Yahoo Tuning Groups Ultimate Backup mills-tuning-list What is consonance?

Gary Morrison's observations, and some chords that Harold Fortuin omitted
from consideration, lead me to review some basics about the psychoacoustics
of consonance.

There are two, somewhat separable components to the sensation of consonance
(relative consonance, as opposed to dissonance). One is roughness, which has
a negative effect on perceived consonance. This is a phenomenon that occurs
when two or more partials are closer in pitch than the critical bandwidth.
Beating is a companion phenomenon to roughness, but the rate of beating is
not a good measure of the amount of roughness. A constant level of roughness
will imply higher rates of beating as the pitch of the stimulus increases.
In many studies with tones with inharmonic spectra, roughness is far and
away the more important determinant of consonance/dissonance judgments.

The other component is tonalness, which is positively related to perceived
consonance. It is the degree to which all the partials approximate a single
harmonic series. As has been discussed, the origin of this sensation is
complex and includes the phenomena of virtual pitch (which replaces a bunch
of partials with a best-fit fundamental and a timbre), combination tones
(artifacts of nonlinear distortion in our ears or in the sound-creating
mechanism), virtual pitch of combination tones and roughness of combination
tones. The relative importance of this component is clearly not a constant
but dependent on pitch level, volume level, etc.

Gary Morrison's observations make some sense when these two components are
viewed individually. Here's another example which I find musically useful
and clears up some of the misconceptions about "undertones."

In my favorite tuning of 22-tET, a dominant ninth chord might be tuned 0 4 7
13 18 22 26. This approximates the just chord 8:9:10:12:14:16:18. Play this
chord in a moderately low register, and use a harmonic timbre. Now try
another dominant ninth chord, tuned 0 4 8 13 19 22 26. This approximates
what Harry Partch would call a "utonal" chord, with frequencies
1/18:1/16:1/14:1/12:1/10:1/9:1/8. Play this in the same register, and with
the same timbre, as the first chord. Which is more consonant? In one sense,
the first chord is more consonant; play it real loud (or with distortion)
and there will be no question. All the partials are within a single harmonic
series, plus or minus nine cents. But there is another, quite audible, sense
in which the second chord is ever-so-slightly more consonant. Observe that
the two chords contain exactly the same intervals. The difference is that
the smallest and most out-of-tune interval, 3/22 oct. (approximating a
10/9), is in a higher register in the second chord. This interval
contributes quite a lot of roughness to the chord, but the critical
bandwidth is wider in low registers, so putting the interval in a lower
register creates more roughness. This fact should be familiar to students of
orchestration.

The second chord, though consonant in a certain sense, does not conform to a
harmonic series. The simplest integral representation of this chord is
280:315:360:420:504:560:630. This is far beyond the range understood by our
virtual-pitch mechanism, as it corresponds to notes in the 8th-9th octave of
the harmonic series. There is no need to invoke an undertone series to
explain this chord, though; as we have seen, all the intervals here are
fairly simple and the roughest interval is in a high register. But if the
chord has nothing to do with a harmonic series, does that make it atonal?

Well, the determinant of the perceived root of a chord, other than the
lowest note in a particular voicing, is the virtual pitch sensation. It
operates perfectly well in the presence of distracting stimuli. Since the
simplest harmonics are usually the loudest, it is not surprising simple
ratios will be the most important for the virtual pitch phenomenon. Remember
that powers of 2 are octave-equivalent to the virtual pitch. A 3/2 (perfect
fifth) above the lowest note will greatly enhance the perception of the
lowest note as the root, while a 4/3 (perfect fourth) will destroy it. In
the absence of a perfect fifth, a 5/4 (major third) will enhance, and an 8/5
(minor sixth) will destroy, the rootedness of a chord.

However, in this case we don't need to go that far, since both chords have a
3/2 above, and no 4/3 above, the lowest note. Therefore, the root of both
chords is well-defined, though it is certainly better-defined in the first
chord. To wit: if the lowest note, and the note a 2/1 (octave) higher, is
removed from both chords, the perceived root of the first chord remains the
now-missing note, while that of the second chord becomes quite ambiguous.
Transposing the note designated as 13/22 oct. or 1/12 (formerly 3/2 above
the lowest note) down a 2/1 makes it into the new root, since a 3/2 above it
occurs at the note 4/22 oct. or 1/16. Thus it is not atonal in that it has a
clear root, although it has at least one other potential root that can be
expressed through omission and octave-shifting.

To my ears, this second chord can be as useful to 9-limit harmony as the
minor triad is to 7-limit harmony, though more care may be required in
voicing, tuning, and orchestrating it. The complexity of its integral
representation signals that any combination tones will serve to turn this
chord into mush. But consonant it is, in a certain sense.

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What is consonance?

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