2008.6.15 Chord shapes
A question that's been floating around in my mind for years is: what is the entire set of chord shapes in the normal 12-semitone octave? By "chord shape" I mean any combination of pitches, where if a combination can be rotated to make another combination, those combinations are counted as the same chord shape. So, for instance, all major chords are grouped as just a single chord shape — The Major Chord.
Well: I finally enumerated all of them in the chart below. Each successive section of the chart contains all the chord shapes with a given number of notes, from zero all the way to 12. Working clockwise, each step represents a pitch one semitone higher than the previous one. How, you ask, do I know whether I got all the chord shapes and didn't repeat any? Well, I know that this is the correct overall number — and that each section has the correct number of shapes — because I worked those values out with some mathematizing, which is appended at the end. It's possible that there are nonetheless repetitions and omissions in the larger sections, but I've checked fairly thoroughly. If you see a mistake, though, let me know at jbdowse at gmail, it is appreciated. Click the chart to toggle between small and large versions.
Interestingly, out of the 352 chord shapes above, there are only 30 (not counting the blank chord) with no semitone intervals. These 30 encompass most of the chords in common use in music (major, minor, diminished, augmented, sevenths, ninths, etc) though not all (e.g. major seventh, most elevenths). Here they are:
If you're interested, here's the math involved in figuring out the overall number of chord shapes. I won't go into how to find the number in each section at this point.
To start out, we can find the total number of pitch combinations. This counts different rotations of a chord shape as different objects. Each pitch can be either included or excluded (2 choices), and because there are twelve pitches, the total number of combinations is 2 multiplied by itself eleven times, i.e. 212 = 4096.
Note that different pitch combinations contain different degrees of rotational symmetry. Most are rotationally asymmetrical (i.e. have 1-fold symmetry), some have 2-fold symmetry, some 3-fold, and so on for all the divisors of 12 (which are 1, 2, 3, 4, 6, and 12). For each symmetry s, we want to find the number of pitch combinations with s-fold symmetry and no higher degree of symmetry. First, though, let's find the total number of pitch combinations with s-fold symmetry without worrying about removing higher symmetries.
As an example, let's let s = 4, so we're finding the number of pitch combinations with 4-fold symmetry. To do this, we can just divide the pitch circle into s = 4 equal portions and look at a single one of those portions for our calculations — since all the other portions will be identical to it because of the symmetry. So when we divide the pitch circle into 4 portions, each portion contains 12 ÷ 4 = 3 consecutive pitches.
Now we find the number of pitch combinations in this portion. As above, each pitch can be included or excluded (2 choices) and since there are 3 pitches, the number of combinations is 23 = 8. This then is the number of pitch combinations (of the whole circle) with 4-fold symmetry.
To get to the number of chord shapes with 4-fold (and no higher) symmetry from this, we must first remove the pitch combinations with higher symmetry. Since the only divisor of 12 that is a multiple of 4 is 12 itself, we just have to remove the pitch combinations with 12-fold symmetry. There are two of these: the blank combination (all pitches excluded) and the total combination (all pitches included). So the number of pitch combinations with 4-fold symmetry only is 8 − 2 = 6.
Now it's quick to find the number of chord shapes with 4-fold symmetry only. Each 4-fold symmetric pitch combination can be rotated 30° to obtain a different pitch combination and again to obtain a third combination; when rotated a third time, the combination returns to itself because of its symmetry. But the three combinations obtained by rotation are, by definition, all considered to be the same chord shape. So we have to divide the total number of 4-fold symmetric pitch combinations by 3 to get the number of 4-fold symmetric chord shapes: 6 ÷ 3 = 2.
We have to do this for all six symmetries and then sum the results to get the overall number of chord shapes. This is shown below in tables for greater clarity.
All pitch combinations of symmetry s:
|p(s) = 212/s|
|p(12) = 2||p(6) = 4||p(3) = 16|
|p(4) = 8||p(2) = 64||p(1) = 4096|
All pitch combinations of symmetry s and no higher:
|x(s) = p(s) − (combinations of higher symmetry)|
|x(12) = p(12) = 2||x(6) = p(6) − p(12)
= 4 − 2 = 2
|x(3) = p(3) − p(6)
= 16 − 4 = 12
|x(4) = p(4) − p(12)
= 8 − 2 = 6
|x(2) = p(2) − p(4) − [p(6) − p(12)]
= 64 − 8 − (4 − 2) = 54
|x(1) = p(1) − p(2) − [p(3) − p(6)]
= 4096 − 64 − (16 − 4) = 4020
All chord shapes of symmetry s and no higher:
|c(s) = x(s) ÷ 12/s = x(s) ⋅ s ÷ 12|
|c(12) = x(12) ⋅ 12 ÷ 12
= 2 ⋅ 12 ÷ 12 = 2
|c(6) = x(6) ⋅ 6 ÷ 12
= 2 ⋅ 6 ÷ 12 = 1
|c(3) = x(3) ⋅ 3 ÷ 12
= 12 ⋅ 3 ÷ 12 = 3
|c(4) = x(4) ⋅ 4 ÷ 12
= 6 ⋅ 4 ÷ 12 = 2
|c(2) = x(2) ⋅ 2 ÷ 12
= 54 ⋅ 2 ÷ 12 = 9
|c(1) = x(1) ⋅ 1 ÷ 12
= 4020 ⋅ 1 ÷ 12 = 335
∑c(s) = 2 + 1 + 3 + 2 + 9 + 335 = 352 — and there we have it.