suspect others are too. Can you suggest an experiment which would tell us which ratio any given person was "hearing it as"?

Regards,
-- Dave Keenan
http://dkeenan.com

>Paul Erlich wrote (See
http://www-math.cudenver.edu/~jstarret/22ALL.pdf):

>"For example, the tritone b-f in 12-equal is ambiguous when heard in
isolation, but can be heard as 7:5, 10:7, 17:12 >in the chords g-b-d-f,
c#-g#-b-f, e-g#-d-b-f respectively."

>These imply the approximations 4:5:6:7, 4:6:7:10, 4:5:7:12:17.

>Paul, I'm rather dubious about what it could mean to hear something
*as* something else in this context. I suspect >others are too. Can you
suggest an experiment which would tell us which ratio any given person
was "hearing it >as"?

>Regards,
>-- Dave Keenan
>http://dkeenan.com

I admit that this is a dubious concept, but there are many situations
where it has meaning, and often these different situations lead to a
unique set of ratios. The first experiment that comes to my mind (and
certainly is one of the important experiments I performed on myself when
trying to understand ratio-interpretations) goes like this: Take the
tempered chord, and alter the tuning of the pitches until maximum
consonance is reached, without ever allowing the quality of the chord to
change (that is, does it sound like a different chord?). Now this
probably still sounds dubious to you, but as another example, take the
minor triad. When I was doing these experiments, I definitely sensed a
change in quality (besides consonance) when it is changed from 12-equal
to 10:12:15, at least for a certain timbre and register. So I would
probably say that under those conditions, the 12-equal minor triad is
"heard" as 16:19:24. However, doing the experiment with the individual
intervals of the chord led unquestionably to 10:12:15 (i.e.,
1/6:1/5:1/4). (BTW, for two-note chords, the most complex ratio I could
tune by ear was 17:13.)

The thinking behind this statement in my paper is a model like
Parncutt's. See his book _Harmony: A Psychoacoustical Approach_ and
you'll get a much better sense of why I think these statements are
meaningful.