Every chord

How many chords are there? The answer depends on what counts as the same chord. This essay starts with the smallest version of the question and adds one complication at a time, ending with every voicing a seven-octave keyboard can play.

All counts use 12-tone equal temperament: 12 keys per octave, each a semitone apart. A chord here is any set of distinct keys played together, whether or not anyone would call it a chord.

Three notes in one octave

Start by ignoring octaves. C is C wherever it sits on the keyboard, so a chord is a choice of pitch names. Choosing 3 of the 12 gives 220 sets.

Most of those sets are the same chord in a different key. C-E-G and D-F#-A are both major triads. If transposing a chord does not make a new one, there are 19 three-note chords.

The arithmetic shows why. Moving a chord through all 12 keys usually gives 12 different sets, so 220 sets would make 18.33 chords. The fraction comes from the augmented triad. C-E-G# moved up 4 semitones is E-G#-C, the same three notes, so it has only 4 distinct transpositions. That gives 18 chords with 12 transpositions each, plus 1 with 4: 18 × 12 + 4 = 220.

Another way to see it: describe each chord by the gaps between its notes, wrapping back to the start, so the gaps always add to 12. C-E-G is 4-3-5. Rotating the gaps (3-5-4, 5-4-3) moves the same chord to start on a different note. There are 55 ways to write 12 as three ordered gaps. Every chord appears 3 times among them except 4-4-4, which appears once: (55 − 1) / 3 + 1 = 19.

The 55 is also the number of three-note sets that contain C. A major triad appears in that list three times, as C major (C-E-G), F major (C-F-A), and A-flat major (A♭-C-E♭), because C can be its root, fifth, or third.

The list of all 19 three-note chords spells each one from C with its gaps and common name.

Every size in one octave

The same counting works for any number of notes:

Notes	Sets of 12 keys	Sets containing C	Chords	Chords, mirrors merged
1	12	1	1	1
2	66	11	6	6
3	220	55	19	12
4	495	165	43	29
5	792	330	66	38
6	924	462	80	50
7	792	462	66	38
8	495	330	43	29
9	220	165	19	12
10	66	55	6	6
11	12	11	1	1
12	1	1	1	1

Summed across every size, there are 351 chords in one octave.

The column is symmetric. A 7-note chord is the 5 keys it leaves out turned inside out, so there are as many 7-note chords as 5-note chords. Six notes, at the middle, has the most.

Each size has its own list: 4 notes, 5 notes, 6 notes, 7 notes, 8 notes, 9 notes, 10 notes, and 11 notes.

The every chord generator lists any size in any of the 12 keys and shows any one chord in every key.

Mirror images

Some chords are upside-down versions of each other. Reading the major triad's gaps backward, 4-3-5 becomes 5-3-4, which is a rotation of the minor triad's 3-4-5. Major and minor are mirror images.

Set theory usually counts a chord and its mirror as one type. These are the Forte set classes, named for Allen Forte's 1973 catalog, and they give the last column of the table: 12 three-note types instead of 19. Chords such as the diminished triad (3-3-6) read the same backward and have no separate mirror.

Merging mirrors is a choice, not a fact about the sound. A major triad and a minor triad do not sound alike, which is why the lists keep them apart.

Adding octaves

Everything so far ignored octaves. On a real keyboard, C-E-G played close together and C-G-E spread over two octaves are different sounds. Counting them separately means counting voicings: sets of specific keys.

The rule for the rest of this essay: a voicing fits within N octaves when its highest key is less than 12N semitones above its lowest, and transposing a voicing still does not make a new one. Fix the lowest key; every other note is a choice among the 12N − 1 keys above it. So the count of n-note voicings is the number of ways to choose n − 1 keys from 12N − 1.

Within one octave this gives the "Sets containing C" column above, which counts C-E-G, E-G-C, and G-C-E as three voicings of one chord. Two octaves already multiply that:

Notes	1 octave	2 octaves
3	55	253
4	165	1,771
5	330	8,855
7	462	100,947
11	11	1,144,066
12	1	1,352,078

The note limit grows with the range

One octave holds 12 keys, so a one-octave chord has at most 12 notes. Two octaves hold 24, and seven hold 84, close to a piano's 88. The largest count no longer sits at 6 notes. It sits at half the range: 12 or 13 notes in two octaves, 42 or 43 in seven.

Counting every size at once has a short answer. Each of the 12N − 1 keys above the lowest one is either in the voicing or not, so there are 2 raised to the 12N − 1 voicings in total.

Octaves	Keys	Most common size	Voicings of that size	All voicings
1	12	6 or 7	462	2,048
2	24	12 or 13	1,352,078	8,388,608
3	36	18 or 19	4,537,567,650	34,359,738,368
4	48	24 or 25	16,123,801,841,550	140,737,488,355,328
5	60	30 or 31	59,132,290,782,430,712	576,460,752,303,423,488
6	72	36 or 37	221,256,270,138,418,389,602	2,361,183,241,434,822,606,848
7	84	42 or 43	839,455,243,105,945,545,123,660	9,671,406,556,917,033,397,649,408

Seven octaves

Across seven octaves, the familiar sizes look like this:

Notes	Voicings within 7 octaves
3	3,403
4	91,881
5	1,837,620
7	377,447,148
11	2,432,332,329,570
12	16,141,841,823,510

Three-note voicings stay countable by hand: 3,403. Forty-two-note voicings number about 8.4 × 10²³, more than Avogadro's number (about 6.0 × 10²³).

What the numbers say

In one octave, harmony is a small, finished catalog: 351 chords, and only 19 with three notes. The explosion comes from register, not from the number of notes. Every octave added multiplies the possibilities by roughly 4,096, because it adds 12 more keys that can each be in or out.

The counts were computed exactly and checked against a brute-force enumeration for one and two octaves.