The keyboard we are all familiar with is the result of a long evolution, in parallel with the development of western music. The set of notes used to create music has not always been 12 notes as we use now (C, C#, D, D#, etc), and even when the 12-note octave became standard, the tuning of the notes was not quite the same as the tuning we use today. If we played a piece of music today using one of these historical tunings, the music would sound out-of-tune to our modern ears, especially if a piece of music changed key somewhere in the middle. The reason for this is the tunings were "lopsided," favoring certain intervals over others. But that's another article.
The even-tempered scale is now the scale used in all the music we hear, no matter what style the music is. How is this scale structured - mathematically?
If a certain pitch is 220 Hz (this means 220 Hertz, or "cycles per second," the unit for measuring sound frequency),
what is the frequency of the note one octave higher? The wrong thing to do would be to add an amount to 220. For
instance, we would not compute
220 = note
220 + 220 = note one octave higher
220 + 220 + 220 = note two octaves higher
220 + 220 + 220 + 220 = note three octaves higher
etc.
In fact, we double the frequency to move up one octave:
220 = note
220 x 2 = 440 = note one octave higher
440 x 2 = 880 = note two octaves higher
880 x 2 = 1760 = note three octaves higher
etc.
Don't ask why - it's simply how our ears work. Doubling the frequency of a pitch has the same effect as halving the length of the plucked string we discussed at the beginning of this article.
Now, how are the pitches of the 12 notes within the octave determined?
Taking 220 Hz as an example (this corresponds to the A below middle C), we know from above that the next higher A is
440 Hz (220 x 2). We need to divide the octave into 12 equal (for equal temperament) sections. It's done like this:
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