Max Size of a Chord

Question

I was recently looking at the the definition of a chord and got me thinking about the max size of a chord.

A chord, in music, is any harmonic set of three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may, for many practical and theoretical purposes, constitute chords.

Typically, chords are built in 3rds and the biggest chord I know is a 13th chord which can have seven notes in it, but usually have some note omitted because of it's size (the 9th and the 11th are optional). The 13th chord uses every scale degree in its construction, but there are 12 named notes and there are also quarter tones not to mention all the frequencies in between each note.

Because of this, I have two questions:

• What is the theoretical max chord size?
• What is the practical max chord size? (A chord that a musician or a group of musicians could play)

Answer 1

I suppose you're asking about what the largest named chord is. You can have a chord with as many pitches as you like, across the whole range of the instruments playing that chord, and of course even more if you are using quarter-tones and micro-tones (as you say in your question). However, as you also say, the largest number of different pitches you could have with the "normal" chromatic system is 12.

You are right that chords being described with common naming conventions (built in thirds for instance) usually have no more than 7 distinct pitches, and so the 13th chord is the limit. Interestingly, one feature of this way of describing chords is that these chords usually have no semitone clusters (i.e. three adjacent semitones).

Beyond this, and for chords with semitone clusters, I would use PC Set analysis to describe sets of pitches.

There is one exception. Chords based upon a full Octatonic (diminished) scale, have no semitone clusters, but do have eight distinct pitches. These can be described using conventional naming, too, and are used in, for instance, Jazz.

To give an example, two Octatonic scales (and so also Octatonic chords) have roots on C: one with a semitone as the lowest interval; one with a whole-tone as the lowest interval:

• C Db D# E F# G A Bb can be described as C13#9b9 - yes it's a bit of a mouthful, but it is completely legitimate, and certainly used. (You always have to decide which letter name is used twice in an octatonic chord; here it seems sensible to use D twice, as we describe it with #9 and b9.)
• C D Eb F Gb Ab A B I'm not sure how you'd describe this… (without forcing it to fit a description; any ideas anyone…?)

So, the first of these two Octatonic chords would be my candidate for largest chord, using the conventional system of building and describing chords.

Answer 2

Falling back on the limiting definition of a "note" as any discrete frequency that can be detected, then the largest theoretical chord is a computer synthesizing noise. (The phase relationships between frequencies must change with time for as long as the "chord" is sounding; fixed phase relations create wave interference, reducing the volume of some specific frequencies below the detection threshold.)

The computer with the fastest processor (capable of synthesizing the finest grid of discrete frequencies) will win, in that it will generate the "chord of the maximum size".

(Between two very fast computers, it may be that the one with the better random-number generator may sometimes win even if it has a slightly slower processor, as it is capable of inserting the highest degree of randomness into phase relations.)

The largest practical chord is the result of the same computer, but where we turn up the volume in order to exploit the nonlinear elasticity of the playback mechanism (e.g. the speaker cone), the medium (e.g. air), or the detector. In this way, nearby frequencies will become mixed and the spectrum will literally become continuous, creating a "chord with infinite size".