How to Modulate a 440 Hz Sine Wave

The clarinet sets the tone for an orchestra by playing a 440 Hz note that each instrument modulates against its own tone.

You thought trigonometric identities were invented just to torture 15-year-old students. Little did you know that a trigonometric identity holds the key to modulating an A. A 440 Hz sine wave is a pure musical note -- specifically, the A above middle C. There’s a very easy way to modulate a 440 Hz note: All you need to do is sing or play a note of another frequency. That’s how harmonies -- and disharmonies -- are created.

Things You'll Need

• 440 Hz tuning fork
• Stringed musical instruments or voices

Instructions

1. Tap the tuning fork and have one instrument or one singer produce an A to match that tone. You can also use a piano to make the 440 Hz tone. Mathematically, the wave is represented by the expression: amplitude x sin (440 x 2 x pi x time), where the time is measured in seconds.

2. Play or sing a note close to 440 Hz. When first trying to identify modulation, you might find it easier to hear if the frequency of the second note is very close to 440 Hz, like 441 or 442 Hz. Mathematically, a 442-Hz wave is amplitude2 x sin(442 x 2 x pi x time).

3. 

   A piano and a violin played together produce a modulated tone created by their individual notes.

   Put the two instruments close to each other so that their sound waves fill the same space. The sound waves in the air are now the sum of the waves from each source. Mathematically, this is sound = amplitude x sin (440 x 2 x pi x time) + amplitude2 x sin(442 x 2 x pi x time).

4. Apply the trigonometric identity that sin A + sin B = 2 sin (A + B) x cos (A - B). For the sound waves, the total sound is now (amplitude + amplitude2) x sin ((440 + 442)/2) x 2 x pi x time) x cos ((440 - 442)/2) x 2 x pi x time) = total amplitude x sin (441 x 2 x pi x time) x cos (2 x pi x time).

5. Listen for the modulation. it’s unlikely you’ll be able to notice the difference between the 440 Hz original note and the new 441 Hz note, but you will notice the 2 Hz frequency. The 2 Hz modulation has the effect of changing the amplitude of the sound wave from zero to maximum and back to zero every second. That’s called a beat frequency, and it’s fairly easy to hear.

Tips & Warnings

• There are even easier ways to modulate a 440 Hz wave — so easy they’re trivial. For example, play a 440 Hz note on a trumpet. Blow harder and the note gets louder, softer and the note diminishes.

• Piano tuners listen for the beat frequency when they strike a tuning fork. They adjust the tension until the beat frequency goes to zero — that is, the tone remains at an even intensity.

• Any time musical notes are mixed, they modulate one another. The sum of the tones produced by an orchestra blend to create notes that are not played by any single instrument.

References

• Clark University; Summary of Trigonometric Identities; David E. Joyce
• Indiana University; What is Phase?; Prof. Jeffrey Hass; 2003