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Treebeard's Stumper Answer
8 November 2002

Sound Tracks

During our afternoon classes at Dunn Middle School, we can't help overhearing Donna's brass band kids practice. At first I thought about barnyards, but now there is some harmony! Here are two stumpers with a curious connection. How can the same note (like middle C) sound so different on a flute, trombone, and sax? And how can any band be recorded on the single sound track of a tape or CD? If a cat and dog walk in line, we can easily tell who made each footprint. But a microphone can only record a single track of sound with everything at once. How can we still pick out each instrument in the recording?

Our beloved Dunn School band teacher Rose Knowles calls the lovely sound of beginning band students the Calling of the Geese. But after just a few weeks our DMS students are producing sweet harmony. This is a two-pronged stumper. All those band instruments sound different even when they are playing the same note. And all those sounds can be recorded in a single (maybe stereo or surround) sound track that looks something like this:

This is really a sample of Bix Beiderbecke's classic Royal Garden Blues, with brass, woodwinds, piano, bass, and drums all at once. The bumps along the groove of a mono 78 or LP record would look something like this up close, as would the magnetic flux variations of a cassette tape or the digital signals of a CD or MP3. How can all that great jazz fit on just one wiggly track on any recording? Which instrument made each bump?

The real question is what's the connection between these two stumpers about playing and recording music?

Toss a few stones into a pool and watch the waves on waves. Sound waves are like that. They have frequency and amplitude that we hear as pitch and volume, but sound waves also have a complex shape that we hear as timbre or tone quality. Every instrument always makes many sounds at once, so every note is already a chord of harmonics. Flutes and trumpets sound so different because they combine different sounds. Since waves combine, we can record a band on a single track, and our amazing ears can still hear each different instrument. Take care of your ears!


Waves in a pool is a useful analogy, but there's an important difference with sound. We only see chop when water waves get complicated, but we can hear the harmony in multiple sound waves. Our ears are an amazing tool for decomposing complex sound waves into their components. Loud sounds (e.g. from live rock bands or gospel choirs or headphones) can damage your ears beyond repair, and the first thing you lose is your upper end high frequency response.

You may think those high frequencies don't matter, since the chart shows that even the highest-pitched instruments have frequencies less than 10kHz. But that ignores the timbre (pronounced "TAM-BER") of all musical sounds, which is produced by high frequency harmonics that we don't usually notice as separate sounds. Without those higher frequencies, all music will sound like RealAudio over a slow modem connection or listening on a telephone. What you lose is tone quality. Take care of your ears!

Tuning forks and cell phones and toy synthesizers make pure tones. Real musical instruments are more complicated and richer because they produce many sounds at once that interact to produce complex waves. Harmonics are the integer multiples (x2, x3, x4, etc.) of any fundamental frequency.

I play folk/blues guitar, and I can play the 2nd harmonic (and above) if I pluck a string sharply and just touch it in the exact middle (at the 12th fret) to damp the fundamental frequency. It takes some practice to sound right. I can play higher harmonics if I touch the string at 1/3rd or 1/4th of the neck, but the higher harmonics are quieter so I have to pluck louder to make it equal. Violins have a shorter neck than guitars, and violinists can play whole melodies by fingering the strings and "playing the harmonics" with their pinky on the midpoint. It's an important technique. The point is that the physical guitar string is vibrating at ALL these different frequencies at once and the sound we hear is the combination of all the waves. You can sometimes see the complex vibrations of the strings if you play in front of a flickering TV or strobe light.

Harmonics (and all waves) combine algebraically by adding and subtracting their values up and down. The result is that simple sine waves can be combine to produce any wave imaginable. Since any and all sounds played together combine to a single wave, they can be recorded on the single track of a CD/tape/LP/DAT/MP3 etc. That's the connection between my two stumpers. Even 5.1 surround sound can be recorded on a single stereo track, though that's another stumper!

The best way to understand harmonics is to play with them. I plundered this fine JAVA applet from Fu-Kwun Hwang, Dept. of physics, National Taiwan Normal University. This applet really is worth a thousand words! Ignore the green dots. The blue dot on the left controls overall volume, and the second blue dot is your fundamental frequency. The rest of the dots add harmonics and you can hear the difference. Slide or right click on a dot to bring it to the top, and right click to bring it back down to the center. You can select a waveform and see the harmonics or add harmonics and hear it. The [Play] and [Stop] buttons control sound output. Sometimes I have to restart my browser when it got stuck on a note.

It's important not to oversimplify. No matter how much I fiddle with the controls in that applet, it doesn't sound like my vintage Gibson LG-1 guitar. Real musical instruments produce many harmonics, but the harmonic texture changes over time while playing any single note; it's different at different volumes; it changes over time; and I vary how I pluck each string. I can change the tone in many ways to suit a particular performance. I would still sound good with a different guitar, but it would be different, and I would hear it as I play, so I would play different. Trying to capture the sound of my guitar is like taking a photo of the reflectons on a pond. There it is, but it won't quite be like that ever again!

"When you hear music, after its over, it's gone in the air.
You can never capture it again.
- Eric Dolphy

I don't take recordings very seriously except for masters like Eric Dolphy who will never jam again on this planet. I think studio mixing gets in the way the the music, and I prefer bootlegs and live recordings. Eric Dolphy's closing words and music on the "Last Date" album really is all we have of that last session. I treasure it, but I don't confuse it with live music!

Real music is always a jam, even played note by note from a score. We don't pay concert hall prices for tickets just to hear the "right notes". Musicians can do that by silently reading the score, and the rest of us can do it with MIDI..

Electronic keyboards and synthesizers and computers may have 100s of instruments in their sound bank, but we rarely confuse a synth with the real thing or a recording with a live performance. Musicians pay big money for fine instruments because of the sound quality. A fine musician will sound good on a cheap instrument, but not vica-versa. We remember a great performance for a lifetime, and it's nothing but combining the right waves. But that's like saying that genius is just firing the right neurons and typing the right keys. It's maybe true, but it doesn't explain anything.

You can produce an almost square wave with the right combination of almost infinitely many odd-harmonic sine waves. Try it! This is where the math model and the physics model diverge. It seems a lot easier to program a fast on-off oscillator like a 555 chip than it is to produce infinitely many sine waves. Are they really the same thing?

Two views of a (almost) square wave.

Combining sine waves to make complex sounds is sometimes called Fourier synthesis, named after French mathematician Jean Baptiste Joseph Fourier (1768 - 1830) who first worked out the math. Reclaiming the pure tones from a complex wave is much more difficult than combining them, but it's always possible with Fourier analysis. The usual method is the math-intensive FFT or Fast Fourier Transform. This takes time even on fast computers. It's the algorithm that powers the spectrum display in WinAmp and other computer music players. (I always it turn off to save CPU cycles.)

I had the idea to record and sample the kids at school one at a time playing the same note on their band instruments. School conferences made that impossible, but I found another source on the web at the University of Iowa collection of Musical Instrument Samples (MIS). These are samples of different instrument carefully recorded in an anechoic (no-echo) chamber. I remember being in an anechoic chamber many years ago at UCLA. It was a strange experience, like the sound was being pulled from my mouth. But it is a standard condition to compare recordings. I collected MIS samples of different instruments and used Cool Edit Pro to cut out just the single note C5, which is C above Middle C, frequency 523.25 hz, played mf. I normalized the volume of all samples to a high level (without clipping) and did a frequency analysis of each sample. I also combined 15 of these samples into a single wave file and ran it through Richard Horne's once-freeware Spectrogram v5.1.7 program. The high-quality WAV file is too big for my server space, but here's a low-quality (24kbs) MP3 version (107k) that's still interesting. In order, the (15+1) instruments (with links to MIS) are:

  1. Bass Flute
  2. Alto Flute
  3. Flute, no vibrato
  4. Alto Sax, no vibrato
  5. Soprano Sax, no vibrato
  6. Eb Clarinet
  7. Bb Clarinet
  8. French Horn
  9. Tenor Trombone
  10. Bassoon
  11. Oboe
  12. Cello bowed D-string
  13. Cello plucked D-string
  14. Piano
  15. Balloon Pop

I was hoping for more dramatic spectral differences between instruments, but maybe that's the point. Sound timbre and quality is a subtle thing that changes over time, and this simple analysis can't capture it. At least I can't see it. I'm reminded of Francis Ford Coppola's movie The Conversation (1974) (and The Hunt for Red October (1990)) where the story hangs on subtle sound analysis...

This is a frequency analysis of the sound of a balloon popping, also from MIS. At least I can see that this noise differs from music because it contains (nearly) all frequencies of sound rather than the integer harmonics of real instruments.

Frequency Analysis of selected instruments made with Cool Edit Pro

Analysis of selected instruments over time made with Spectrogram v5.1.7.

This is starting to sound like a fun science fair project: record and sample as many different instruments as possible all playing the same note/pitch, and use software to try to understand the difference.

It's remarkable that our ears can do this FFT analysis in realtime so that we can pick out different instruments in a one-track recording. I don't know a computer program that can just record the flute out of a combo, but I can easily listen to just that flute and ignore the rest.

I understand this complex math, but music is still a stumper for me. How can my ears pick out just the flute and all its harmonics as single notes, but still separate the sax and all its harmonics as different notes? Why do some combinations of waves combine to one instrument in my ears and mind, but I hear other waves separately, like the famous Tristan Chord at the start of Wagner's Tristan und Isolde? How can I hear sound from all directions with just my two ears when it takes many surround sound speakers to play it back? Why are some instruments literally worth a forture?

Here are some Web links for your own research on my stumper.

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