TEMPERAMENT: The Idea That Solved Music’s Greatest Riddle.

By Stuart Isacoff. Knopf. 259 pp. $23

The lyre of the mythic Orpheus, enchanting even to trees and rocks, and the spirithealing harp of the psalmist David have everything to do with the story of Temperament. Into modern times, musicians and instrument makers have been driven to break the code that gave that ancient music its magical powers. The tuning of notes and their spacing in the octave were thought to be the heart of the matter.

From the sixth century b.c. on, Pythagoras ruled the discussion with his general dictum that the right relationships were mathematical. The top note of the octave resonates exactly twice as fast as the bottom note (in "Over the Rainbow," the vibrational leap in the opening notes on the word "somewhere" is 1:2, as also in Duke Ellington’s "Daydream"). Equally important to Pythagoras was the 2:3 ratio between the tones of the "perfect fifth" (the opening notes of "My Favorite Things") and the 3:4 ratio in the "perfect fourth" (the start of "Auld Lang Syne"). The exaltation of the "major third" (the mi-mi-mi-do! opening of Beethoven’s Fifth Symphony) came later; with a 5:4 ratio of vibrations, it extended the rule of small integers over the definition of musical value.

Beauty lay not just in the ears but in the clarity of numbers. In the 17th century, Johannes Kepler argued that the courses of the several planets corresponded precisely with the harmonious proportions of musical thirds, fourths, and fifths. That is, music was the language both of man’s singing inner life and of the celestial spheres in their orbits. "From earliest times," writes Isacoff, the editor of Piano Today magazine, "number, sound and virtue wrapped themselves like intertwining vines around the trunk of Western culture."

The problem with the number mysticism in music was spotted early on, even by Pythagoras. Intervals deemed "perfect" on their own were not neat nesting blocks that fitted perfectly together. Rather, like weeks fitting into months fitting into calendar years (or like mates in blessed matrimony), pleasing musical units had to be stretched here and whittled there to make a workable whole. The expressive voice and the tolerant ear had no trouble with the compromises, but the development of fixed-note instruments and especially keyboards forced hard choices and fierce arguments starting in the Renaissance. This is the terrain of Isacoff’s chatty survey, an anecdotal, name-dropping slide show of the evolution that made pianos and orchestras possible.

Many paths led eventually to the system that dominates today, "equal temperament," which divides the octave into 12 uniform "half-tone" intervals. Another approach, "just intonation," preserved perfect fifths and major thirds more adamantly. "Mean-tone temperament" sacrificed the sanctity of the fifth in favor of the third. "Well temperament," which J. S. Bach loved, preserved clear differences of color and character among the scales built on each of 12 keys. Finding the right temperament sounds like an engineering challenge, but innumerable churchmen, sages, scientists, and musicians deemed it vastly more. Vincenzo Galilei, father of the astronomer, made a humanist cause of liberating music (as singers naturally do anyway) from "the tyranny of inviolable number." Descartes, by contrast, maintained that equal temperament’s altered proportions were a violation of nature.

The spirit of Temperament owes more than a little to Dava Sobel’s marvelous little volume Longitude (1995). Temperament, unlike Longitude, has no breakthrough moment and no single hero (the Baroque composer Jean-Philippe Rameau comes close). Though the book is light sailing, the pleasure here is that it gives readers a glimpse of the oceanic depths of musical metaphors and mysteries still unsolved by cognitive science and evolutionary psychology. Please, God, may we never know just where music comes from, or why it moves us so!

—Christopher Lydon