### random thoughts on classifying chords

I’m going to attempt to classify chords from first principles, forgetting about the restrictions imposed by existing terminology. Chords are essentially a partial harmonic series. Therefore, they can be indicated as a series of ascending integers indicating ratios of frequencies of elements in the chord, such as 1:2 (octave), 2:3 (fifth), 4:5:6 (major triad), and so on. This is for pure tone combinations. Real instruments contain overtones in each note, so the total effect is more complicated (or collapsed, depending on the view). We will just deal with pure tones for now.

To start with, we classify all the viable consonant dyads. “Viable” means integers that are not too far apart and “consonant” means integers that are not too close together. If the ratio is wider than 1:2, the dyad tends to break into its components. If the ratio is narrower than 10:11, the dyad beats in dissonance. So we are looking for dyads between these ranges. Some of these intervals have common names, so they are given:

1:2 (8)

2:3 (5)

3:4 (4), 3:5 (major 6)

4:5 (major 3), 4:7 (harmonic 7)

5:6 (minor 3), 5:7 (augmented 4), 5:8 (minor 6), 5:9 (minor 7)

6:7 (septimal minor 3), 6:11

7:8 (*augmented 2), 7:9 (*augmented 3), 7:10 (diminished 5), 7:11, 7:12, 7:13

8:9 (major 2), 8:11, 8:13, 8:15

The dyads with stars are tuning dependent, and any dyad with an integer larger than 10 basically do not have names since they don’t naturally occur in low order tunings.

Next, we classify consonant triads, by combining intervals

1:2:3 (P8+P5)

2:3:4 (P5+P4)

3:4:5 (P4+M3, major)

4:5:6 (M3+m3, major), 4:5:7 (M3+A4, partial 7), 4:5:8 (M3+m6), 4:6:7 (P5+sm3, partial 7)

5:6:7 (m3+sm3, diminished), 5:6:8 (m3+P4, partial major), 5:6:9 (m3+P5, partial dim7), 5:7:8 (A4+A2, partial 7), 5:7:9 (A4+A3, partial dim7), 5:8:9 (m6+M2, partial 9)

6:7:8 (sm3+A2, partial m7), 6:7:9 (sm3+A3, harmonic minor), 6:7:10 (sm3+d5, diminished), 6:8:9 (4+2, quartal triad)

7:8:9 (A2+M2, getting weird…), 7:8:10 (A2+M3, partial 7), 7:9:10 (A3+M2, partial m7)

8:9:10 (still stranger)

10:12:15 (m3+M3, “standard” minor)

Though there are countless more of these, we stop here. From the above, we begin to realize that most chords are just partial suggestions of the natural harmonic series. If we only take integers no larger than 10, we will get the harmonic series with the indicated intervals between harmonics:

1:2:3:4:5:6:7:8:9:10

(P8+P5+P4+M3+m3+sm3+A2+M2+M2)

The simplest chords just build from the base up. The more harmonics taken, the higher order the chord would appear to be. In practice, modifications are made by omitting (and implying) the lower harmonics, especially the zeroth, which is almost always omitted.

To generalize even further, omitting harmonics is a special case of applying weighting (or a spectral filter) to all the harmonics. This is perhaps what people aptly call “color” of the sound. Thus all chords and chord colors can actually be identified entirely by the weight vector on the harmonic series. (This may not be as trivial as it sounds, because it implies additional degrees of freedom in musical activity, by varying weights dynamically and continuously, for example, and generalizing discrete progression of chords.)

Depending on the psychoacoustic circuitry, these weight vectors somehow can be partitioned into equivalence classes, so there is more complication here. I haven’t thought this one through, but for instance, not only are 3:4:5 and 6:8:10 equivalent, but 3:4:5 and 4:5:6 are also somewhat equivalent as they are inversions of each other. The latter case is possibly due to psychoacoustic masking of 6 in the implied chord 3:4:5:6 in 4:5:6. Not sure.

In any case, choices not taken from the series below 10 will tend to be dissonant, because the higher harmonics, if they are not clearly resolvable in frequency psychoacoustically, will tend to combine as colored noise in the limit. In particular, the (non-harmonic) minor triad is exceptionally dissonant, but I suppose we have learned to accept it, if only for contrast.

The first paragraph of this old paper is intriguing. Don’t agree with the rest of the conclusion.

[...] Following up on a side note from this post, it is possible to construct continuous changes from one chord to the other by linear interpolation of their frequency envelopes. [...]