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Subharmonic or Undertone Series

The harmonic series, or the overtone series is pretty simple. You have a fundamental frequency. The harmonics go upward in whole number multiples of that frequency. All of the overtones fit neatly "inside" the fundamental.

If you think of it as a rhythm, the fundamental is a beat, and each overtone is 2 per beat, 3 per beat, 4 per beat going up and up. As you continue to add overtones eventually you end up with a sawtooth wave.

The subharmonic series, or the undertone series is way more problematic. You have a fundamental frequency. The subharmonics go down from there, first is 1/2 the fundamental, then 1/3, 1/4, 1/5 and so on.

If you think if this as a rhythm, the fundamental is a beat. Then you have 2 beats (1/2 rate) then 3 beats (1/3 rate) then 4 beats (1/4 rate), etc. As you continue to add undertones, you don't end up with a pure wave at all. You end up with a strong attack, where all the waves are initially aligned (starting together), but they they spread out and disperse, never looping, ever.

I built a 48 wave subharmonic tone, and did some experiments with it. At a high frequency, it sounds like an 808 cymbal.

I really love the sound of subharmonic tones. I was messing with Reaktor's Subharmonic. The dumb thing is that it only gives you 4 sines. Would love to see an additive synth with a subharmonic mode. I'll add it to the list.
Just had an interesting thought to take the undertone series, and "invert" it. In essence, this would involve taking each subharmonic and moving it up in octaves until it is roughly the same distance above the fundamental as it was below.

Did a little test. Sounds cool, and actually does sound pretty similar-- especially in that it has a kind of minor-triad kind of sound.
If any of you have seen Jacob Collier's fascinating little mini theory "lectures"-- you will have seen an interesting explanation of how following the cycle of 4ths is "darkening" and following the cycle of 5ths is "brightening". That is, going down 5ths (same as up 4ths), is darker, and going up 5ths brighter. This is perhaps similar to saying that major chords are "happy" and minor chords are "sad".

In a similar way, the harmonic/overtone series produces a major triad-- (and extends to form a dominant 7th chord), and it sounds bright.

The subharmonic/undertone series SORT-OF produces a minor triad but the fundamental is not the root. For example, if your fundamental frequency/note is C, you get a Fm/C (Fm with a C bass). Then the next low note in the series is a D, so you also get a half-diminished chord, built in the 2 of the scale-- which is bizarre-- and yes, darker than the Dominant 7th.

This brings me to another observation: If the undertone series is in some ways arbitrary-- yes, it sounds like a synthesized cymbal at high notes, but it's not necessarily an "a-ha!!" kind of tone-- then perhaps some additive tones can be built from scratch, and derived with more of a "circuit-bending" aesthetic. Of course, it should include lots of octaves and simple intervals, but perhaps even the tones could be "bent" to fit equal temperament more, and to create some new chord sounds that sound new and thick but more "in tune".
Now that I've played around with the subharmonic series, I'm getting a sense that there is more room to play than the natural physics would lead us to believe.

I'd like to play with tempered over and/or under tones. The idea is that with (sub)harmonics being tempered, that some chords will sound nicer than their pure harmonic versions. This could be applied to other divisions of an octave (non-12 equal temperaments), as well.

It might also be cool to mess with sounds where the overtones morph more dramatically through their envelope.
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