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semitone. We will leave aside the question why it is divided into twelve steps rather than some other number. But given that it is so divided, why does the keyboard not look like this: |
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with the notes lettered from A to L? Why the distinction between white and black keys, with the white ones looking as if they are trying to squeeze the black ones out? The answeror part of the answeris that our music, on the whole, is based on scales that go by tonal rather than semitonal steps. It is much more convenient for the fingers to make these steps by striking adjacent keys than to have to keep omitting alternate ones. |
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Very well,. then, why is the keyboard not like this: |
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with six white notes alternating with six black ones? Why is it seven and five, in an asymmetrical arrangement, so that the white notes are not always a tone apart but, at two points (E-F and B-C), only a semitone? |
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Answer: it is a consequence of the outstanding importance in melody and harmony of the two most consonant intervals after the octave, the ones corresponding to the vibration-ratios 3: 2 and 4: 3, in which, while not every peak of the lower note's wave coincides with one of the higher note's, every second or third one does. These intervals are the ones we know respectively as the fifth and the fourth.13 Added together, they make exactly an octave; the mathematical aspect of this is that  , since adding two intervals corresponds to multiplying their ratios. |
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Because of the musical importance of these two intervals, there is |
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13 The fifth is what you get if you start from C (or any other white note except B) and. counting that as no. 1, go up to white note no. 5; the fourth is what you get if you start on a white note (other than F) and go up to no. 4. Other intervals (third, sixth, eleventh, etc.) are named on the same principle. They are of the major type (major third, etc.) when equivalent to the span from C to the relevant white note, minor when one semitone less. Thus C to A, or D to B, is a major sixth (nine semitones), E to C is a minor one (eight semitones). |
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