The blues and the harmonic series – a visual guide

Does the harmony of the blues come from the natural overtone series? Is it a just intonation system that later got shoehorned into Western twelve-tone equal temperament?

Whether the blues comes from just intonation, or just intonation happens to sound like the blues, this is a rich and promising avenue of inquiry, both for understanding the blues and for creating new music inspired by it. In this post, I use MTS-ESP, Oddsound’s amazing microtonal tuning plugin, to visualize the possible harmonic basis of the blues.

The circle below represents one octave, which both starts and ends at the 12 o’clock position. Let’s call our starting note middle C, and say that it has a frequency of 1 Hz. (Middle C’s actual frequency is 261.626 Hz.) If you double the frequency of middle C, you get another C an octave higher, and that is also at 12 o’clock on the circle.

We’ll start building our blues scale using the natural harmonics of middle C. These are integer multiples of C’s frequency, which we can octave normalize (divide by two enough times to bring them within the same octave). The image below shows the first thirteen harmonics of C.

Note that MTS-ESP only shows prime-numbered harmonics, because all the other intervals can be derived from them. Also, it doesn’t show any even-numbered harmonics, because those are higher-octave copies of the odd-numbered ones.

The first harmonic of middle C is middle C itself. The third harmonic of C is the just intonation perfect fifth (G) at 3/2 Hz. The fifth harmonic of C is the just intonation major third (E) at 5/4 Hz. The seventh harmonic of C is the harmonic seventh (a very flat B-flat) at 7/4 Hz. Finally, the eleventh harmonic of C is the harmonic eleventh (about halfway between F and F-sharp) at 11/8 Hz. (Is that last one really a blue note? I’m not sure. It’s audible if you twang your string hard enough.)

Next, we’ll find more bluesy-sounding intervals by examining the subharmonics of C. These are notes whose own harmonic series include C. They are shown below in red:

The third subharmonic of C is the just intonation perfect fourth (F) at 4/3 Hz. (This is another way of saying that the third harmonic of F is C at 1 Hz.)

Now we can make more potentially useful blues intervals by looking at the harmonics of F.

The fifth harmonic of F is the just intonation major sixth (A) at 5/3 Hz. The seventh harmonic of F is the subminor third (between D and E-flat) at 7/6 Hz.

This is already enough notes to make a good blues scale. The closest twelve-tone equal temperament approximation is C, D, E-flat, E, F, F-sharp, G, A, and B-flat. That’s the union of the Aebersold blues scale (C, E-flat, F, F-sharp, G, B-flat) and the major blues scale (C, D, E-flat, E, G, A). You could also think of it as Mixolydian mode with an added minor third and sharp fourth.

Let’s find some more intervals. How about the subharmonics of F?

The third subharmonic of F is the Pythagorean minor seventh (B-flat) at 16/9 Hz. This is a noticeably different pitch from the harmonic seventh at 7/4 Hz. Blues musicians do a lot of pitch play around the flat seventh; this might be why.

While we’re at it, let’s take a look at the subharmonics of G at 3/2 Hz.

The fifth subharmonic of G is the just intonation minor third (a little above E-flat) at 6/5 Hz. This is a completely different interval from the subminor third at 7/6 Hz, and blues musicians treat them as such. It’s also a possible origin of the “blue” or neutral third.

Now let’s see the harmonics of that just intonation E-flat.

The third harmonic of the just intonation minor third is the just intonation minor seventh (a bit above B-flat) at 9/5 Hz. Yet another version of B-flat!

What might we get from the subharmonics of the 7/4 harmonic seventh?

The fifth subharmonic of the harmonic seventh is the narrow tritone (a bit below F-sharp) at 7/5 Hz. Once again, this is a different note from the harmonic eleventh at 11/8 Hz. I have found that there are two different good-sounding spots to bend my Fs up to, and these might be they.

What can we find in the subharmonics of the just minor seventh?

The fifth subharmonic of the just minor seventh is the just intonation diminished fifth (between F-sharp and G) at 36/25 Hz. This is yet another distinct pitch in that zone between F and G. This is probably why bending your F-sharps up or your Gs down both sound good.

If we take all these intervals together, we have three pitch zones: the one between D and E, the one between F and G, and the one between A and B-flat (and slightly beyond.) Jeff Titon describes these zones as E complex, G complex, and B complex.

Hear this “scale” in action:

So how do you play all these spicy microtonal pitches (other than a DAW plugin)? If you’re a blues, jazz, gospel, R&B or rock singer, you may have been singing them unconsciously all along. Players of continuous-pitch instruments like violin, trombone and slide guitar are also likely to seek out just intonation intervals instinctively. On (non-slide) guitar, you play the fretted note below your desired blue note and then bend up to it. On harmonica, you play the note above your desired blue note and then bend down to it. On a synth or MIDI controller with a pitch bend wheel, you can bend either way. On piano, you can’t play the just intonation blue notes at all. However, Thelonious Monk found a way to imply them, by playing the keys on either side and then quickly releasing one of them.

I am not making an argument here about the actual historical origins of the blues. Those origins are unknown and probably unknowable. The blues was created and transmitted aurally by people who were legally forbidden to read and write. White people didn’t show much interest in documenting Black American music before the twentieth century, except in the most general and stereotyped terms. We can therefore only guess about blues history before the invention of audio recording. It’s reasonable to assume that the blues is a hybrid of various African, Native American and European folk influences. It’s also reasonable to assume that the musical predecessors of the blues had some features in common: metronomic grooves, swung and syncopated rhythms, drones and pedals, and tuning systems that favored the first seven harmonics of I and IV. But it’s hard to be more specific than that.

I do feel confident in asserting that the blues is not originally a modification of European tonal harmony. If blues musicians began by flattening the third and seventh degrees of the major scale, or by mixing major and minor modes together, why did they do it? Why those specific departures from European tradition? Did they just stumble on the idea of using dominant seventh chords on I and IV at random? No, the music is too beautiful and internally consistent for that. The harmonic series might not be the basis of blues harmony, but there must be some organizing principle behind it.

If blues tonality does come from the overtone series, it doesn’t mean that the first blues musicians sat down and worked out the mathematical ratios on paper. It’s more likely that they discovered and transmitted the characteristic intervals of the blues by ear. It’s not difficult to investigate the natural overtone series; all you have to do is stretch a wire between two nails and pluck it. We don’t know which specific overtones or combinations of overtones gave rise to the blues, but the ones in this post are all plausible candidates.

Ultimately, it’s not important whether we definitively understand the origins of the blues, or whether we work out some systematic theory of blues tonality. It is important, however, to understand that the blues has order and structure unto itself. The blues is not Western European harmony played “wrong” or “out of tune”. Contrary to the ill-informed cliches of music theorists, blues is not willfully dissonant or weird. While the first blues musicians may have been poor and lacking in formal education, they were not naive or unsophisticated. I want music theorists to understand that the blues is beautiful, that it has a harmonic logic of its own, and that Western Europe is not the only salient influence on the “Western” music of the present. American music is saturated with the blues, not just its grooves and timbres, but its harmonies too. There is no understanding the musical world we live in without understanding the blues.

Update:

I was watching this @12tone video about Harry Partch and there is surprising relevance to the blues https://t.co/ahC31qOHd6

— Dr Ethan Hein (@ethanhein) October 25, 2021

Further update: this excellent series of YouTube videos seems to support my hypothesis, because most of the destinations in the pitch zones could plausibly be attributed to the just intonation intervals described here.

References

Aebersold, J. (1967). Volume 1 – How to play jazz & improvise. Jamey Aebersold.

Chodos, A. T. (2018). The blues scale: Historical and epistemological considerations. Jazz Perspectives, 11(2), 139–171.

Cutting, C. B. (2018). Microtonal analysis of “blue notes” and the blues scale. Empirical Musicology Review, 13(1–2).

Gann, K. (1997). Just intonation explained.

Gioia, T. (2013). The guitar and the New World: A fugitive history. SUNY Press.

Jaffe, A. (2011). Something borrowed something blue: Principles of jazz composition. Advance Music GmbH.

Kubik, G. (2008). Africa and the blues. University Press of Mississippi.

Titon, J. T. (1977). Early downhome blues: A musical and cultural analysis. University of North Carolina Press.