Cross-associations between melodic and harmonic data in categorical perception

1) It's possible, in certain musical contexts in 12-tet meantone, for
an augmented second to sound drastically more dissonant than a major
third.
2) 300 cents is the largest second in 12-tet meantone[5], and is
consonant. 300 cents is the largest second in 12-tet harmonic minor,
where it is dissonant. 300 cents is the smallest third in meantone[7],
and it is consonant.
3) Thus, the perceived consonance of this interval within a western
tonal context is not correlated with the number of degrees that it
subtends in a scale.
4) However, the lowest-complexity meantone mapping for this interval,
independent of MOS or MODMOS, -IS- perfectly correlated with its
perceived dissonance and consonance.
5) Or, in other words, if within some scalar context the 300 cents
would be mapped to a consonant interval such as 6/5 in meantone, it's
consonant. If it would be mapped to something more dissonant or
complex like "75/64," which may be heard as "junk" as far as the ear
is concerned, it's dissonant.
6) #4 holds even in a population of listeners that has never heard
music in non-12 tunings of meantone.

From this, we can infer that the generalized meantone pattern of
consonances can in some capacity be non-trivially internalized by the
listener. For now it should be known that the full picture of this
process is poorly understood, but from this its existence can be
inferred on a basic level.

It should be noted that the consonance judgments implicated by these
associations seem to override those implicated by psychoacoustics.
This mechanism can change our perception of the consonance and
dissonance of intervals that are exactly equal in size and harmonic
entropy. It can also cause intervals of low harmonic entropy to sound
much more dissonant than they would otherwise.

Most significantly, this process happens even if the listener has
never heard a tuning in which the ambiguous intervals differ in real
size. This proves that a listener does not require initial exposure to
a tuning in which the diminished fourth is higher in harmonic entropy
to perceive it as being more dissonant. This shows that the consonance
values of more complex intervals in this map are -inferred- by the
listener, and not -remembered-.

If the meantone organizational pattern can be internalized by the
listener such that major thirds and diminished fourths differ in
perceived consonance, then this suggests that interval categories
don't have to be limited to containing only melodic data, but can
contain some form of harmonic data as well. It implies that listeners
of harmonic music form cross-associations and expectations between
melodic and harmonic percepts.

Any such series of cross-associations between melody and harmony can
be represented by a wedgie of a list of vals. This theory thus
suggests that the wedgie is a representation of an actual, low-level
cognitive procedure that listeners form after continued exposure to
music in a certain tuning system. It implies that the associations
that a regular temperament forms between melody and harmony can be
predicted by the listener to such an intuitive extent so as to impart
"functions" onto the temperament's structure, and that these functions
can be distinguished even in tunings where intervals in different
categories are tempered together in size. Most generally, it suggests
that the regular mapping paradigm is "very successful" in describing
music cognition, rather than "partially successful."

Lastly, it suggests that interval categories are most wholly
represented by a set of coordinates in a projective tempered space.

From all this we might make a prediction: if intervallic consonance
can be inferred, and if this inference can override the psychoacoustic
consonance of an interval as modeled by harmonic entropy, then a piece
of common practice music should be recognizable, with all of the
consonances remaining ordered in the same way relative to one another,
as long as the melodic structure of the piece can remain intact. It is
further predicted that, as far as the diatonic scale is concerned,
this perception will "break down" only when the size of the generator
actually reaches 3\5 or 5\7, as this will be the point at which the
scale will start to present inconsistent categorical information.

So we might test this by loading up a piece of common practice music,
perhaps a composition by Schubert, altering the size of the generator,
and seeing what happens.

-Mike

I just had a streaming audio sesh with Gene and Ryan, both of whom
stated that they heard the dim4 as being equal in consonance to an M3,
even with a quite lengthy shredfest in the relevant scales preceding
it, and then ending on the bare dyad in question. So now I have no
idea how music works at all anymore.

I think I'm going to make some musical examples and get people to vote
in a poll. The goal, of course, isn't to "validate" any one perception
over others, but to see what sorts of perception are most common, and
study what factors might cause it to differ.

A few additional observations, which at least apply to me:
1) It's possible for a diminished fourth to sound consonant as well,
for example when it's used as part of a dom7 chord in the altered
scale.
2) After experiencing the diminished fourth as consonant in this
context, it's possible to then experience it as dissonant again when
you go back to common practice music.

FWIW, if the consonance of the first one is related to "5/4" in some
capacity, then that's schismatic temperament.

On Thu, Sep 8, 2011 at 8:53 AM, Mike Battaglia <battaglia01@...> wrote:
> 1) It's possible, in certain musical contexts in 12-tet meantone, for
> an augmented second to sound drastically more dissonant than a major
> third.