The Augmented Fourth
I’ve described eleven notes now, and each one has a piano key to go with it, an equal tempered equivalent.
The one remaining black key has a lot of names. It’s the note between the 4 and 5, right in the middle of the octave — the tritone, devil’s interval, flatted fifth, augmented fourth.
In ET there’s only one tritone, and it precisely splits the octave in half. In JI, there are several tritones, with different tunings, that sound and function differently from each other.
One tritone, that nicely fills out the set of 12 notes, is the augmented fourth:
This note is not like the other black keys. It’s completely overtonal, that is, it is generated entirely by multiplication — x3, x3, x5, or 45/1. It does appear in the Chord of Nature, but so far up that it wouldn’t be audible in the harmonics of a vibrating string. I think the fact that we can hear any harmony at all with this note shows that we can hear compounds of simple ratios, even when the numbers are getting pretty big. If pure ratios were all that mattered, 13/1 would be far more harmonious than 45/1 — the numbers are smaller. But 13/1 is almost nonexistent in the musics of the world, and even 11/1 is very rare.
So the harmonic connection with the tonic is tenuous, but it’s there. I hear a different kind of dissonance than the b6 or b2-, more harmonically distant, but without as much of that urgency-to-move that the reciprocal notes have.
It’s natural to resolve it melodically to the 5:
Or once again we can travel through harmonic space to get back home.
Can you hear yourself getting closer to home with each step?
We now have a set of 12 notes, one for each key of the keyboard. Next, the prime number 7, and then some notes between the keys. Oh, the places we’ll go!