The ugly, the good and the minor third

Why does a melody sound good to us, to some of us but not to others? Why does a song sound sad while another one sounds glad or awfully dissonant? I don’t know, of course, but it seems to have to do with evocation of known melodies, cultural conventions and something in our brain that is still to be discovered.

Some time ago I was curious about what kind of sound resulted from the addition of two notes in a chord or a melody.

Let’s consider a root note, let’s say A440, known to have a frequency of 440Hz. The sound will be a sinusoidal wave,  something that would seem like a flute.


Now let’s add a perfect fifth. We’ll consider a perfect fifth to have a frequency = 3/2 of its tonic.


In black we have the tonic, in red perfect fifth and in blue the sum of both signals. What happened? The resulting sound is a signal with a frequency of 1/2 * 440Hz = 220 Hz.

Now let’s consider our A flute and an added major third. There are several standards for major and minor third frequencies related to tonic (see but for this exercise I will take the just intonation, in which the major third has a frequency of 5/4 multiplied by its tonic frequency.


In black the tonic, in red the major third and in green the sum. Now we have a resulting signal of  1/4 * 440Hz = 110 Hz.

Now let’s add a minor third to the tonic. In just intonation, minor third has a relationship of 6/5 with its tonic frequency.


In black the tonic, in red the minor third and in green the addition of both. Now the resulting frequency is 1/5 * 440Hz = 88Hz

The last plot will consider a very dissonant note, a minor second, with a relationship with the tonic of 16:15.


The frequency of the resulting signal is 1/14 * 440Hz = 31.42Hz

Obviously, as the second note’s frequency approaches the tonic’s frequency, the resulting signal has a lower frequency.

Now let’s take a chord. A major chord will be something like this:


In black the tonic and in red the addition of tonic, major third and perfect fifth. The resulting signal has a frequency of 1/4 * 440Hz=110Hz.

Now with A minor:


Now the resulting signal has a frequency of 1/10 * 440Hz.

Let’s summarize the results:

interval/chord frequency note
tonic 440Hz A4
perfect 5th 220Hz A3
major 3rd 110Hz A2
minor 3rd 88Hz ~F2-F#2
minor 2nd 31.42Hz ~B0-C1
A major 110Hz A2
A minor 44Hz ~F1-F#1

Perfect fifth and major 3rd have a curious property: when you add them to the tonic note, you produce a signal whose frequency has the same frequency of the same note in a different octave. However, when you add a minor 3rd or a minor second to the tonic, the resulting signal frequency has no relationship with any note.

I hope you have enjoyed this mathematical experiment and that it will make you think about intervals in a different way. But above all, please enjoy your favorite music, either thinking about intervals or not thinking at all.


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