Consonance and Dissonance in Music

Beware: Science and Math Ahead!


A Facebook friend posed the following question:

Here’s a bit of music theory to dabble with…

The fifth in a scale is called the “perfect fifth”. The ear loves it with a passion: So much so that it has a vile hatred of the diminished fifth, or tritone. The fifth is even called the “dominant”. Musical tuning and key signatures revolve around the fifth…literally. (circle of fifths)

But why is that? I wondered about the underlying math. The ear accepts an octave as the same note because it is double the frequency. A4 is 440 Hz. A5 is 880 Hz. A3 is 220 Hz. The chromatic scale has 12 progressive semitones (minor seconds), and the fifth happens to be 7 semitones above the tonic. (12 and 7 are Divine numbers.) What’s 7 the number of? Divine completeness. Six is the number of man. The tritone is only six semitones above the tonic.

Does your ear really understand that? It might. But what is the fifth in terms of actual frequency?

Well, a minor second is 2^(1/12) Hz above its tonic. So it stands that the fifth is 2^(7/12) Hz above. And that power is almost exactly 1.5. (It’s actually 1.49something). Incidentally, the tritone, being six semitones above tonic is exactly root two. That’s right, the square root of two times tonic. Since A major has 3 sharps (F, C, G), and the fifth is E natural, it stands to reason that E4 is just under 440 x 1.5 Hz, or almost 660 Hz. The ear likes that interval.

My answer follows:

You don’t ask simple questions, do you? First of all, consonance (the opposite of dissonance) is entirely subjective. Yes, we have centuries of study that show harmonics are pleasing to the ear of the average person, but that’s still only the average person’s opinion. Consider that different rules apply across the world; we are mostly familiar with Western music. Also, consonance would be boring without dissonance, which gives us the ability to color our music with emotion and expression.

Nonetheless, the perfect fifth is considered pleasing to the ear by general consensus. Mathematically, the perfect fifth has the simplest frequency ratio (see “pitch ratio“) except for the unison and octave intervals. If you would like to conjecture about the spiritual basis for the consonance of the perfect fifth, I certainly enjoyed reading your theories. I can only add that if you believe that God made all things beautiful and all things unpleasant, then it follows He is responsible for the design of audio frequencies and their relative consonance or dissonance. I find it entirely believable that He would have a sense of humor or at least irony about the whole thing.

In your question, you correctly stated that the pitch ratio of the perfect fifth is 1.5, but consider it this way instead: 3/2. The octave is 2/1.The perfect fourth is 4/3, also considered a highly consonant interval, and the major third is 5/4. The augmented fifth, according to this chart of pitch intervals, is 25/16.

When multiple audio frequencies are heard together, they do not remain independent. They interfere with one another. So the relative frequencies, and how simple or complex they are with respect to one another, will impact the resulting frequency that you hear. For the perfect fifth, independently, the fifth oscillates three times for every two oscillations of the root. This is why we consider pitch ratios in whole number fractions instead of decimals to evaluate their simplicity (where “simplicity” is an indicator of consonance.)

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