Most music students, though practicality, are taught that the seventh in a dominant chord corresponds to the seventh partial of the harmonic series, and extensions also correspond to the same-named partial of the harmonic series. This is wrong. First, for completeness, a bit of background:
Cents refer to percent differences of a semitone away from 12-note equal temperament. 12ET is the model for how modern pianos are tuned (for the most part; inharmonicity is beyond the scope of this essay) such that the octave is divided into exactly 12 equally spaced intervals.
The harmonic series represents the relationships between all of the different frequencies that will happen within an actively vibrated string – as on a bowed violin. It is in theory an infinite series, but fairly simple: the frequencies generated will depend on the full length of the string, half of the length, one-third of the length, one-quarter, and so on. When I refer to partials, each is named for how many times it divides the string: the first partial divides it by one, the second by two, et cetera.
Now, consider this C13(#11) chord and the tuning tendencies of these harmonic series partials:
C : fundamental, +0 cents
E : 5th partial, -14 cents
G : 3rd partial, +2 cents
Bb : 7th partial, -31 cents
D : 9th partial, +4 cents
F# : 11th partial, -49 cents
A : 13th partial, -59 cents
Things do not line up that well. For starters, we see that the pattern is probably accidental, because the primary triad, C-E-G, does not represent sequential notes on the harmonic series, but rearranged. It does so happen that a second-inversion triad does use sequential harmonic series-related notes; we can look at G-C-E as 3rd partial-4th partial-5th partial. But this is not how we name them when we spell out chords in the usual way. And look at the 7th partial. 31 cents away from equal temperament is not really that close. It’s certainly far enough to make an auditory difference, and that’s not the worst offender. Therefore, I submit that this method of correlating the common scalar names for notes do not, in fact, correspond with the harmonic series notes of the same count.
Now, for those notes which have some significant amount of variance from equal temperament, look at the next-simplest harmonic series note that is closer than those I listed above:
E : 81st partial, +8 cents
Bb: 57th partial, -1 cent
F#: 45th partial, -10 cents
A : 27th partial, +6 cents
Two of these are, in fact, Pythagorean tunings: the E and the A. But, this also is the point where I should bring up Harry Partch’s idea that simpler intervals sound more consonant. This is particularly clear on the E: a 5th partial tuning of E is much more consonant than the 81st partial tuning, and 8 cents sharp is really not that much closer to 12ET than 14 cents flat, so I think it’s a safe assumption that 12ET approximates the 14-cent flat major third. But, it may be that 12ET implies the other, closer approximations of the Bb, F#, and A than those in the first list. It may also be that 12ET implies other consonant relationships that are not directly part of this harmonic series. For example, the A might also approximate the major third of the related IV chord, which, when we add up all the relationships, works out to be 16 cents flat, and in just-intonation-ratio speak, 5/3.
The point is that the convention of naming 9ths, 11ths and 13ths makes sense only when counting thirds on a piano keyboard, and does not make sense when we extend those meanings beyond that.
And now, the proof, which is in the hearing. Below are the three differently tuned versions of this C13(#11) chord as represented by a clarinet sample thanks to Sibelius.
57th, 45th, and 27th Overtones
The first and third definitely sound similar. The second, which actually uses those wonderful 7th, 11th, and 13th overtones, is a very strong, colorful chord – but that’s definitely not even close to what performers play.