The blues and the harmonic series

In this post, I’m going to expand on an idea in my blues tonality treatise: that the distinctive scales and chords of the blues are an approximation of African-descended tuning systems based on the natural overtone series. Gerhard Kubik argues in his book Africa and the Blues that blues tonality comes from the overtone series of I and IV, and can only be approximated using instruments tuned to standard twelve-tone equal temperament (12-TET). Let’s unpack what that means!

First, you need to know what the overtone series is, and for that you need to know some physics. When you pluck a guitar string, it vibrates to and fro. You can tell how fast the string is vibrating (its frequency) by listening to the pitch it produces. Shorter strings vibrate faster and make higher pitches. Longer strings vibrate slower and make lower pitches. You measure the rate at which a string vibrates in Hertz (vibrations per second.) The standard tuning pitch, A440, is the note you hear when a guitar string vibrates back and forth 440 times per second.

Strings can vibrate in many different ways at once. In addition to the entire length of the string bending back and forth, it can also vibrate in halves, in thirds, in quarters, and so on. The vibrations of these string subsections are called harmonics (or overtones, or partials, they all mean the same thing.)

Each string harmonic produces a different pitch. So when you play a note, you’re actually hearing many different pitches at once as the harmonics all vibrate together. This cool web tool shows the interaction of the first three harmonics of a string. Read more about harmonics here.

Imagine that you have a guitar string tuned to play a note called “middle C,” which has a frequency of 1 Hz. (Middle C’s actual frequency is 261.626 Hz, so if you want to think in terms of actual frequencies, just multiply all the numbers below by 261.626.)

• The first harmonic of the string is its vibration along its entire length. You already know that the string is vibrating at 1 Hz, producing the note C.
• The second harmonic is the string vibrating in halves. Each half of the string vibrates twice as fast as the whole string, so the second harmonic’s frequency is 2 Hz. This note is also called C. The two-to-one relationship between the first and second harmonics is very important in music. We hear notes whose frequencies are related by powers of two as being the same note. So if 1 Hz is C, then 2 Hz is also C, but an octave higher.
• The third harmonic is the string vibrating in thirds. Each third vibrates three times as fast as the whole string, so the third harmonic’s frequency is 3 Hz. It produces the note G.
• The fourth harmonic is the string vibrating in quarters. Each quarter vibrates four times as fast as the whole string, so the fourth harmonic’s frequency is 4 Hz. It produces another C, an octave higher than the second harmonic and two octaves higher than the first harmonic.
• The fifth harmonic is the string vibrating in fifths, so its frequency is 5 Hz. It produces the note E.
• The sixth harmonic produces another G at 6 Hz.
• The seventh harmonic produces an approximate B-flat at 7 Hz.

Now imagine that your guitar has another string that’s tuned so that its first harmonic has a frequency of 1/3 Hz, so it plays the note F. This string has its own overtone series. Its second harmonic is another F at 2/3 Hz. Its third harmonic is C at 3/3 Hz (so, 1 Hz.) This is why the notes F and C feel so closely related – they share this very important harmonic in common. The other harmonics of the F string include A at 5/3 Hz and an approximate E-flat at 7/3 Hz.

You can turn these pitches into a scale by moving them up or down in octaves (multiplying and dividing their frequencies by two). You can also invert and combine them to make additional related notes. Here are some candidates for just intonation blue notes in C, listed by their distance to their closest 12-TET neighbors in cents –  thank you to the Xenharmonic Wiki, and to Kyle Gann for your handy just intonation guide. Read more about the sources of these intervals.

E-flat

• 7/6 (33 cents flat): Subminor third. The 7th harmonic of F. I use this in my own playing a lot, but I think of it more as a very sharp D.
• 6/5 (16 cents sharp): Just minor third. The inversion of A at 5/3. Possible candidate for the “blue” or neutral third.

E natural

• 5/4 (14 cents flat): Just major third. The 5th harmonic of C. Another candidate for the “blue” third.

F natural

• 4/3 (2 cents flat): Just perfect fourth. Inversion of G at 3/2. Basically identical to F in 12-TET.

G-flat/F-sharp

• 11/8 (49 cents flat): Harmonic eleventh. The 11th harmonic of C. That is very high in the overtone series, but it’s audible on an acoustic instrument if you listen closely. It’s just about halfway between F and G-flat/F-sharp in 12-TET.
• 7/5 (17 cents flat): Narrow tritone. The interval between the just major third and the harmonic seventh; also the sum of the just minor third at 6/5 and the subminor third at 7/6.
• 36/25 (31 cents sharp): Just diminished fifth. Two 6/5 minor thirds stacked up.

G natural

• 3/2 (2 cents sharp): Just perfect fifth. The 3rd harmonic of C. Basically identical to G in 12-TET.

A natural

• 5/3 (16 cents flat): Just major sixth. The 5th harmonic of F. There is some debate about whether this counts as a blue note or not. I use it on harmonica sometimes.

B-flat

• 7/4 (31 cents flat): “Blue seventh” or harmonic seventh. The 7th harmonic of C. You hear it in barbershop quartet music.
• 16/9 (4 cents flat): Pythagorean minor seventh. Two perfect fourths stacked up. Essentially identical to the B-flat from 12-TET.
• 9/5 (18 cents sharp): Just minor seventh. The sum of the perfect fifth at 3/2 and the minor third at 6/5.

You can combine the 12-TET approximations of these notes to make chords that are ubiquitous in the blues: C7, F7, and Cdim7. (You can also make Eb and Gm7, but these are less common in the blues.) Here’s a graphical view.

You notably can not make anything resembling G7 with these notes. You do hear V7 chords in the blues, but Kubik argues that they aren’t “native” to it, that they were imported from European-derived music in the US. Furthermore, when blues songs do use V7, they usually don’t have it resolve directly to I7, they go to IV7 first. Kubik thinks that blues musicians avoid V7-I resolutions on purpose because they “lack oxygen” as he puts it.

Court Cutting digitally analyzed the vocal melodies of fifteen classic blues tunes. He found that if you transpose all the songs in his sample to C, the most commonly sung pitches in them are C, F, G, the “E-flat” at 6/5, the “G-flat” at 7/5, and “B-flats” near 7/4, 9/5 and the 12-TET pitch between them. Cutting points out that the tonic, the 6/5 minor third, the 7/5 flat fifth and the 9/5 flat seventh together form a “harmonic half diminished chord.” When you arpeggiate this chord, it sounds convincingly “bluesy,” and when you add the fourth and fifth, it makes for a similarly convincing blues scale. Listen to the arpeggio and scale here. Here’s a track I made using a just intonation blues scale.

There have been a few other studies of the pitches in the blues, but they have all been done by ear. Cutting’s study is the first one I know of that measures the pitches digitally. It has a small sample size and some methodological shortcomings, but it’s a start. I would love to see a bigger and more systematic study like this. What do you say, computational musicologists?

I have been playing white-guy blues for thirty years or so, and have been playing all my blue notes by ear without ever doing much reflection on them. Since reading about blue notes and just intonation, I went back to the guitar to bend strings with more intention. There are some definite sweet spots in those bends that sound deliciously right, and I do believe that I’m experiencing an intuitive attraction to just intonation. It certainly is a more appealing explanation than the conventional idea that blue notes are “out of tune.” If that’s true, why do some “out of tune” intervals sound so much better than others? If you play slide guitar, you find out very quickly that most microtonal intervals sound terrible. Hitting the right ones consistently takes years of practice.

Whether or not the blues historically comes from just intonation intervals, it does seem uncontroversial to locate its central axis between I7 and IV7. Blues doesn’t need any chord changes, and many blues classics stay on the I7 chord the whole time. If you are going to add a second chord to a blues groove, it is most likely going to be IV7. I hear the tonic diminished seventh chord as a kind of functional hybrid of I7 and IV7. That’s how Robert Johnson uses it throughout “Kind Hearted Woman Blues“–it’s a departure from I7, but not as dramatic a departure as IV7. I hear a similar effect in the minor thirteenth chords that funk musicians use. Cm13 sounds like an F7 chord being played on top of a C7 chord. For examples, check out “The Payback” by James Brown and the guitar solo in “Kiss” by Prince.

If you wanted to grossly oversimplify, you could say that Western tonality derives from the first five harmonics of I, IV and V. There’s more to it than that, lots more, but I don’t think that’s a bad place to start.

In practice, though, we don’t use these harmonics-based intervals, we use approximations of them. That’s what the “temperament” in twelve-tone equal temperament refers to. It’s conventional to say that blue notes are “out of tune”, but that’s incorrect. It’s really 12-TET that’s out of tune! Octaves are perfect in 12-TET, and fourths and fifth are pretty close to being in tune, but the rest of the intervals are all varying degrees of sour. This is deliberate; we make everything a little out of tune so that all the keys sound equally bad. Otherwise, you would have to retune your instruments every time you wanted to play in a different key. In some musical traditions, they decided that temperament is not a worthwhile trade-off. Hindustani classical music uses just intonation, and everything is based on an unvarying drone. It’s significant that blues songs are also often drone-based, and that they hardly ever change keys.

The blues is my favorite music in the world. I think it sounds better than anything else, and I value other kinds of music based mainly on how much blues influence they display. And while the music itself is wonderful, I also think the blues has profound significance for all other Western musics as well. A main theme of twentieth century music in the US and Europe represents the rejection of conventional tonal harmony. Classical musicians expressed that rejection through approaches like atonality, serialism, musique concréte and so on. In doing so, they lost the majority of their listeners.

Meanwhile, blues musicians did something different: they kept tonality, but they redefined and reoriented it. Rather than staying locked into 12-TET like the serialists and atonalists, they brought in a richer and more diverse collection of intervals, and they slid and glided between them. They did all this in a way that connected intensely with their audience, and then that audience grew exponentially until it spread across the entire world. This is a breathtaking accomplishment, made all the more impressive by the fact that these musicians were working under conditions of extreme oppression and poverty.

In all of this writing, I haven’t talked about rhythm at all. The rhythms of the blues are also rich and profound, and they have their own equivalents to blue notes that are worth exploring in depth. But even if you ignore them and only think about harmony, the blues is one of the most important foundational components of our musical culture. If it can violate the basic conventions of the dominant harmonic system and still win over a massive audience, then what other conventions should we be questioning? Given its importance, the blues has received shockingly little attention from music educators. I hope we can start to do better.

By the way, hear me play a lot of blue notes on the harmonica:

Update: check out this response post from Wenatchee the Hatchet

Further update: People are pointing out that IV is not found in the overtone series (at least not until you get very high up in it.) This is true, but Kubik isn’t arguing that IV comes directly from the overtone series. Instead, the idea is that people figured out that there was another note whose third harmonic meshed with the drone they were playing.

Further further update: this Adam Neely video shows what happens when you pitch correct the blue notes out of vocals by Led Zeppelin, Aretha Franklin etc. The results are… not good.

Further further further update: I used MTS-ESP to visualize the information in this post in a more visually compelling way.

Further further further 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.