Chords have these things called inversions. If you play the chord CEG, the frequency ratio of the notes is 4:5:6. But if you play the G an octave lower, you get a ratio of 3:4:5. Somebody once said that lower integer ratios are more consonant, so the second inversion of a major triad is the most consonant inversion. o_O
The minor triad has the ratio 10:12:15. Since 15 is an odd number, inverting the chord doesn't improve its consonance.
If you look at the note that ought to the be the "1" in that ratio, it's not even in the chord at all. This is the "implied fundamental". It and all of its harmonics sound good with the chord. This is why people spam the flat 6th so much in minor.
The minor triad is best thought of as 1/6:1/5:1/3 which has the same amount of consonance as 3:4:5 but its 1 is up above. That's one of two reasons why minor is evocative of downwardness. The other is that in the circle of the fifths it's mostly downwards.
I always felt that those reciprocal harmonics were some kind of drunken math nerd fantasy that was totally made up. Real physical vibrating objects (and periodic functions in general) only have overtones.
Chords with one low note and a bunch of clustered-together high notes can sound good (sometimes cool and jazzy). Chords with one high note and a bunch of clustered-together low notes always sound like mud. It's not symmetrical at all. Other asymmetries include the fact that very fast melodies played on low notes have super weak pitch clarity (thanks Heisenberg).
The harmonics still line up, they're just spread out among multiple values instead of all being clustered around a single pitch. In the 1/6:1/5:1/4 example the pairwise ratios are 6/5, 5/4, and 6/4, exactly the same as they are for 4:5:6. It's the pairwise ratios which matter, not the thing as a whole, but them being spread out does make minor feel more ambiguous and less tight. In general anything major can be made minor and vice versa by flipping it upside down in frequency space, but the symmetry is broken because the human brain wants melodies to initially go up from the tonic.
My understanding (mostly from reading the psychoacoustics chapters from Donald Hall's book) was that the thing as a whole *does* matter, and your brain is doing a GCD-like calculation to determine the implied fundamental whenever it hears a chord, because it's trying to disambiguate between [one sound with multiple harmonics] vs [multiple sounds], and chords are the thing that screw with your ability to do so precisely because they are the latter masquerading as the former.
The reason the GCD calculation works in the first place is that things in real life vibrating at frequency X will cause other things to resonantly vibrate at frequency 2X or 3X. But they will never cause sympathetic resonances at X/2 (that would be like trying to get a swing to go high by pushing it forwards when it is moving backwards).
No it's based off dissonance curves which don't even have to have harmonics line up. Usually local minima of dissonance happen on precise harmonic alignments but that isn't always the case.
I guess you're technically right for percussion instruments where the frequencies don't follow a harmonic series because of the multiple vibrational modes. But I think human psychoacoustics have been more-or-less evolutionarily overtrained on human voice which has a pretty linear series of overtones.
Nice post Bram.
Things nobody will tell you in music school...
Chords have these things called inversions. If you play the chord CEG, the frequency ratio of the notes is 4:5:6. But if you play the G an octave lower, you get a ratio of 3:4:5. Somebody once said that lower integer ratios are more consonant, so the second inversion of a major triad is the most consonant inversion. o_O
The minor triad has the ratio 10:12:15. Since 15 is an odd number, inverting the chord doesn't improve its consonance.
If you look at the note that ought to the be the "1" in that ratio, it's not even in the chord at all. This is the "implied fundamental". It and all of its harmonics sound good with the chord. This is why people spam the flat 6th so much in minor.
The minor triad is best thought of as 1/6:1/5:1/3 which has the same amount of consonance as 3:4:5 but its 1 is up above. That's one of two reasons why minor is evocative of downwardness. The other is that in the circle of the fifths it's mostly downwards.
I always felt that those reciprocal harmonics were some kind of drunken math nerd fantasy that was totally made up. Real physical vibrating objects (and periodic functions in general) only have overtones.
Chords with one low note and a bunch of clustered-together high notes can sound good (sometimes cool and jazzy). Chords with one high note and a bunch of clustered-together low notes always sound like mud. It's not symmetrical at all. Other asymmetries include the fact that very fast melodies played on low notes have super weak pitch clarity (thanks Heisenberg).
The harmonics still line up, they're just spread out among multiple values instead of all being clustered around a single pitch. In the 1/6:1/5:1/4 example the pairwise ratios are 6/5, 5/4, and 6/4, exactly the same as they are for 4:5:6. It's the pairwise ratios which matter, not the thing as a whole, but them being spread out does make minor feel more ambiguous and less tight. In general anything major can be made minor and vice versa by flipping it upside down in frequency space, but the symmetry is broken because the human brain wants melodies to initially go up from the tonic.
My understanding (mostly from reading the psychoacoustics chapters from Donald Hall's book) was that the thing as a whole *does* matter, and your brain is doing a GCD-like calculation to determine the implied fundamental whenever it hears a chord, because it's trying to disambiguate between [one sound with multiple harmonics] vs [multiple sounds], and chords are the thing that screw with your ability to do so precisely because they are the latter masquerading as the former.
The reason the GCD calculation works in the first place is that things in real life vibrating at frequency X will cause other things to resonantly vibrate at frequency 2X or 3X. But they will never cause sympathetic resonances at X/2 (that would be like trying to get a swing to go high by pushing it forwards when it is moving backwards).
No it's based off dissonance curves which don't even have to have harmonics line up. Usually local minima of dissonance happen on precise harmonic alignments but that isn't always the case.
I guess you're technically right for percussion instruments where the frequencies don't follow a harmonic series because of the multiple vibrational modes. But I think human psychoacoustics have been more-or-less evolutionarily overtrained on human voice which has a pretty linear series of overtones.
Thanks fpr your detailed explanation, it save me a lot of headaches.