EJ=0 as a special case of contrary motion

Yeo Doglas (Singapore, Independent Researcher)

Published in musi.philica.com

Abstract
This article reconciles a mathematical observation with a known rule of counterpoint — that of contrary motion.

(A) Observation:

I have observed that the relationship $y=\sum{J}$=0 (algebraic sum of "jumps" equals to zero) occurs quite frequently in classical music.

A "jump" is defined as the number of semitones apart, taking into account direction, between two notes. For instance, a progression of C to C# is equivalent to a "jump" of +1. A progression of C to B, on the other hand, is equivalent to a jump of -1.

I cite two examples of $y=\sum{J}$=0 :

1) In Debussy's Premiere Arabesque, the famous first two bars are:i) C#,E,A,C#,E,F#,G#,D#,B,G#,D#,B

ii) A,C#,F#,A,C#,D#,E,B,G#,E,B,G#

The first bar is actually made up of two sub-phrases, each consisting of 6 notes:

ia) C#,E,A,C#,E,F#

ib) G#,D#,B,G#,D#,B

Assuming that only the outer parts are essential in the harmony, and those in between are passing notes, it yieds the following progression:

C#—>B J=-2

F#—>G# J=+2

$y=\sum{J}$=0

Similarly for the second bar,

A—>G# J=-1

D#—>E J=+1

$y=\sum{J}$=0

2) The second example is the popular chord progression of V7 to I. (FGB—> EGC)

Separating the individual notes, there are actually 3 note progressions:

i) F—>E J=-1

ii) G—>G J=0

iii) B—>C J=+1

$y=\sum{J}$=0

(B) Analysis

It took me a while to realise that the relationship $y=\sum{J}$=0 is actually a special case of the rule of counterpoint that notes of a chord progression should ideally proceed in contrary motion. This is the consequence of the four fundamental rules of Johann Joseph Fux's (1660-1741) Gradus Ad Parnassum.

There are only two possible forms of contrary motion: The upper part moves up, while the lower part moves down. Another possibility is the upper part moves down, while the lower part moves up.

The case where one note does not "move" is defined as oblique motion, not contrary motion.

Hence the mathematical significance of contrary motion is that

J1>0

J2<0 or vice versa.

In contrary motion, one "jump" will definitely be negative, while the other would be positive.

In the special case that J1=-J2 , then $y=\sum{J}$=J1+J2=0.

Hence $y=\sum{J}$=0 is a special case of contrary motion.

(C) Possible drawbacks of this observation.

i) The idea that notes must proceed in contrary motion is at least 300 years old, and hence is not exactly the most modern or advanced theory of harmony. However, it must be noted that many later composers like Beethoven, Chopin and even Berlioz subscribed to the ideas of Johann Joseph Fux, the Baroque theorist of counterpoint.

ii) The significance of $y=\sum{J}$=0 is at most a mathematical one. I believe that the best theory of music is a biological one, followed by physical, mathematical, then metaphysical. However, I could not attach any physical significance to EJ=0, much less a biological one.

(D) References: