Sep

11

My young son, now 18 months old, has been "singing" for almost a year. Since both his parents sing a great deal both to him and to each other, it seemed natural that he would pick it up early. This month, though, he started for the first time to mimic pitch correctly — very exciting.

What about musical tones make some sound lovely together and others clash terribly? Musicians call two notes consonant or dissonant depending on whether they blend or clash. Pythagoras discovered that when two strings were equally stretched, they would be consonant if their lengths were in the ratio of two small numbers such as 2:1, 3:2, etc. So important was this discovery of a law of nature being ruled by integers that it was extended to all the sciences, especially astronomy.

The simplest example of two notes that sound good together is when they are the same note, played one or more octaves apart. An octave separation between notes corresponds to a doubling of the frequency.

When a musical note is played on a stringed instrument, it has a primary frequency and numerous overtones, which are integer multiples of the frequency: for example the A below middle C has its primary frequency at 440Hz (by convention), while the E just above middle C is 660 Hz. The overtones of the A are 880, 1320, 1760, 2200, 2640 and so on. The overtones of E are at 1320, 1980, 2640, and so on. We see that the overtones line up with each other, and the result is a harmonious, or consonant sound.

A   440         880          1320         1760         2200         2640
E         660                1320                1980               2640

If on the other hand, we play the A with a G (freq ~ 785Hz, then the overtones clash with each other.

A   440         880          1320         1760         2200         2640
G             785                  1520                  2355

The result is a disharmonious or dissonant sound. Moving the frequencies of the note around to reflect integer ratios brings a sense of relief and correctness to the listener. This is true even if the notes are not real notes: for example, moving the G's fundamental up 100 Hz, the first harmonic down 200 Hz, and the third harmonic down 150 Hz is not a note found in nature, but can be produced by computer. It is consonant with the A because of its integer-bound relationship to it.

Musicians use these integer relations to great effect manipulating our emotions: they can elicit discomfort, relief, tension, and even laughter by moving further and more aggressively into non-integer ruled domains (dissonance) and resolving back into the more natural integer ratios that sound so lovely.

We know that the Mistress is a musician. Where in her symphony can we find the sources of consonance and dissonance that cause such anxiety and give such relief? An analogue to frequency in plucked strings might be found in volatility. As markets vibrate around whole numbers — fundamental frequencies — are other markets in consonance or dissonance with them? When SPX is ringing out the pure tone of 1300, are the vibrations of DAX part of a small-integer ratio? What has to happen to bring a dissonant market into consonance?

As a final note (yes, intended), one should probably be aware of the changing fashion of music. The notes that Bach wrote had mathematical progressions and rarely used dissonance: today's composers use dissonance to extract maximum effect and may never even resolve into consonance. You have to know your composer if you want to know when to applaud.


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