Ivars Peterson's MathTrek

Medieval Harmony

At a Mathematical Sciences Research Institute meeting last October, number theorist Hendrik W. Lenstra Jr. of the University of California at Berkeley provided a glimpse of one of these gems, dating back to the 14th century.

Philippe de Vitry (1291-1361) was one of the most prominent figures in medieval music. He was the author of an important music theory text, Ars Nova, which introduced new rhythmic schemes and musical notation. He had a deep knowledge of philosophy, rhetoric, and mathematics.

In many ways, de Vitry's interests and accomplishments reflected the Pythagorean view that music is a subdivision of arithmetic, as shown, for example, in the simple mathematical relations between pitch and length of a string (see Circles of Dissonance, Nov. 24, 1997). His work honored the dictum of the Roman philosopher Boethius (480-524) that "music is number made audible."

One of de Vitry's observations concerned numbers that can be written as a power of 2 multiplied by a power of 3. In modern terms, such numbers would be expressed as 2ⁿ x 3^m, where n and m are integers. He called them harmonic numbers.

Here's a table of all harmonic numbers less than 1000.

1	3	9	27	81	243	729
2	6	18	54	162	486
4	12	36	108	324	972
8	24	72	216	648
16	48	144	432
32	96	288	864
64	192	576
128	384
256	768
512

De Vitry focused on pairs of harmonic numbers that differ by 1: 1, 2; 2, 3; 3, 4; and 8, 9. As it happens, those particular pairs correspond to musically significant ratios, representing an octave, fifth, fourth, and whole tone. He wondered whether there are any other such pairs of harmonic numbers.

In 1342, Levi ben Gerson, also known as Gersonides (1288-1344), proved that the original four pairs of harmonic numbers are the only ones that differ by 1. Here's how his proof went (using modern notation).

If two harmonic numbers differ by 1, one must be odd and the other even. The only odd numbers are powers of 3 (top row of chart, above). Hence, one of the two harmonic numbers must be a power of 3 and the other a power of 2 (first row and first column of chart). The basic task then involves solving two equations:

2ⁿ = 3^m + 1 and 2ⁿ = 3^m - 1.

Gersonides had the idea of looking at remainders after division of powers of 3 by 8 and powers of 2 by 8. For example, 27 divided by 8 gives a remainder of 3. For powers of 3, the remainders all turn out to be 1 or 3, depending on whether the power is even or odd. The remainders for powers of 2 are 1, 2, 4, then 0 for all powers higher than 2.

For 2ⁿ = 3^m + 1, when m is odd, 3^m has remainder 3, and 2ⁿ = 3^m + 1 must then have the remainder 4, so n = 2 and m = 1. That gives the consecutive harmonic numbers 3, 4. When m is odd, the equation gives the consecutive harmonic numbers 1, 2.

For 2ⁿ = 3^m - 1, when m is odd, 3^m has remainder 3, so 2ⁿ = 3^m - 1 has remainder 2; as a result, n = 1 and m = 1, to give the consecutive harmonic numbers 2, 3. The final case, when m is even, is a little trickier and requires substituting 2k for m, then solving the equation 2ⁿ = 3^2k - 1 = (3^k - 1)(3^k + 1). That gives the consecutive harmonic numbers 8, 9. QED.

Interestingly, the theorem about harmonic numbers proved by Gersonides has connections with Fermat's last theorem and with the ABC conjecture, a topic presently of considerable interest among number theorists (see The Amazing ABC Conjecture, Dec. 8, 1997).

Copyright 1999 by Ivars Peterson

References:

Grattan-Guinness, I. 1998. The Norton History of the Mathematical Sciences: The Rainbow of Mathematics. New York: W.W. Norton.

Hendrik Lenstra's lecture notes on harmonic numbers and a video of his talk are available at the MSRI Web site (http://www.msri.org/publications/video/contents.html). For a directory listing of pages corresponding to Lenstra's overhead transparencies, go to http://www.msri.org/publications/ln/msri/1998/mandm/lenstra/1/slides/ .

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.