How many possible piano chords are there?

Randomly hit notes on a keyboard and see if that combination of notes sounds like a chord you might like to use. Remember there are 4017 possible chords before we ever even get into voicing!

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How many possible chords are there in music? This is a question I remember thinking about when I started learning music theory but have never answered. Until now! Warning: This is some pseudo-intellectual shizz! But still, it might be fun to do some math and discuss the chords of music.

This article will run through the calculations to answer our question: How many possible chords are there in music?

First, Let's Define A Chord

A chord is defined by three or more unique notes sounded together. Unique in the sense that they are not the same pitch or an octave of a pitch already present in the chord (Three C notes spread across three octaves do not create a chord). Usually, these 3-note chords are built by stacking thirds and are called triads. However, any three unique notes will build a three-note chord! There are 12 notes in the chromatic scale, and a chord is made of 3 or more notes. Therefore, we have 3-note chords, 4-note chords, 5,6,7… all the way up to 12-note chord(s).

So How Many Possible Chords Do We Have?

If we're not at all concerned with voicing (or only concerned with notes within a single octave), we can use the following combination formula to find how many chords are possible:

n = the set//notes to choose from (12)

k = the subset (number of notes in a chord)

The ! in the formula is called a factorial. And it means to multiply an integer by each whole number that is less than the integer (that's a strange way of writing it out, so let's look at an example). If we have n! and n=12, then: n! = 12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x = 479001600 n will always be 12 in the case of musical notes because we'll always be drawing from the chromatic scale. If we were looking for the possible combinations of chords in a heptatonic scale, n would be 7. If it was a pentatonic scale, n would be 5. And so on. k is the subset (or the number of notes in the chord). Since we've defined a chord as being 3 or more notes played together, we will run with the above equation from k=3 all the way to k=12. Note that in the above equation, the order of the notes does not matter. Therefore, all the different voicings of a set of notes are considered to be one chord. It's not perfect, but it's a start. Another note for the above equation is that the subset will not account for duplicate notes. Which is what we want! Since, for example, 7 D notes do not make up a chord.

So now that we've defined the equation, let's crunch some numbers!

The number of 3-note chords:

The number of 4-note chords:

The number of 5-note chords:

The number of 6-note chords:

The number of 7-note chords:

The number of 8-note chords:

The number of 9-note chords:

The number of 10-note chords:

The number of 11-note chords:

The number of 12-note chords:

Math note: 0! = 1

And if you're one of those people who think that two notes can make a chord, then we add

The number of 2-note chords:

to our number and have 4083 unique chords.

So we have 4017 possible chords.

That's if we disregard voicing. Or if we look at building chords only within one octave. So sure, we have a number. But that's not really a satisfactory answer since voicing is very important in music and music has way more than one octave.

Let's Talk About Voicing

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We'll take a C major triad to help explain this.

C major in root position is made of C E and G all within the same octave. The C is the lowest note and the G is the highest note. What if we made E the lowest note with G above it and C the highest note within one octave? We'd have a C major triad in first inversion. And if G was the lowest note with C above it and E at the top within one octave? That would be C major in second inversion. But what if we have an E in the bass, a C two octaves above it, and a G an octave above the C? It is still the same three notes as the C major triad, so we can argue that it is still the same chord, but it's certainly different. We could call this a C major triad. When I play this in Logic Pro, it tells me the chord is an “E no3♭3/♯5” And this is only a simple triad. Chords get much more complicated than this!

Let's Look At A More Complex Example

To get further into my point, we'll take a C major 9 chord instead of a C major triad. In root position, the C major 9 chord is built with C E G B (in the first octave) and a D (in the second octave).

What if we took that C and placed it two octaves above its original position?

We'd have E G B D (in the first octave starting on E) and a C in the second octave, 20 semitones from the root. If you were to ask me what this chord is called, I'd tell you something like E minor 7 flat 13. Even though it has the same exact notes as C major 9. Are E minor 7 flat 13 and C major 9 the same chord? I wouldn't say so. Voicing does matter, making “how many possible chords are there?” even more difficult to answer.

Let's Look At The 12-note Chords

It seems at first glance that there's only one way to fit all 12 notes into a 12-note chord, and that's true within one octave. But music is made with multiple octaves and therefore we have plenty of voicings for this 12-note chord. Take for example:

The Pyramid Chord

The Mother Chord

The Grandmother Chord

Each one of the above chords contains all 12 notes of the chromatic scale and is thereby defined by its voicing rather than by which notes it contains.

And How About Symmetrical Chords?

The augmented triad is built by stacking 2 major third intervals. Stacking a 3rd major third interval would give us the octave. The full diminished seventh is built by stacking 3 minor third intervals. Stacking a 4th minor third interval would give us an octave. That's why there are only truly 4 augmented triads and 3 full diminished seventh chords. Take a look at which augmented triads contain exactly the same notes as other augmented triads:

C aug = E aug = G♯/A♭aug

C♯/D♭aug = F aug = A aug

D aug = F♯/G♭aug = A♯/B♭aug

D♯/E♭aug = G aug = B aug

And check out which full diminished seventh chords contain exactly the same notes as other full diminished seventh chords:

C♯/D♭dim7 = E dim7 = G dim7 = A♯/B♭dim7

D dim7 = F dim7 = G♯/A♭dim7 = B dim7

These are some things to think about when considering all the different chords in music. The above examples would not be considered as different chords with our initial combination calculations.

So How Many Different Voicings Are Possible?

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This is beyond me. Are there any mathematicians who'd like to solve this?

First, we'd have to figure out how many total notes there are in music. The Organ has the largest range of any instrument, but electronic synthesizers can be programmed to go beyond those ranges.

Are high-end frequencies still considered notes or are they simply “sibilance” and “air” like we're taught in EQ class?

And can we build chords in the sub-bass frequencies and still hear and distinguish them as chords rather than low-end rumble?

What if we get really ridiculous and build an E minor triad out of sine waves (only one harmonic with no overtones) with the following pitches:

E = 20.60 Hz

G = 392 Hz

B = 15804 Hz

Here, E is in the sub-bass range, the very extreme lower limit of possible human hearing. We feel this frequency way more than we hear it. But at great enough sound pressure level (dB SPL), we theoretically should be able to hear it.

G is in the mid-range. No issues there!

B is so high up that it's hard to hear it as a note. It's still in the human range of hearing, but rather than hearing a note, we hear it as an annoying ringing in our ears.

Is this still a triad? Here's where the conversation turns ambiguous…

A Solution?

Rather than thinking about how many possible chords there are in music, I like to think of voicings and how they can be used effectively. This post is called “How Many Possible Chords Are There In Music?” but perhaps it should be called something like “A Brief Look Into Chord Voicings Without Getting Too Into The Details Of Voicing Chords.” That doesn't have a very good ring to it haha

So here are some thoughts if I may provoke them:

Perhaps the next time you're composing with a C major 9, substitute it for an E minor 7 flat 13. Alternate through all the full diminished chord voicings on each beat of a bar where, originally, there was only one full diminished chord. Try inverting your chords in ways you never have before. Spread voices out in your instrument, or across multiple instruments in your compositions. Randomly hit notes on a keyboard and see if that combination of notes sounds like a chord you might like to use. Remember there are 4017 possible chords before we ever even get into voicing! There's so much variety that sometimes it's good to forget about theory for a second and just experiment.

Conclusion

We can count the combinations of possible chords from subsets of notes from the chromatic scale, but to name all the possible chords would take ages. Even if two chords contain the same exact notes, they can sound totally different and have wildly different functions in music. These different functions, voicings, and names for chords are part of what makes music and music theory so beautiful. I hope that this article has taught you something new, or has at least shown you something you already knew in a different light. I urge you to experiment with different chords and voicings. Don't be afraid to ditch theory for a moment to build a chord, and then reapply theory to understand it! As always, I'd like to thank you for reading and for your support.

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