Several of the audio demos use properties of the western 12-tone musical scale. Here, we give a very brief introduction to the structure of this scale.

The human ear is extremely sensitive to harmonic relationships between sounds when these sounds are combined. For example, two tones whose frequencies are a factor of two apart are perceived as being the same musical note, although one is clearly higher than the other. These two notes are said to be one octave apart. Thus, for example, 440 Hz is the musical note A. So is 220 Hz and 880 Hz.

The twelve distinct notes in the musical scale are evenly distributed over one octave. Note, however, that how we "evenly distribute" these tones is important. The arithmetic distance between A-440 and A-880 is not the same as the distance between A-220 and A-440. The notes need to be distributed geometrically, rather than arithmetically. How do we do this?

One simple mechanism is to start with some note, say A-220, and multiply it by the 12-th root of two to get the next note. If we do this 12 times, we will have multiplied by two, thus arriving at the note one octave up, A-440. The 12-th root of two is 1.0594631. If we multiply this successively, starting with A-220, we get the following frequencies:

A 220 A# 233.082 B 246.942 C 261.626 C# 277.183 D 293.665 D# 311.127 E 329.628 F 349.228 F# 369.994 G 391.995 G# 415.305 A 440This scale is said to have "equal temperament" and is the scale commonly used in western music.

Why twelve? It turns out that this scale has very special properties. In particular, it can be constructed approximately by a very different procedure that emphasizes the fact that the human ear perceives harmonic relationships very clearly. This procedure is called the "circle of fifths."

Start with A-220 and double it to get A-440. These two frequencies will be percieved as being the same note, and will sound very pleasing together. If we triple it to get 660 Hz, this is no longer an A. However, this new frequency will also sound good in combination with an A. So will half of this frequency, 330 Hz. Notice that 330 Hz is very close to the note E in the above scale. It is said to be a fifth above A.

We can repeat this procedure. Triple E-330 to get 990 Hz, and divide by two. The result, 495 Hz, lies above the octave shown above, so we divide by two again to get 247.5 Hz. This is very close to B, which is the note a fifth above E.

If we continue this procedure, we get the following sequence of notes: A, E, B, F#, C#, G#, D#, A#, F, C, G, D, A.

A 220 vs. 220 A# 234.932 vs. 233.082 B 247.5 vs. 246.942 C 264.298 vs. 261.626 C# 278.438 vs. 277.183 D 297.335 vs. 293.665 D# 313.242 vs. 311.127 E 330 vs. 329.628 F 352.397 vs. 349.228 F# 371.25 vs. 369.994 G 396.447 vs. 391.995 G# 417.656 vs. 415.305 A 446.003 vs. 440Notice that once the circle has closed, there is an error of only 6 Hz! The well-tempered scale distributes this error uniformly through all the notes.

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