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I’ve added a new scale to the microtonal synthesizer for your enjoyment.  The preset is called Harmonic (C1), and it’s the first 127 notes of the harmonic scale beginning at C1.  I think it’s pretty cool!

Before they invented valves, trumpets were limited to only playing certain notes.  A trumpet in C could maybe get a low C3 out (it sounds kind of terrible), but it could play its first few harmonics fairly easily.  Middle C (C4), G4, C5, E5 (a bit flat), G5, Bb5 (very flat), C6, D6, E6, Ft6 (that’s F half-sharp), G6, Ab6, A6, Bb6, B6, C7, …  Actually, it’s REALLY not easy to play that high, but you’ll notice the notes are closer together as you go up.  What’s going on?

A sound with definite pitch at frequency f has a sound wave that’s periodic with frequency f.  What does this mean?  It means that the sound wave repeats f times every second (if f is in Hz).  A sound wave propagates in space and in time, so if you assume a speed of sound of c, the wavelength is y = c/f.  To put this in understandable terms, the wavelength of A440 (A4, which is 440 Hz) at standard temperature and pressure (speed of sound is 343 m/s) is (343 m/s)/(440/s) ≈ 78 cm (Hz is the same as 1/s).  So if you have a pipe that’s 78 cm long and you make the air inside it vibrate (it’s more complicated than that), it’ll produce an A4, the A above middle C (which is C4).  But more to the point, if you take a snapshot of an A4, it’ll repeat every 78 cm.  (Note that this depends on temperature and pressure; at 0 °C, for example, the speed of sound drops to 331 m/s, so the wavelength is shorter.  This is why the pitch of wind instruments, whose pitch is dependent on the length of the tube of air, goes down when it’s cold.)

If you draw the A4 sound wave, it’ll repeat every 78 cm, but beyond that, you can do whatever you want.  You can make it a smooth sine wave, or you can make it squares, or triangles, or sawteeth, or whatever you want, so long as it repeats every 78 cm.  The shape of the wave is what gives a sound its characteristic timbre.  It’s what makes the sustained sound of the trumpet different from that of the clarinet, for instance.  (The shape I use in the Offtonic Microtonal Synthesizer is a fairly simple one called a pulse wave.  Real instruments have much more nuanced shapes.)  In particular, you can make a wave that repeats twice every 78 cm, and it’ll still repeat every 78 cm.  You can make it repeat three times.  You can make it repeat four times.  You can make it repeat n times, where n is any positive integer.  So if your wave repeats three times every wavelength, doesn’t it secretly have frequency 3*440 Hz = 1320 Hz, and is therefore an E6 (real frequency: 1318.5 Hz, easy to check on the synth using the default chromatic preset)?

Yes.  So you wouldn’t call it a 440 Hz wave anymore.  However, part of the 440 Hz signal can be at 1320 Hz.  Part can be at 2*440 Hz = 880 Hz.  Part can be at 4*440 Hz = 1760 Hz.  In fact, whatever the shape you have, if the whole thing repeats only at 440 Hz, you can break it down into parts that repeat at 880 Hz, 1320 Hz, 1760 Hz, and so on.  Let’s say you have some shape F(t) with frequency f.  If you define w = 2πf (just to make things easier), you can always break down F(t) into A0 + A1sin(wt + ø1) + A2sin(2wt + ø2) + A3sin(3wt + ø3) + … + Ansin(nwt + øn) + …, forever, where Ai and øi are constants (øi are just phases, so they don’t really matter).  It’s a bunch of sine waves!  In fact, this is what the ear does.  When you hear a sound, any sound, the ear breaks it up into a sum of sines like this and feeds the output to your brain.  (I’m not going to explain how to do it by hand, maybe another time, but it’s called a Fourier transform.)  The collection of frequencies, f, 2f, 3f, 4f, …, nf, … is called the overtone series.  It’s a natural physical phenomenon.

In particular, when you have a long tube of air, that air can only vibrate at the natural frequencies of the tube.  Since that tube is long, the natural frequencies are the overtone series!  If your tube is 78 cm long, for example, it can produce an A4 at 440 Hz, an A5 at 880 Hz, an E6 at 1320 Hz, and so on.  A bugle is such a tube, so it can only produce those notes.  That’s why bugle calls, like taps, only have a couple of different notes — the bugle can’t play any others without going high.  A trumpet in Bb, for example, has tubing of length 148 cm, which corresponds to Bb3.  However, the fundamental is actually an octave below that, at Bb2 due to the physics of the instrument.

The notes of the Harmonic (C1) scale on the synth are the notes that a bugle pitched at C1 can play, in theory.  They’re also the overtones that you can hear when you play a C1 sound on an instrument.  Since the sound waves line up, the lower notes on this scale sound very good together.  As you go up, the pitches get closer and closer together.  You can see that the first few pitches are C1 (the fundamental), C2, G2, C3, E3 (a bit flat), G3, Bb3 (flat), C4, D4, E4 (a bit flat), F#4 (very, very flat), G4, Ab4 (very sharp), Bb4 (flat), B4 (a bit flat), C5, and so on.  Since multiplying a frequency by 2 ups the pitch by an octave, there is 1 note in octave 1, 2 notes in octave 2, 4 notes in octave 3, 8 notes in octave 4, 16 in octave 5, 32 in octave 6, 64 in octave 7, and to make sure there aren’t too many notes to fit — they’re high enough as it is — I didn’t put 128 in octave 8, 256 in octave 9, and 512 in octave 10, but they exist!  With this scale, you can play around with perfect intervals and overtones.  You can especially hear what a perfectly in-tune major chord sounds like, with C3, E3, and G3.  The E is a bit low compared to 12-tone tuning, but in reality, 12-tone tuning’s E is a bit high compared to the overtone E!  You get a very crunchy sound when you add in the Bb.  Interestingly, you can get a fairly close chromatic scale using the octave with 16 notes.  It’s not a perfect fit, of course, but you can get a passable C at the 16th harmonic, C# at the 17th, D at the 18th, Eb at the 19th, E at the 20th, F at the 21st, F# at the 23rd (22 is way too flat), G at the 24th, G# at the 25th, A at the 27th, Bb at the 28th or 29th, B at the 30th, C at the 32nd.

Note that this is by no means symmetrical.  You can change keys but it won’t sound very good!  (But an infinite overtone series contains the overtone series of any of its members; it just gets too high to be useful.)  The overtone scale is the premier example of frequency increasing linearly while pitch goes as the log of frequency.  There are also lots of patterns.  Go ahead and play around with it, because it has some nifty sounds!

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