decreasing dissonance with more notes

Following the thread of Joe Monzo and Paul Erhlich:

> That's true in the context of higher-limit intervals and chords -- for
> example, the rather dissonant dyad 11:9 'sounds fine' in the context of
> 4:5:6:7:9:11 because of tonalness... Perhaps it's the average roughness
> per interval, rather than the total roughness, that better represents
> that component of dissonance. What do you think, Bill Sethares?

I suspect that what may be going on here (at least in part) is
the "masking" of certain partials. Consider an extreme case: white noise.
If you just apply the sensory dissonance calculations naively, you
would need to conclude that white noise is the most (sensory) dissonant sound
possible, yet this is clearly not the case. One sensible resolution
is to exploit the psycho-acoustic work on masking, in which larger
partials drown out smaller nearby partials. Following this, white noise
is no longer all that dissonant.

To apply this to the example you cited: the 11:9 dyad may be
quite dissonant when played alone, but when played in the
context of a 4:5:6:7:9:11, many of the partials may be masked by the
lower tones, and so the whole has considerably less sensory dissonance
than it might at first appear.

bill sethares

William Sethares wrote:

> -- for
> > example, the rather dissonant dyad 11:9 'sounds fine' in the context of
> > 4:5:6:7:9:11 because of tonalness... Perhaps it's the average roughness
> > per interval, rather than the total roughness, that better represents
> > that component of dissonance. What do you think, Bill Sethares?
>
> I suspect that what may be going on here (at least in part) is
> the "masking" of certain partials. . . . .
>
> To apply this to the example you cited: the 11:9 dyad may be
> quite dissonant when played alone, but when played in the
> context of a 4:5:6:7:9:11, many of the partials may be masked by the
> lower tones, and so the whole has considerably less sensory dissonance
> than it might at first appear.
>
> Might it be that the 9:11 difference tone (2) is heavily reinforced by the 4:6,
> 5:7, and 7:9 dyad component's difference tones, as well as the octave below by
> the 4:5, 5:6, and 6:7 difference tones. This strongly implies a harmonic
> series which is a readily recognizable psycho-acoustic pattern in frequency
> space. Human beings seem to like these harmonic series patterns.
>
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I wrote,

>> That's true in the context of higher-limit intervals and chords -- for
>> example, the rather dissonant dyad 11:9 'sounds fine' in the context of
>> 4:5:6:7:9:11 because of tonalness... Perhaps it's the average roughness
>> per interval, rather than the total roughness, that better represents
>> that component of dissonance. What do you think, Bill Sethares?

Bill Sethares wrote,

>I suspect that what may be going on here (at least in part) is
>the "masking" of certain partials. Consider an extreme case: white noise.
>If you just apply the sensory dissonance calculations naively, you
>would need to conclude that white noise is the most (sensory) dissonant
sound
>possible, yet this is clearly not the case. One sensible resolution
>is to exploit the psycho-acoustic work on masking, in which larger
>partials drown out smaller nearby partials. Following this, white noise
>is no longer all that dissonant.

That may be part of the answer for lower-limit examples, but:

>To apply this to the example you cited: the 11:9 dyad may be
>quite dissonant when played alone, but when played in the
>context of a 4:5:6:7:9:11, many of the partials may be masked by the
>lower tones, and so the whole has considerably less sensory dissonance
>than it might at first appear.

I wouldn't apply this to the example, since lots of dissonant dyads would
still sound dissonant over 4:5:6:7. In this example, 9 and 11 form a
harmonic-series segment with the other notes, and thus the entire chord
"blends" and has much of the character of a single tone. All fundamentals,
partials, and combination tones form a harmonic series over a fundamental
two octaves below the root of the chord. Because of the psychoacoustic
virtual pitch phenomenon, this complex will be heard as a very clear pitch
sensation. Although part of dissonance is the roughness (what you call
sensory dissonance), part of it is the complexity of the virtual pitch
sensation, which my harmonic entropy concept provides mathematical models
for. If you believe in the pattern-matching theories of virtual pitch, this
component might be described as "cognitive dissonance" of trying to
understand the heard stimulus in termps of a learned pattern of harmonics.
But periodicity mechanisms in hearing are well-established for periods
between 20 and 1000 Hz. So it could very likely be that hearing a
4:5:6:7:9:11 chord under the right circumstances, the brain actually detects
the exact repetition of a signal with a period corresponding to the
frequency of the 1 (two octaves below the root).

Either way, the sensory component roughness is somehow counteracted by
another component, while in a 1/11:1/9:1/7:1/6:1/5:1/4 chord, although the
sensory dissonance is virtually the same, even considering masking, there is
no equivalent phenomenon to "simplify" the sensation, and it sounds very
dissonant.

What we were talking about, Bill, is the fact that a 720-cent fifth sounds
kind of off when the dyad is played alone, but in the context of 15-tET
triads it doesn't sound so bad. Do you think masking is responsible? How?