Download Music Theory Combination Tones

Musical Notation of Combination Tones for
Chromatic, Equal-Tempered Dyads
Glen Halls ©All Rights Reserved
Abstract
All chromatic intervals within the octave are tested for flute with clarinet, and for viola with alto
saxophone, with the intent of prediction and aural confirmation of combination tones for that interval.
Multiple transposed samples of the same interval are compared to arrive at generalized and reasonably
predictable, actually perceivable, combination tones for that interval. The validity of combination tones
has ramifications in the areas of consonance and dissonance, and of harmony in general, both within and
outside of the tonal system.
Introduction
Combination tones may arise whenever two sounds are presented simultaneously. Combination tones are
not "physically" present, but are generated within the ear of the perceiver due to the "non-linearity" of
auditory processing. For two tones of frequencies x and y, a first order "summation tone" will be
generated at frequency x+y , and a first order "difference tone" at x - y . (always subtracting the smaller
frequency from the larger). Theoretically, frequency may be predicted for a given interval. Also, If
harmonics are present and of sufficient amplitude the 'second and third order' combination tones
corresponding to possible combination tones of second and third harmonics respectively, are also
predictable. This is where instrumental timbre comes into play, possibly effecting the net combination
tones for a given interval. Rough spectrographs of the intervals sampled attest to the rich harmonic
content of the sounds, but this is an empircal limitation. In some instances combination tones will overlap
with and reinforce harmonics. In the present study, perceived pitches which match harmonics of either
tone are not included In any event, if other tones corresponding to predicted combination tones are
proven to be consistently present, there are large implications for our conceptions of chordal quality, of
consonance and dissonance and chord stability, for function, for orchestration, and for composition in
general.
Methodology
In a sense this project is a kind of psychological , or literally a psychoacoustic, inquiry. It is normal in
this realm to employ a large sample of subjects given the same stimuli, and then to make conclusions
based on statistical trends. On the contrary and for both musical and practical reasons, the present study
takes as its sample pool a single individual ( me ) , someone who might be termed an 'expert witness' or
'authority' on sounds. As in jazz music where the novice listener might hear only a pounding ride symbol
and a stream of notes, the experienced listener hears a vast number of nuances and relationships. So it is
with sonority, timbre, and consonance. One requires an educated ear and only with practice and patience
and an absence of conflicting stimuli can one actually hear these details within the sound as specific
pitches.
With the assistance of graduate students in music,Cchromatic intervals were recorded in different registers
for flute and clarinet, and for viola and alto saxophone. These sounds were later edited into individual
samples, looped, and auditioned. The data gathering proceeded in three stages. First, the sound was
simply listened to, over and over, with no predicted outcome, and any additional pitches ( with the
exception of 8ves and P5ths rationalized as 2nd, 3rd, and 4th harmonics) notated in music notation. Note,
in many cases the perceived additional tones were quarter tones out from equal tempered pitches, but at
this stage they were simply notated as what I subjectively interpreted as the closer equal tempered tone. (
I did this work at the piano) Secondly, using cycles per second values corresponding to equal tempered
pitches, first , second, and third order combination tones were listed in spreadsheet form, and also
transferred to musical notation. This in itself is of some interest, as it present a kind of factored tree
branching off from a given internal. It is a kind of related pitch set which may in itself present an
interesting compositional strategy down the road. Finally, each interval was auditioned again but this
time with the knowledge of predicted pitches.
For Example: First the interval F3C4 is identified, and in the following measure perceived combination
tones E2 and F2 are notated.
Secondly, predicted combination tones, corresponding in imperfect fashion to the closest equal tempered
pitches, are notated, left to right, as a) original pitches b) first order combination tones ( the sum tone and
the differences tone), c) second order combination tones ( as half notes) and d) third order combination
tones ( as quarter notes)
Finally the second 'guided' listening is notated, with the original pitches in the first measure, the strongest
combination tones in the second measure, ( and then in many instances barely audible and uncertain
combination tones in a third measure.) Plus and Minus signs beside the note are used to indicate pitches
which are clearly lay in between equal tempered pitches.
Note: There are many instances were perceived tones DID NOT correspond either to harmonics or to the
predicted range of combination tones. A later study will consider the origin of these mystery tones.
Perhaps they are simply combination tones from higher harmonics or some untried combination of
harmonics, or some other phenomena.
In many instances an informal commentary on the sample follows, but the idea is really to see if the same
interval gives some of the same results for different instruments and in transposition.
Part One. Sequential/chromatic Testing of intervals for combination tones.
First Set of Samples. Viola moves against a stationary Alto saxophone C4
(note-numbers refer only to recorded samples stored on CD, available by request
(but I'm not sure why one would want them)
Second Set of Samples: Flute moves against a stationary Clarinet C5
Next Sample. Alto Saxophone Descends chromatically against sustained Viola A4
5
Next Sample. Clarinet ascends chromatically against Flute sustained A4
Special Thanks to Glen Halls for his invaluable insight and assistance on this project.