Combination tones. Hindemith wrote about them in "Craft of Musical Composition."
If I sing frequency A while you sing lower frequency B, then the first-order combination tone produced is frequency (A-B).
Perfect 5th reinforces the bottom note, P4th the top (but octaves down).
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I can make them on my cello, if I play the right two notes at once.
The new pitch created by the combination tone is physically produced by the interaction of two notes, not by the overtones within one note.
(But the overtones of two notes can also interact.)
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! I think I see… with P5th 200Hz + 300Hz, an implied 100Hz tone (i.e. 8vb the root) would produce both as overtones; with P4th 300Hz + 400Hz, an implied 100Hz (i.e. 15vb the top) would produce both.
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Exactly! That's why fifths sound "rightside up" and fourths sound "upside down."
The combination tone sets the frame.
Fun, fun stuff.😀
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It's also why m6 sounds like a first inversion, but M6 sounds like a second inversion.
First and second order combination tones for m6 give root and 5th, but give octave roots for M6.
The better the timbral blend, the more pronounced the effect.
(Sorry about the jargon, heh.)
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Realized I'm back to being confused. With P4th 300Hz + 400Hz, an implied 100Hz has 300Hz as its second harmonic and 400Hz as its third. They're both included—so why does the combination tone reinforce the top over the bottom in this case?
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First-order combination tone is 100Hz, or two octaves below the top.
Second order is the interaction of the first-order with EACH of the original pitches, giving 200Hz and 300HZ respectively.
The double reinforcement of that upper pitch with low octaves anchors it as the root.
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The overtones themselves are something of a red herring. They affect perceived timbre, mostly.
The combination tones are what set the perceived inversions.
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Interesting. I notice that singing with a bright, forward tone is more likely to create a "ringing" effect when in just harmony. That difference is about the overtones I'm producing, right? Or do you think the ringing is about combination tones *produced from* the overtones?
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Barbershop quartet singing has the idea of the "fifth voice."
I suspect it is combination tones produced from the fundamentals, rather than overtones. Probably.
Page 16 of this essay Google found for me has a useful description of combination tones:
uh-ir.tdl.org/bitstream/hand.
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Thanks for the link! I worry that this theory doesn't explain why the effect is much stronger with a bright vocal timbre. Maybe that's illusory, though, or it's doing something else orthogonal to the fifth voice?
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The brightness given by the high overtones may simply make it easier to "lock" the tuning of the chords. But that's strictly a guess.🤔
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I find that fairly persuasive! I do have an easier time tuning to people singing with bright tone.
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