I enjoy listening to Jacob Collier explain his music more than I enjoy the music itself. His arrangement of “Moon River” is mostly exhausting. However, Miles Comiskey pointed me to an interesting moment in this explainer video at the 1:04:22 mark where Jacob talks about how Kontakt enables you to change your instrument tuning on the fly.
Jacob takes a dominant seventh chord and plays it in two different tuning systems: twelve-tone equal temperament, the system we’re all used to, and just intonation, which is a more “pure” harmonics-based system. The chord sounds very different in the two systems. That is a profound musical concept that is not easy to understand! Jacob buries it in his song under five thousand other ideas, but I thought it would be helpful if I built a whole track around it:
To make sense of what you’re hearing in my track, the first thing you need to know is how to build a C7 chord using the natural harmonic series. Imagine that you have a guitar string tuned to play C2 (the note that’s two octaves below middle C). When you pluck the string, it vibrates to and fro 65.41 times per second (65.41 Hz in science terms). The note C2 is the name we give to the sound of a string (or any other object) vibrating at 65.41 Hz.
Here is where things get interesting. Guitar strings don’t just vibrate back and forth along their entire length. They also vibrate in many other ways at once. These vibrational patterns are called harmonics, and they are crucial to the sound of musical instruments.
- The vibration of the string along its entire length is called the first harmonic. It produces the note C2.
- The string’s second harmonic is the sound it makes as its two halves vibrate separately. The halves of the string each vibrate twice as fast as the string’s entire length. The second harmonic produces the note C3, an octave higher than C2.
- The string’s third harmonic is the sound it makes as it vibrates in thirds. Each third vibrates three times as fast as the whole string. The third harmonic produces the note G3, a fifth higher than C3.
- The string’s fourth harmonic is the sound it makes as it vibrates in quarters. Each quarter vibrates four times as fast as the whole string. The fourth harmonic produces the note C4, an octave higher than C3.
- The string’s fifth harmonic is the sound it makes as it vibrates in fifths. Each fifth vibrates five times as fast as the whole string. The fifth harmonic produces the note E4, a major third higher than C4.
- The string’s sixth harmonic is the sound it makes as it vibrates in sixths. Each sixth vibrates six times as fast as the whole string. The sixth harmonic produces the note G4, an octave higher than G3.
- The string’s seventh harmonic is the sound it makes as it vibrates in sevenths. Each seventh vibrates seven times as fast as the whole string. The seventh harmonic produces the note Bb4, a flat seventh higher than C4.
Here’s a helpful animation of the first five harmonics of a string from Wikipedia:
When you pluck a guitar string, you are hearing the mathematical sum of all of its harmonics at once. The specific loudness and decay of each harmonic gives the guitar its unique tone.
You can make a C7 chord by combining the first, third, fifth and seventh harmonics of C. That’s C, the G whose frequency is three times C’s frequency, the E whose frequency is five times C’s frequency, and the B-flat whose frequency is seven times C’s frequency. Simple! That said, it will sound better to move all the notes into the same octave. You can bring G3 down to G2 by dividing its frequency by two. You can bring E4 and Bb4 down to E2 and Bb2 respectively by dividing their frequencies by two, and then dividing by two again. This gives you the following pitches:
- C2: 65.41 Hz
- E2: 65.41 Hz * 5/4 = 81.76 Hz
- G2: 65.41 Hz * 3/2 = 98.12 Hz
- Bb2: 65.41 Hz * 7/4 = 114.47 Hz
This elegant graph, which I screencapped from MTS-ESP, shows the frequency ratios in our C7 chord, derived from the first, third, fifth and seventh harmonics .
When you tune your notes based on the natural harmonics, you are working within a system called just intonation. Many world cultures use just intonation, because many (though not all) humans prefer the sound of natural harmonics. Western Europeans used just intonation until the late 15th century or so. The thing is that just intonation systems have some big problems. If you tune using the harmonics of C, then all the notes you generate will be beautifully in tune with C, but they will not all be in tune with each other. This has some awkward consequences: you can’t change keys unless you retune all your instruments, and you can’t get a guitar to be in tune with itself.
Some cultures decided, big deal, we just won’t use multiple keys in the same piece of music. However, Western Europeans really wanted to be able to switch keys at will. So they started exploring alternatives to just intonation. The system they eventually settled on is twelve-tone equal temperament (12-TET for short). In just intonation, remember that all the notes are in tune with your starting pitch, but they aren’t all in tune with each other. 12-TET solves that problem by spreading the out-of-tuneness around so that none of the notes are perfectly in tune with each other, but nor are any of them unbearably out of tune. 12-TET makes every key sound exactly the same, so moving around between them is no problem. You can only approximate harmonics-based intervals in 12-TET, but, well, that’s life.
12-TET became the global tuning standard in Europe and its colonies (and the United States, a former colony) at the same time that Europe’s global hegemonic power reached its peak. As a result, 12-TET plays the same role in music that the English alphabet plays in computing and the internet: it’s a standard whose saturation and reach is very difficult to resist. Every piano, every guitar, every synthesizer, every tuner, every major piece of software: they all use 12-TET. If you want to tune to something else, you are going to need some specialized knowledge and equipment. To play synths in 12-TET, all I have to do is load Ableton Live, but to play them in just intonation, I have to use MTS-ESP, a complex and esoteric plugin. Meanwhile, most musicians I know spend their entire lives working within 12-TET without ever knowing that alternative tuning systems exist.
Let’s quantify the difference between the just intonation C7 and the 12-TET C7. In the graph below, the numbers inside the circle represent 12-TET semitones, divided into 100 cents per semitone. So 100 cents above C is C-sharp, 200 cents above C is D, 300 cents above C is D-sharp, 400 cents above C is E, and so on.
So how close are 12-TET intervals to just intonation intervals?
- The just intonation E is 386 cents above C. That is noticeably flatter than the 12-TET E at 400 cents.
- The just intonation G is 702 cents above C. That is very close to the 12-TET G at 700 cents; so close as to be indistinguishable for most listeners.
- The just intonation B-flat is 969 cents above C. That is much flatter than the 12-TET B-flat at 1000 cents.
In my track, the tuning shifts back and forth between just intonation (harmonics) and 12-TET every two bars. As it does, the Cs stay the same (by definition), and the Gs stay almost the same. However, the Es get noticeably sharper in 12-TET (or noticeably flatter in just intonation), and the B-flats get wildly sharper in 12-TET (or wildly flatter in just intonation). The harmonic B-flat and the 12-TET B-flat are so far apart as to practically feel like different notes.
So why should you care about any of this? Why does Jacob Collier make such a big deal about it? The 12-TET intervals sound okay and all, but there’s a reason why people prefer just intonation. When you play a just intonation C7, the harmonics in the E, G and B-flat line up precisely with the harmonics in C, creating a beautiful shimmery resonance. In 12-TET, the harmonics in E and B-flat fight the harmonics in C, creating a duller, messier sound. Singers tend to instinctively adjust their tuning to produce just intonation intervals. Players of instruments with fine pitch control, like violins and trumpets, do the same thing. Jacob Collier is so excited about the Kontakt synth plugin because it can automatically retune itself to play just intonation intervals relative to any chord root. Then he can match his singing to the synth. I made myself a track that plays just intonation blue notes and have been practicing bending strings to match it. Once you get used to just intonation, you start wanting to hear more of it. That makes your musical life more complicated, but also more satisfying.