clarinets vs violins for M3 demos

Gary and Paul have been discussing the

>> Best Timbre for Demonstrating 12TET M3s

>> Huh? I would have thought the opposite, since the main source of
>> beating/roughness of the 400-cent M3 is due to the 4th partial of the
>> higher note clashing with the 5th partial of the lower note.

> Yip, that's what I would have thought too, but that's not what ex-
>periment seems to bear out. I have also managed to convince more people
>that a 13-some-odd-cent sharp M3 sounds raunchy more often with a
>clarinet timbre than with any other.

In terms of sensory dissonance (roughness), I think this can go either
way depending on the absolute pitch of the fundamental.

To see this, suppose you have a timbres built on a fundamental at
200 Hz.
The partials will be at:

200 400 600 8000 1000 (idealization of a violin spectrum)
252 504 756 1008 1260 (idealized violin at M3)
200 600 1000 1400 1800 (idealization of a clarinet
spectrum)
252 756 1259 1764 2268 (idealized clarinet at M3)

Here there is beating primarily between the 1800 and the 1764
in the clarinet spectrum and between the partials at 1000 and 1008
for the violin. Since the 1800-1764=36 is in the range for maximum
roughness, the clarinet will be relatively dissonant, while none of the
partials in the violin are particularly rough.

On the other hand, suppose the fundamental is at 600 Hz.
Then the partials are at:

600 1200 1800 2400 3000 (idealization of a violin
spectrum)
756 1512 2268 3024 3779 (idealized violin at M3)
600 1800 3000 4200 5400 (idealization of a clarinet
spectrum)
756 2268 3779 5292 6804 (idealized clarinet at M3)

Now there is little beating between the partials of the clarinet and its
version at the M3 because all the partials are well separated, while
the violin has beating near the region of max roughness between the
partials at 3000 and 3024 - I suspect this is the case that Paul was
thinking about in his post. Hence I am guessing that you were
playing the clarinet and violin samples at fairly low pitch when
making these observations.

Here again we see

>timbre unequivocally affects perception of tuning

(thanks Gary - this is one of my favorite themes!)
The caveat (or warning) here is that *absolute* frequency also has
an impact. With all else equal, a lower sound will be more dissonant
than a higher sound, but unfortunately, all else is not always equal!

William Sethares wrote,

>suppose you have a timbres built on a fundamental at
>200 Hz.
>The partials will be at:

>200 400 600 8000 1000 (idealization of a violin
spectrum)
>252 504 756 1008 1260 (idealized violin at M3)
>200 600 1000 1400 1800 (idealization of a clarinet
>spectrum)
>252 756 1259 1764 2268 (idealized clarinet at M3)

>Here there is beating primarily between the 1800 and the 1764
>in the clarinet spectrum and between the partials at 1000 and 1008
>for the violin. Since the 1800-1764=36 is in the range for maximum
>roughness, the clarinet will be relatively dissonant, while none of the

>partials in the violin are particularly rough.

questions:

1. Won't the violin also have the 1800 and 1764 partials?

2. Tuning the clarinets in this register to a 5:4 would only speed up
the beating, lowering the roughness only slightly. Tuning them to a 9:7,
however, would eliminate the beating entirely. So is a 9:7 more
consonant than a 5:4 for low clarinets? (Gary, you made the original
comment, so tell us what you think.)

> 2. Tuning the clarinets in this register to a 5:4 would only speed up
> the beating, lowering the roughness only slightly. Tuning them to a 9:7,
> however, would eliminate the beating entirely. So is a 9:7 more
> consonant than a 5:4 for low clarinets? (Gary, you made the original
> comment, so tell us what you think.)

Perhaps it has something to do with other than calculable roughness of
beating.

I may be remembering incorrectly, but I think I also concluded that
clarinet is better at showing up temperament errors in 12TET vs. JI
4:5:6:7, and just 5:6:7. Maybe that's a hint of some sort, perhaps the
clarinet timbre's hollow character just accentuates tuning errors in
general.

It's definitely tied to range too. I don't hear the same effect with
a bass clarinet, which of course has a somewhat similar timbre in a lower
range.