Portrait of Portrait of Tracy

While I was writing about fretless bass, I briefly mentioned Jaco Pastorius’ “Portrait of Tracy.” Now it’s time to dig into this tune properly.

Here’s a live performance that also brings in Jaco’s tune “Okonkolé y Trompa”.

Extremely technically skilled bassists, like extremely technically skilled guitarists, frequently compose these kinds of showy solo pieces for themselves. However, they are rarely as musical as “Portrait of Tracy.” I am not going to try to teach you how to play it, since I can’t even approach playing it myself. If you do want to learn it, there are plenty of transcriptions and tutorials out there: this video has nice tab and notation, and this one explains further. I’m more interested in the tune as a recording.

In the process of warping out Jaco’s recording to the metrical grid in Ableton, I tried putting a beat under it, and ending up making a full remix. Enjoy:

The big idea in “Portrait of Tracy” is harmonics. Understanding what that means requires some physics. A plucked string’s movement is much more complicated than just wobbling back and forth along its entire length, and its vibration produces many pitches at once. If you listen carefully to the sound of a plucked guitar or bass string, you can hear these other pitches. You have been hearing them in many musical sounds for your entire life, whether or not you were conscious of it.

The very first thing you hear when you pluck a string is a short, sharp burst of noise, that is, many frequencies blended together. This burst of noise is called a transient. If you bang on a miscellaneous metal object, the transient (a plunk or clunk) is probably all you’ll hear. If you bang on a more flexible piece of metal, however, it might resonate, and that resonance might have a distinct pitch to it. (There are a couple of metal mixing bowls in my kitchen that make perfect bells or gongs if you tap them in the right place.) Bass strings are designed to vibrate easily, and the bass is designed to maximize the strings’ resonance.

When you pluck a string, you send waves of displacement cascading down the string to its ends. The waves then bounce off the string’s ends, racing back and forth, mixing together, and interfering with each other and themselves. This interference quickly self-organizes into standing waves. The animation below shows a simplified version of the process.

The green wave is traveling to the right, and the blue wave is traveling to the left. They interfere with each other to produce the red wave. When the peaks of the green and blue waves line up, the red wave reaches its maximum displacement. Where the green and blue waves cancel each other out, the red wave doesn’t move at all. These cancelation points are called nodes. The red dot in the center of the animation is a node.

A bass string produces standing waves at each of its resonant frequencies. The largest-amplitude resonance is the string’s first harmonic, otherwise known as its fundamental. The low E string on a bass in standard tuning has a fundamental frequency is 41.20 Hz, a pitch called E1. To make the math easier in the rest of this post, however, I’m going to pretend that E1 is a frequency of 1 Hz. If you want to convert back to actual frequencies, multiply everything that follows by 41.20.

We conventionally describe a string’s fundamental frequency as “the note” produced by that string. However, while E1 is the E string’s loudest audible resonance, it is far from the only one. These frequencies have a surprisingly simple mathematical relationship: they are all whole-number multiples of the fundamental.

• If you pluck an E string on a bass and listen as the note decays, you will hear the sound change. As the fundamental fades out, you will hear a higher E still ringing. It was always there, but the fundamental masked it at first. The higher E is the string’s second harmonic, its vibration in halves, with a frequency of 2 Hz, twice that of the fundamental. The second harmonic produces the note E2, an octave higher than E1.
• Keep listening to the sound of the string as it decays, and you should be able to hear an even higher pitch emerging out of the sound as the first and second harmonics fade out. This is the third harmonic, the string’s vibration in thirds, with a frequency of 3 Hz. It produces the pitch B2, a fifth plus an octave higher than E1.
• The next higher (and quieter) sound to listen for is the fourth harmonic, the string’s vibration in quarters, with a frequency of 4 Hz. The fourth harmonic produces E3, two octaves higher than E1.
• If you really focus deeply, you may be able to detect the fifth harmonic, the string’s vibration in fifths, with a frequency of 5 Hz. It produces G#3, a major third plus two octaves higher than E1.

There are many more audible harmonics of the low E string, though you are unlikely to be able to pick them out just by listening. The specific loudnesses of the bass string’s harmonics and their different rates of decay depend on a lot of factors: the instrument’s construction, the makeup of the string and its age, how hard you pluck, where on the string you pluck, whether you are using a pick or your fingers, the angle of the pick or your fingers, and so on. This is a lot of physical parameters, and they result in a highly complex sound! You can see why it’s so difficult to synthesize the bass believably.

You may not be able to easily hear individual harmonics when they are blended together in the sound of a plucked string, but you can isolate them so you can hear them by themselves. To hear a specific harmonic, all you have to do is lightly touch the vibrating string at one of its nodes at the same time that you pluck. This lets your harmonic of choice keep ringing, while damping all of the other harmonics that normally obscure it.

• To isolate the second harmonic, touch the string at its halfway point, which is at the twelfth fret. (This means right above the fret itself, not the place where you would normally finger the string.)
• To isolate the third harmonic, touch the string a third of the way along its length, at the seventh fret.
• To isolate the fourth harmonic, touch the string a quarter of the way along its length, at the fifth fret.
• To isolate the fifth harmonic, touch the string a fifth of the way along its length, between the third and fourth frets. This spot will take some trial and error to find.

It’s possible to find harmonics beyond the fifth one with some practice. I can reliably find the first eleven, but it gets harder as you go higher because they are spaced more closely and are quieter.

How do these individual harmonics combine to make the rich and complex timbre of the bass? This is one of those physics questions where intuition is not much help, and you really need to approach it mathematically. The concept you need is called the Fourier transform, named for the French mathematician Jean-Baptiste Joseph Fourier. Fourier’s insight was that you can represent any repeating waveform (like a bass string’s vibrations) by adding together simple sine waves (the harmonics). The animation below shows how you can add sine waves together to make a square(-ish) wave. Basses don’t produce square waves, but oboes (sort of) do.

The Fourier transform made more sense to me when I learned to understand sine waves as the curve being swept out by a clock hand going around in a circle, as in these animations:

These clock hands are called phasors. You can experiment with summing phasors together using this beautiful interactive visualization by Jack Schaedler. Understanding phasors is not necessary for playing the bass, but they are a window into a wide range of physical systems, including subatomic particles.

What follows is a list of the natural harmonics you can most easily play on a four-string bass. The strings are tuned in fourths, a ratio of 4/3. So if the E string is tuned to 1 Hz, then the A string is tuned to 4/3 Hz, the D string is tuned to 16/9 Hz, and the G string is tuned to 64/27 Hz.

The E string’s first eight harmonics create a just intonation E7 chord.

• 1st harmonic: E at 1 Hz
• 2nd harmonic: E at 2 Hz
• 3rd harmonic: B at 3 Hz
• 4th harmonic: E at 4 Hz
• 5th harmonic: G-sharp at 5 Hz, flatter than you’re used to
• 6th harmonic: B at 6 Hz
• 7th harmonic: D at 7 Hz, much flatter than you’re used to
• 8th harmonic: E at 8 Hz

The A string’s first eight harmonics create a just intonation A7 chord.

• 1st harmonic: A at 4/3 Hz
• 2nd harmonic: A at 8/3 Hz
• 3rd harmonic: E at 12/3 Hz, so, 4 Hz, same as the E string’s 4th harmonic
• 4th harmonic: A at 16/3 Hz
• 5th harmonic: C-sharp at 20/3 Hz, flatter than you’re used to
• 6th harmonic: E at 24/3 Hz, so, 8 Hz, same as the E string’s 8th harmonic
• 7th harmonic: G at 28/3 Hz, much flatter than you’re used to
• 8th harmonic: A at 32/3 Hz

The D string’s first eight harmonics create a just intonation D7 chord:

• 1st harmonic: D at 16/9 Hz
• 2nd harmonic: D at 32/9 Hz
• 3rd harmonic: A at 48/9 Hz, so, 16/3 Hz, same as the A string’s 4th harmonic
• 4th harmonic: D at 64/9 Hz
• 5th harmonic: F-sharp at 80/9 Hz, flatter than you’re used to
• 6th harmonic: A at 96/9 Hz, so, 32/3 Hz, same as the A string’s 8th harmonic
• 7th harmonic: C at 112/9 Hz, much flatter than you’re used to
• 8th harmonic: D at 128/9 Hz

The G string’s first eight harmonics create a just intonation G7 chord:

• 1st harmonic: G at 64/27 Hz
• 2nd harmonic: G at 128/27 Hz
• 3rd harmonic: D at 192/27 Hz, so, 64/9 Hz, same as the D string’s 4th harmonic
• 4th harmonic: G at 256/27 Hz
• 5th harmonic: B at 320/27 Hz, flatter than you’re used to
• 6th harmonic: D at 384/27 Hz, so, 128/9 Hz, same as the D string’s 4th harmonic
• 7th harmonic: F at 448/27 Hz, much flatter than you’re used to
• 8th harmonic: G at 512/27 Hz

You can also produce harmonics tuned to other notes using a technique called “false” or “artificial” harmonics. If you press down on the A string at the 2nd fret, you effectively get a B string. You can then play that B string’s fifth harmonic to get D-sharp, which isn’t available from the natural string harmonics. In practice, this is not easy to do, unless you have huge hands like Jaco.

Now you’re ready to understand how “Portrait of Tracy” works. Jaco combines natural and false harmonics together with regular fretted notes to make various lovely chords. I’ve listed some examples with timestamps.

0:07 Cmaj9 (no 3rd)

• G string: B (5th harmonic)
• D string: D (4th harmonic)
• A string: C (3rd fret)

0:09 Gmaj9 (no 3rd)

• D string: F-sharp (5th harmonic)
• A string: A (4th harmonic)
• E string: G (3rd fret)

0:11 Emaj7 (no 3rd)

• G string: B (5th harmonic)
• A string: D-sharp (false 5th harmonic of the B on the 2nd fret)
• E string: E (open)

0:18 Eb7#9

• G string: G (2nd harmonic)
• D string: F-sharp (5th harmonic)
• A string: C-sharp (5th harmonic)
• E string: E-flat (11th fret)

0:42 E13 (no 3rd)

• G string: D (3rd harmonic)
• D string: F-sharp (5th harmonic)
• A string: C-sharp (5th harmonic)
• E string: E (open)

0:53 C6/9 (no 3rd)

• D string: D (4th harmonic)
• A string: A (4th harmonic)
• E string: C (8th fret)

1:18 Ab13 (no 3rd)

• G string: F (7th harmonic)
• D string: F-sharp (5th harmonic)
• E string: A-flat (4th fret)

1:42 Fmaj7(no 3rd, add6)

• D string: D (8th harmonic)
• A string: E (6th harmonic)
• E string: F (1st fret)

2:07 Emaj7#11

• G string: G-sharp (false 5th harmonic of the E on the 9th fret)
• D string: D-sharp (false 5th harmonic of the B on the 9th fret)
• A string: A-sharp (false 5th harmonic of the F-sharp on the 9th fret)
• E string: E (open)

The tuning of some of these chords is very strange. First of all, Jaco is using some incorrect enharmonics. The fretted notes are tuned in equal temperament, while the harmonics are just intervals. The difference isn’t that noticeable for fourths and fifths, but for thirds and sevenths, it’s obvious even to untrained listeners. Also, Jaco is treating enharmonics as equivalent. In the Ab13 chord, he’s using F-sharp rather than G-flat, which is fine in equal temperament, but in just intonation, those are two completely different pitches. (Alternatively, you could think of it as a G#13 chord, in which case Jaco is problematically using F rather than E-sharp.) But, honestly, who cares? The chords sound strange, but they don’t necessarily sound bad. After you listen to the track enough times and you get used to the blend of equal tempered and just intervals, it all starts sounding beautiful.

The distinctive timbre of all those harmonics is appealing to hip-hop and dance music producers. Here are some tracks that sample the tune.

“Who’s Listening” by Ghetto Children (1996)

“Tubuluka Beach Resort” by Amon Tobin (1997), briefly at the end

“Pigeon” by Cannibal Ox (2001)

“Mr Mukatsuku” by Wagon Christ (2011)

The main melody is also frequently interpolated by R&B and rap producers who are unwilling or unable to pay for sample clearance.

“Rain” by SWV (1998)

“Ghetto Love” by Master P (1998)

“Pullin’ Me Back” by Chingy ft. Tyrese (2006)

“How Can I Not Feel Love?” by PM Dawn (2019)

So what do you say, virtuosos? We know you can shred. How about following Jaco’s example and writing some more tunes that show new colors of the instrument, new timbres, new pitch shadings? I would love to hear it.