Dissonance is word used to describe how unpleasant something sounds; particularly in psycho-acoustic studies. In one article I read by Normal Cook, in the Music Perception Journal, it can be calculated numerically from the magnitude of tones that make up a sound.

The formula for calculating dissonance (D) from tones is

where:

- ν = the product of the relative amplitudes of the two tones (0.0 to 1.0),
- x = the interval size defined in semitones.
- β
_{1 }= -0.8 is the interval of maximum dissonance
- β
_{2 }= -1.6 is the steepness of the fall from maximal dissonance
- β
_{3 }= 4.0

When we plot this for 2 pure tones we get a nice little plot like this:

However, real instruments do not produce pure tones. They produce harmonics, that is, lots of extra tones that are roughly integer multiples of the frequency of the lowest tone. Interestingly, if you add up all the dissonances from all the pairs of tones between 2 “notes”, the plot starts looking a lot more interesting.

There are tantalising dips at important musical intervals; like the 5th, 4th, and octave.

I did my own plot, using the first 8 harmonics with similar magnitudes as a piano sound and got this plot:

This graph has some very interesting features. I have plotted the equal temperment semitone points with small grey lines. These miss the kinks in the dissonance plot slightly. But, when I plot the Just Intonation semitone points [1, 16/15, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8, 2] in red, they match the troughs and kinks almost perfectly.

I wonder what this graph looks like if we extend it into the next octave?

Interestingly, this calculation can be extended to cover any combination of any number of notes and is an interesting way to compare chords of any kind. The part that appeals to me is that a simple relationship between tones and their perception can generate parts of music theory, without requiring “the edifice of music theory”. This is *Lateral Music Theory*.

You can calculate your own dissonance for chords at a web page I wrote a while ago: here.