[Smt-talk] Harmonics

Whitcomb, Benjamin D whitcomb at uww.edu
Thu Jul 24 08:17:58 PDT 2014


The reason why we frequently find M3, m3, and other intervals altered in musical contexts is best explained, in my opinion, by the "two channel" theories of pitch processing, combined with ideas like Boomsliter and Creel's "extended reference" theory (see the short bibliography below). Major thirds are often expanded and minor thirds contracted in order to better differentiate them from each other. Experiments like Balzano's show this tendency well. The longer a chord is held, the more likely it is to be tuned justly (i.e., according to the harmonic "channel"); the shorter it is, the easier it is to bend the note in the direction of its voice leading tendencies (i.e., according to the melodic "channel"). 

The "roughness" theories fall flat (so to speak): if they were true, why are we just as able to tune a chord whose pitches are separated by an octave or more (thus, not close on the basilar membrane) as those within an octave? And, why can I tune intervals binaurally (i.e., one pitch in one ear and one in the other) as well as when I hear both pitches in both ears?

Here is a list of some of my favorite articles on these ideas:

Balzano, Gerald J.  “Musical vs. Psychoacoustical Variables and their Influence on the Perception of Musical Intervals.”  Bulletin of the Council for Research in Music Education 70 (1982): 1-11.

Boomsliter, Paul, and Warren Creel.  “Extended Reference: An Unrecognized Dynamic in Melody.”  Journal of Music Theory 7 (1963), 2-22.

__________.  “The Long Pattern Hypothesis in Harmony and Hearing.”  Journal of Music Theory 5 (1961): 2-31.

Deutsch, Diana.  “The Processing of Pitch Combinations.”  In The Psychology of Music, 2nd ed., ed. Diana Deutsch, 349-411.  New York: Academic Press, 1999.

Hajdu, Georg.  “Low Energy and Equal Spacing; the Multifactorial Evolution of Tuning Systems.” Interface 22 (1993): 319-333.

Noorden, Leon van.  “Two Channel Pitch Perception.”  In Music, Mind, and Brain, ed. Manfred Clynes, 251-67.  New York: Plenum Press, 1982.

Terhardt, Ernst.  “The Two-Component Theory of Musical Consonance.”  In Psychophysics and Physiology of Hearing  ed. E. F. Evans and J. P. Wilson, 381-390.  London: Academic Press, 1977.


Dr. Benjamin Whitcomb
Professor of Cello and Music Theory
University of Wisconsin-Whitewater

From: Smt-talk [smt-talk-bounces at lists.societymusictheory.org] on behalf of Marcel de Velde [marcel at justintonation.com]
Sent: Wednesday, July 23, 2014 7:20 PM
To: smt-talk at lists.societymusictheory.org
Subject: Re: [Smt-talk] Harmonics

I have researched just intonation full time for the past 9 years or so,
and can share the following on the major and minor triad and harmonics.
First of all, these days we have computers which can reproduce perfectly
the tuning of 5/4 "major third" and 6/5 "minor third" time and time
again without error and with a variety of acoustical (sampled), and
synthetic sounds with perfect harmonic spectrum etc.
I've tuned hundreds of common practice period pieces to just about every
form of "just intonation" ever written about and several more invented
by myself. A luxury non of the music theorists and writers on the
subject in the past had.
And I've come to a very clear conclusion. 5/4 is NOT the "just" major
third and 6/5 is NOT the "just" minor third.
We distinguish intervals solely according to a chain of perfect 3/2
fifths and 2/1 octaves.
Though playing / singing a major third as 5/4 in certain places will
give a nice acoustic effect with most timbres because of many
interlocking harmonic overtones, and more rarely practised the same goes
for the 6/5, it is ultimately "out of tune".
And it is true that many string quartets, trombone quartets and choirs
(especially barbershop quartets) will often play a major third as 5/4,
many others do not, and those that do often get themselves into trouble.
I have analysed a lot of polyphonic music with Melodyne Direct Note
Access which allows one to detect the pitches of individual instruments
within a polyphonic recording.
There is a great difference between overtones and chords as has been
mentioned in this discussion before. The confusion is and potential for
error is made even bigger in part because one can play for instance a
dominant 7th chord as 1/1 5/4 3/2 7/4 in isolation without trouble and
most ears will not hear it as out of tune instead only marvelling in
it's highly synchronous sound. Yet once one starts to tune actual music
this way, for instance in a piece by Beethoven or Bach or Chopin it will
sound very out of tune to most ears.
In addition there are mathematical problems with tuning major thirds as
5/4 etc that cannot be overcome. For instance circle progressions. It is
because 4 perfect fifths (or fourths) reduced by octaves do not make a
5/4 major third, but an 81/64 major third instead (this 81/64 major
third is in fact the correct tuning for the major third).
The 5/4 major third is functionally completely incompatible with western
music theory and western notation.
Western music theory and notation functions according to a chain of
perfect fifths and octaves, in other words Pythagorean tuning (which one
can call Pythagorean just intonation).

As for the fundamental nature of the major and minor thirds I agree.
But I can add that the pentatonic scale is arguably as fundamental.
Especially in world music where polyphony hasn't developed.
The pentatonic scale is of course 5 tones connected by 4 perfect fifths
and contains both the major and minor triad within itself (whereas the
western major and natural minor scales are of course 7 tones connected
by 6 perfect fifths).
I don't see how one can sell the idea of a pentatonic scale based on
harmonic overtones.
For an easy demonstration let's play for instance the chord E-A-D-G-C.
Surely one would tune these fourths as 4/3, a wolf fourth would sound
very out of tune.
Now play in addition a C in the bass, we have contained herein a C - E -
G major chord tuned as 1/1 81/64 3/2. It doesn't matter if we play the
additional A and D it's still a major triad in there.
Same if we play an additional A in the bass, we get an A - E - C minor
chord tuned as 1/1 32/27 3/2.

The question then remains why is a 1/1 81/64 3/2 major triad perceived
as "consonant" and as more consonant than a 1/1 9/8 3/2 suspension even
though the latter has more simple ratios.
There are several reasons for this I belief but one of them is to be
found in the "roughness" theory, which very roughly states that the
closer 2 pitches (and one should take into account the most prominent
harmonics of the timbre as well like the 2nd and third harmonics) the
more "dissonant" they become. The 81/64 major third and 32/27 minor
third are simply the only simple intervals that are of maximum distance
between the 1/1 unison and 3/2 perfect fifth.

Hope this is of help to some here.

Kind regards,

Marcel de Velde
Zwolle, Netherlands
marcel at justintonation.com

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