Web Audio: Hz and Cents

Question

I'm trying to make a simple Oscillator program where I can change the Octave kind of the way the Massive VST shows it with positive and negative numbers:

_{(source: massivesynth.com)}

Now, I know there is 1200 cents to an octave (100 cents per semitone). The problem I'm running into is, when making the Osc code I realized that the pitch of it was measured in Cents.

ctx = new webkitAudioContext(); 
function osc1(pitch){ 
osc = ctx.createOscillator(), 
osc.type = 2; //0 = sine, 1 = square, 2 = saw, 3 = triangle, 4 = custom
osc.frequency.value = pitch; //in cents
gainNode = ctx.createGainNode(); 
osc.connect(gainNode); 
gainNode.connect(ctx.destination);
gainNode.gain.value = 1; 
osc.noteOn(0); 
};

osc1 (20);

and since Pitch changes the Frequency of a note, what I'm confused about is, without a MIDI keyboard how I could know

1. What note is being played?
2. What frequency is the note being played?

Furthermore, how could I get the bassy-er sounds from these waveforms? I've done a couple of tests generating sounds at 1 cent, 2 cents, 5 cents, 20 cents, etc. to see how they sound and when the Osc generates the Pitch at 1 cent, all I get is a low click whereas with 2 cents, I get almost the same click in a 4/4 beat. From my understanding, you can look at Frequency like points on a map and likewise, cents like the distance in between those points. That being said, how can cents determine the frequency of the note since the sound is being generated directly from the browser? Also, if it's as simple as just moving the pitch of the Oscillator, what note does the oscillators start on? In other words, what note are you "pitching" per say?

I hope what I wrote makes sense considering I'm pretty confused myself.

Thanks for any help and feedback!

Answer 1

Pitch is measured in Hz (cycles per second), and can also be given as a musical note and deviation from that note in cents. As you correctly noted, 1 cent is 1/100 of a semitone or 1/1200 of an octave.

The key to translating cents is to realize that the progression of pitches is not linear but geometric. A is defined as 440 Hz. The A one octave above that is 880Hz, and the next A above that is 1760Hz, and so on. Note that the difference in Hz between two A notes doubles for each octave upwards.

Mathematically, starting at a given pitch, the next semitone is larger not by a fixed number of Hz but by a ratio, which is the twelfth root of 2 (2^1/12). Since a cent is 100 times smaller, its ratio is the 1200th root of 2, or 2^1/1200. Remember, for each octave you double the Hz. For each semitone you multiply the frequency by 2^1/12. If you do that twelve times you'll have doubled the original number (440 * (2^1/12)¹² = 440 * 2 = 880). Likewise, (440 * (2^1/1200)¹²⁰⁰ = 880).

Try it in Excel.

Cell A1: 440
Cell A2: =A1*2^(1/1200)
Cell A3: ...copy A2 down...
.
.
.
Cell A1201:

If you do this right, cell A1201 will magically contain the value 880 and all the cells in between will give you the frequencies of each cent (but see important caveat just below).

Unfortunately, it's not quite so simple. What I described is the Pythagorean scale based on mathematics, which nobody uses today. If you tuned a piano this way C major and A minor would probably sound pretty good, but other keys would sound badly out of tune. The reasons for this are subtle having to do with the physiology of hearing. There are entire scholarly books on temperament, and it was hotly debated during the Baroque period, with several systems competing for mind-share, pretty much until Bach "decided" for everybody to use well-temperament because the keyboard couldn't easily have its temperament changed.

Musicians today use a modification called the well-tempered scale, where some notes are adjusted slightly up or down from mathematically perfect so that all keys are usable without retuning. It's a compromise but we're all used to hearing it. I won't discuss it firther except to say that, since the intervals aren't exactly 2^1/12, the definition of the cent depends on the actual width of each interval slightly for every interval. Think of cents as a set of points between 0 and 100, spaced equally between two notes but based on a fixed ratio (2^1/1200) instead of a fixed amount of Hz.

If you want to make an accurate oscillator to provide musical notes you are going to have to study temperaments and adjust the Hz values to match current musical convention. You will need a good book on temperament, and/or a tuner that does different temperaments. There are several iPhone apps that will do different temperaments, both input and output to the speaker.

Good luck and have fun.

Answer 2

Or do it the easy way - oscillator has a "detune" AudioParameter, which allows you to adjust the pitch up or down in cents (based on the oscillator's frequency param; in short, frequency is the baseline (as you noted, default is A-440Hz), detune is an offset in cents). :)