The Mathematics of Jazz
Ward Douglas Maurer · Bridges: Mathematical Connections in Art, Music, and Science · 2004
BRIDGES Mathematical Connections in Art, Music, and Science There is a sense in which all music has underlying mathematics. There is also, however, a widespread perception that this mathematics is associated only with symphonic music, not with free-flowing, improvisational music such as jazz. It is our premise here that jazz has its own mathematics, which is just as fascinating as the mathematics of a Bach fugue. This underlies such musical fundamentals as the way in which note durations are expressed, the number of bars in a passage, and the actual notes, chords, and key signatures used. The differences between jazz and rock in these respects are also interesting; and jazz musicians have even been known to play sly mathematical games. 1. Durations Of Notes In Symphonic Music We first take up the ways in which the duration of a note is expressed. In symphonic music there are several conventions for this; and, mathematically, the most important of these is based on binary fractions. Let t be the duration of a whole note in a specific piece of music. Then, in the simplest case, every other note in that piece has (absolute) duration t1 where f may be expressed in the binary number system, well known in computer science, as follows: /=0.1 /=0.01 /=0.001 / = 0 .. 0001 f = 0.0000 1 f = 0.000001 In the decimal number system, f is 1/2 (half note), 1/4 (quarter note), 1;8 (eighth note), h6 (sixteenth note), 1/32 (thirty-second note), or 1/64 (sixty-fourth note). The specification of t here is made by the metronome marking, of the form N = B where N is one of the musical notes above (usually a quarter note) and B is an integer, called the number of beats per minute. Here t, expressed in fractions of a minute, is 1/jB where f is associated with N as above. For example, if N is a quarter note and B is 120, there are 120 beats per minute. Each beat is therefore 11120 of a minute (or 1/2 of a second); a whole note is four beats, or 4/120 of a minute, and here this is 1I( 1/4 '120) since f, in decimal, is 1/4. All of the mathematics of durations is independent of t, and indeed, in what follows, we will refer to f, rather than tJ, as the (relative) duration. One or more additional one-bits may be added to the binary representations above by the use of dotted notes, with every such one-bit being represented by a dot. Thus, for eighth notes, we have: 274 2004 Bridges Proceedings ~ ~. ~ .. /=0.001 /=0.0011 /=0.00111 /= 0.001111 Here these appear more elegantly expressed in the binary than in the decimal, number system, where the relative durations, as above, would be 1/8, 3/16, 7/32, and 15/64 respectively. All other durations representable as terminating binary fractions may be specified by using ties, which add the tied durations. Thus, forJ= 0.10101, or 21/32 in decimal, we may write r /=8..1 ~ /=0..001 P /=0.00001 'p 1= 0.1 + 0.001 + 0.00001 = 0.10101 Fractional durations with non-terminating binary representations may be expressed by means of triplets and their generalizations, although here the scheme is not completely general. Let the duration of a note be tfwhereJis a fraction ilj.lfj is a power of two, we have the case above; more generally,j = k'2d, where k is odd. In the simplest case, i is 1 and k is 3, and the note is represented as part of a triplet. However, this triplet must be completed, meaning that there are either three notes, all of duration lIj, or one of duration I/j and another one of duration 21j (in either order). The total duration in either case is 31j = l/2d, and the three notes, taken together, take up the space of one note of that duration., The notes of actual duration lIj or 2/j are written as if they had duration lI2'2d or l/2d, respectively, like this: