What happens when you multiply a bass clarinet with a sine tone?
Let's start this year at the beginning. In 1971 Dutch composer Ton Bruynèl wrote Intra for bass clarinet, soundtracks and ring modulator. To my knowledge this is the first piece for bass clarinet and live electronics in history. I worked together with digital composer Jessica Aslan on re-implementing this piece in the digital domain, making a version for bass clarinet and laptop. Here is a small excerpt:
Now where are those sounds coming from? Except from some air sounds at the beginning, all I am playing is a single note. But, I am also controlling the ring modulation. The ring modulator, also called a product modulator, takes the input from the bass clarinet and multiplies it with a series of so-called control signals, which are basically a series of sine tones. A funny thing happens when multiplying two signals. The multiplication actually results in adding and subtracting the frequencies of the two signals. Remember the trigonometric identities? Here is the one that shows how multiplying two signals leads to addition and subtraction of the frequencies of those signals:
sin(αt) x cos(βt) = 1/2[sin((α+β)t) + sin((α-β)t)]
So let's say I am playing an A at 440Hz, and the control signal is a sine tone an octave higher than my A, so 880Hz. Then the ring modulator creates a sound at 880Hz+440Hz=1320Hz and at 880Hz-440Hz=440Hz. That means you should hear the same pitch as I was playing coming from the ringmod (A@440Hz), as well as a pitch an octave plus a fifth higher (E@1320Hz). Now what happens if the control signal moves to 660Hz? 660Hz+440Hz=1100Hz (that's a C#, a third plus an octave above what I am playing), and 660Hz-440Hz=220Hz (that's the A an octave lower than what I am playing).
Don't trust it? Here is a couple of little soundfiles to illustrate what I just described:
No need to be distracted by the theory though. Ton Bruynèl was not a theorist, technician, or mathematician. In fact it could be more appropriate to think of him as a musical cook, mixing his ingredients and tasting his sounds.
Till next time, MVG
ps. Yes I did cheat a little. The example files showing the adding and subtracting of frequencies don't actually use the bass clarinet as input. Thing is that a bass clarinet doesn't sound at just a single frequency and things get a little more complex.