ABSTRACT
Twelve is a very special number. ere are 12 months in a year. e Chinese use a 12-year cycle in their calendar. Near the equator, there are 12 h of daylight, and 12 h of darkness. ere are 12 days of Christmas, and we buy a dozen eggs. To a mathematician, 12 is the sum of the three smallest integers a = 3, b = 4, and c = 5, which satisfy the famous eorem of Pythagoras (a2 + b2 = c2). Here, 3 + 4 + 5 = 12, and 32 + 42 = 52 or 9 + 16 = 25. From the musical point of view, 12 has the important property that it is a small number that contains many divisors other than one and 12, in particular two, three, four, and six. For comparison, the larger number 16 only has two, four, and eight as divisors other than 1 and 16. Twelve is also the number of dierent pitches in the chromatic scale or octave of the modern piano keyboard that consists of 12 pitch intervals called semitones (see Figures 11.1 and 11.2). Since a note that is transposed by an octave may be considered to be the same note, we may wrap the piano octave onto a circle consisting of 12 intervals, as in Figure 11.3 using the labels C, C#, D, Eb, E, F, F#, G, G#, A, Bb, B.* e white keys on the modern piano keyboard correspond to the seven pitches without the sharps and ats, that is, C, D, E, F, G, A, B, as shown in polygon notation† in Figure 11.3 (le ), and have been immortalized in song with the words Do-Re-Mi-Fa-Sol-La-Ti, which will bring us back to Do. Note that this polygon is identical to the bembé ternary rhythm timeline polygon of Figure 10.1 (right). us, the bembé rhythm and the diatonic scale are isomorphic to each other, they are the same pattern of long and short intervals, one expressed in time intervals and the other in pitch intervals.‡
e diatonic scale is a heptatonic (seven-note) scale.* For this reason, the interval pattern [2-2-1-2-2-2-1] is sometimes called the “diatonic pattern,” even when referring to rhythm.† e pentatonic (ve-note) scale is found throughout the world, prompting Simha Arom to suggest that it is a musical universal.‡ One of the favorites contains no semitones, and consists of the tones C, D, E, G, and A, as shown in polygon notation§ in Figure 11.3 (right). In ancient classical Chinese music, this was the only scale used until the Chou dynasty
more than 3000 years ago.* Observe that this scale is isomorphic to the fume-fume timeline of Figure 10.1 (le ). Also worthy of note is that if this scale is rotated by a half-circle, one obtains the complement of the diatonic scale, or the black keys of the modern piano keyboard, corresponding to the white circles in Figure 11.3 (le ).
