Pitch Space
Let
The universal lifting property for coverings states that, given a covering, every path in the base space lifts to a unique path in the covering space.
Octave Equivalent Mappings
We call a mapping out of the universal covering space of
Note that not all mappings are octave equivalent. The purpose of this writing is to suggest that there are musically relevant mappings which fail to respect octave equivalency.
Intermodulation Distortion
Let
But what about signals more complicated then a single pure tone? Let
So we can see that pushing two pure tones through this nonlinear system produces: the original tones, tones at double the original frequencies, and tones at the sum and difference of the two input frequencies. In general, upto phase differences, the result of running a collection of tones through a nonlinear system is a collection of harmonics (integer multiples of the input frequencies), and sums and differences of the input tones and their harmonics. This is called intermodulation distortion.
For a component
Chords
Given an
Second order intermodulation products can be computed pairwise. Higher order components can be computed by recursing on the union of the lower order components.
I don’t have a good unifying scheme for analyzing these products from a musical perspective, so for the moment I present here a table of illustrative computations.
Each entry in the table lists the second order intermodulation products which result from combining the row note with the next note up of the column pitchclass. These distortion products are listed as a pitchclass name along with octave number, and a signed error measured in halfsteps; e.g. A4 intermodulated with A5 (the next A ‘up’) results in a somewhat sharp E6 (sharp by ~20 cents), and an exact A4.
A | Bb | B | C | Db | D | Eb | E | F | Gb | G | Ab | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
A | E6 1.955e-2; A4 0.000e0 | Bb5 -4.928e-1; Ab0 1.377e-1 | Bb5 2.887e-2; A1 -3.551e-1 | B5 -4.351e-1; E2 1.765e-1 | B5 1.153e-1; Bb2 -3.263e-1 | C5 -3.201e-1; D3 5.852e-2 | C6 2.586e-1; Gb3 -2.586e-1 | Db6 -1.486e-1; A3 -5.872e-2 | Db6 4.580e-1; C3 -2.110e-1 | D6 7.837e-2; D4 3.689e-1 | Eb6 -2.878e-1; F4 -2.616e-1 | Eb6 3.593e-1; G4 -6.132e-2 |
Bb | E6 3.593e-1; Ab4 -6.132e-2 | F6 1.955e-2; Bb4 0.000e0 | B5 -4.928e-1; A0 1.377e-1 | B5 2.887e-2; Bb1 -3.551e-1 | C5 -4.351e-1; F2 1.765e-1 | C6 1.153e-1; B2 -3.263e-1 | Db6 -3.201e-1; Eb3 5.852e-2 | Db6 2.586e-1; G3 -2.586e-1 | D6 -1.486e-1; Bb3 -5.872e-2 | D6 4.580e-1; Db4 -2.110e-1 | Eb6 7.837e-2; Eb4 3.689e-1 | E6 -2.878e-1; Gb4 -2.616e-1 |
B | F6 -2.878e-1; G4 -2.616e-1 | F6 3.593e-1; A4 -6.132e-2 | Gb6 1.955e-2; B4 -8.882e-16 | C5 -4.928e-1; Bb0 1.377e-1 | C6 2.887e-2; B1 -3.551e-1 | Db6 -4.351e-1; Gb2 1.765e-1 | Db6 1.153e-1; C2 -3.263e-1 | D6 -3.201e-1; E3 5.852e-2 | D6 2.586e-1; Ab3 -2.586e-1 | Eb6 -1.486e-1; B3 -5.872e-2 | Eb6 4.580e-1; D4 -2.110e-1 | E6 7.837e-2; E4 3.689e-1 |
C | F6 7.837e-2; F4 3.689e-1 | Gb6 -2.878e-1; Ab4 -2.616e-1 | Gb6 3.593e-1; Bb4 -6.132e-2 | G6 1.955e-2; C5 4.441e-16 | Db6 -4.928e-1; B0 1.377e-1 | Db6 2.887e-2; C1 -3.551e-1 | D6 -4.351e-1; G2 1.765e-1 | D6 1.153e-1; Db3 -3.263e-1 | Eb6 -3.201e-1; F3 5.852e-2 | Eb6 2.586e-1; A3 -2.586e-1 | E6 -1.486e-1; C3 -5.872e-2 | E6 4.580e-1; Eb4 -2.110e-1 |
Db | F6 4.580e-1; E4 -2.110e-1 | Gb6 7.837e-2; Gb4 3.689e-1 | G6 -2.878e-1; A4 -2.616e-1 | G6 3.593e-1; B4 -6.132e-2 | Ab6 1.955e-2; Db5 8.882e-16 | D6 -4.928e-1; C1 1.377e-1 | D6 2.887e-2; Db2 -3.551e-1 | Eb6 -4.351e-1; Ab2 1.765e-1 | Eb6 1.153e-1; D3 -3.263e-1 | E6 -3.201e-1; Gb3 5.852e-2 | E6 2.586e-1; Bb3 -2.586e-1 | F6 -1.486e-1; Db4 -5.872e-2 |
D | Gb6 -1.486e-1; D4 -5.872e-2 | Gb6 4.580e-1; F4 -2.110e-1 | G6 7.837e-2; G4 3.689e-1 | Ab6 -2.878e-1; Bb4 -2.616e-1 | Ab6 3.593e-1; C4 -6.132e-2 | A6 1.955e-2; D5 -8.882e-16 | Eb6 -4.928e-1; Db1 1.377e-1 | Eb6 2.887e-2; D2 -3.551e-1 | E6 -4.351e-1; A2 1.765e-1 | E6 1.153e-1; Eb3 -3.263e-1 | F6 -3.201e-1; G3 5.852e-2 | F6 2.586e-1; B3 -2.586e-1 |
Eb | Gb6 2.586e-1; C3 -2.586e-1 | G6 -1.486e-1; Eb4 -5.872e-2 | G6 4.580e-1; Gb4 -2.110e-1 | Ab6 7.837e-2; Ab4 3.689e-1 | A6 -2.878e-1; B4 -2.616e-1 | A6 3.593e-1; Db5 -6.132e-2 | Bb6 1.955e-2; Eb5 -8.882e-16 | E6 -4.928e-1; D1 1.377e-1 | E6 2.887e-2; Eb2 -3.551e-1 | F6 -4.351e-1; Bb2 1.765e-1 | F6 1.153e-1; E3 -3.263e-1 | Gb6 -3.201e-1; Ab3 5.852e-2 |
E | G6 -3.201e-1; A3 5.852e-2 | G6 2.586e-1; Db4 -2.586e-1 | Ab6 -1.486e-1; E4 -5.872e-2 | Ab6 4.580e-1; G4 -2.110e-1 | A6 7.837e-2; A4 3.689e-1 | Bb6 -2.878e-1; C4 -2.616e-1 | Bb6 3.593e-1; D5 -6.132e-2 | B6 1.955e-2; E5 0.000e0 | F6 -4.928e-1; Eb1 1.377e-1 | F6 2.887e-2; E2 -3.551e-1 | Gb6 -4.351e-1; B2 1.765e-1 | Gb6 1.153e-1; F3 -3.263e-1 |
F | G6 1.153e-1; Gb3 -3.263e-1 | Ab6 -3.201e-1; Bb3 5.852e-2 | Ab6 2.586e-1; D4 -2.586e-1 | A6 -1.486e-1; F4 -5.872e-2 | A6 4.580e-1; Ab4 -2.110e-1 | Bb6 7.837e-2; Bb4 3.689e-1 | B6 -2.878e-1; Db5 -2.616e-1 | B6 3.593e-1; Eb5 -6.132e-2 | C7 1.955e-2; F5 0.000e0 | Gb6 -4.928e-1; E1 1.377e-1 | Gb6 2.887e-2; F2 -3.551e-1 | G6 -4.351e-1; C3 1.765e-1 |
Gb | Ab6 -4.351e-1; Db3 1.765e-1 | Ab6 1.153e-1; G3 -3.263e-1 | A6 -3.201e-1; B3 5.852e-2 | A6 2.586e-1; Eb4 -2.586e-1 | Bb6 -1.486e-1; Gb4 -5.872e-2 | Bb6 4.580e-1; A4 -2.110e-1 | B6 7.837e-2; B4 3.689e-1 | C6 -2.878e-1; D5 -2.616e-1 | C7 3.593e-1; E5 -6.132e-2 | Db7 1.955e-2; Gb5 0.000e0 | G6 -4.928e-1; F1 1.377e-1 | G6 2.887e-2; Gb2 -3.551e-1 |
G | Ab6 2.887e-2; G2 -3.551e-1 | A6 -4.351e-1; D3 1.765e-1 | A6 1.153e-1; Ab3 -3.263e-1 | Bb6 -3.201e-1; C4 5.852e-2 | Bb6 2.586e-1; E4 -2.586e-1 | B6 -1.486e-1; G4 -5.872e-2 | B6 4.580e-1; Bb4 -2.110e-1 | C7 7.837e-2; C5 3.689e-1 | Db7 -2.878e-1; Eb5 -2.616e-1 | Db7 3.593e-1; F5 -6.132e-2 | D7 1.955e-2; G5 0.000e0 | Ab6 -4.928e-1; Gb1 1.377e-1 |
Ab | A6 -4.928e-1; G1 1.377e-1 | A6 2.887e-2; Ab2 -3.551e-1 | Bb6 -4.351e-1; Eb3 1.765e-1 | Bb6 1.153e-1; A3 -3.263e-1 | B6 -3.201e-1; Db4 5.852e-2 | B6 2.586e-1; F4 -2.586e-1 | C6 -1.486e-1; Ab4 -5.872e-2 | C7 4.580e-1; B4 -2.110e-1 | Db7 7.837e-2; Db5 3.689e-1 | D7 -2.878e-1; E5 -2.616e-1 | D7 3.593e-1; Gb5 -6.132e-2 | Eb7 1.955e-2; Ab5 0.000e0 |
Chord Voicings
Consider a CMaj triad in close voicing,
If we neglect tuning/temperment issues, we can see that the close voiced major triad produces intermodulation products corresponding to the ninth, the third an octave up, the flat five (flatter than normal) an octave up, the minor second, the root two octaves down, and the major seventh three octaves down.
The wide voiced major traid produces intermodulation products corresponding to the minor seventh, the root two octaves up, the major third one octave up, the fifth, the root, and the root two octaves down.
On this account then, it is no surprise that wide voiced triads are less dissonant than their close voiced cousins. Wide voiced major triads hint at the presence of the major seventh, but are otherwise consonant with themselves. Close voiced major triads on the other hand produce a minor second, flat five, and a major ninth!
Consider now Cmin triads voiced similarly.
A close voiced Cmin triad produces the following intermodulation products
A close voiced Cmin triad produces the following intermodulation products