
"TEMPERAMENT" 

Excerpts
from Stuart Isacoff, Temperament, Alfred A. Knopf, New York, 2001
Discovering that equal temperament was possible did not make it acceptable, however. Indeed, this tuning must have sounded very odd to many of Willaert's contemporaries. It is wondrously free of "wolves," inconsistencies, or unpleasant surprises. To our modern ears, attuned to today's pianos, it is a perfectly beautiful tuning. Yet, its sound was a radical departure at the time. Creating twelve equal steps in an octave changed the proportions that had been used for the various musical intervals, some more drastically than others. Equaltempered fifths, for example, are closer to pure (3:2) than the fifths used in meantone tuning. But major thirds in equal temperament are tempered seven times as much as the fifths. Minor thirds are tempered eight times as much. In the end, these alterations are not intolerable: None of these intervals sounds bad. But in comparison many seem robbed of their original character. This was readily noticed in the sound of equaltempered major thirds: Lustrous and calm in their pure form, they were now slightly rough and somewhat bland. [p118] Newton had already begun a mathematical analysis of various musical scales in his student days. In a tiny, dogged scrawl his notebooks outline the formulas used to divide the most basic musical span, the octave, into various arrangements of the usual twelve tones; then, in precise columns of numbers, he compared other scales that divide the same span into 20, 24, 25, 29, 36, 41, 51, 53, 100, 120 and 612 parts. [p24] Man's senses have become accustomed to the new sounds, wrote Newton, but the alterations of the modernists rob music of its real power. To those who know better, equal temperament's compromised tuning is as ungrateful to the ear as "soiled and faint colors are to the eye." . . . [pp2425] Another scientist keenly interested in the problem was Christiaan Huygens, the son of Dutch musician, poet, and statesman Constantijn Huygens. In addition to proposing the wave theory of light, inventing the pendulum clock, and discovering the rings of Saturn, Christiaanwho was proficient on the lute, flute, and harpsichordused logarithms to calculate the division of the octave into thirtyone equal parts, which, he claimed, satisfactorily produced all the pitches anyone might need to play any piece. [p185] ... For Newton, who wrote in 1675 that "perhaps the whole frame of nature may be nothing but various contextures of some certain ethereal spirits," music, light, and the planets were simply different constituents of one eternal, divine harmony. Why shouldn't the laws of gravity be the same as the laws governing musical strings.? [p194] 

Temperament [G. Temperatur]. The term denotes those systems of tuning in which the intervals deviate from the "pure," i.e., acoustically correct intervals as used in the Pythagorean system and in Just intonation. . . . compromise[s] . . . which, instead of being perfect in the simple keys and intolerably wrong in the others, spread the . . . inaccuracy over all the tones and keys. [Apel, Willi, Harvard Dictionary of Music, Harvard University Press, Cambridge, MA 1964, p734] 

Alain Danielou, Music and the Power of Sound, Inner Traditions, Rochester, Vermont, 1995


Defects of Equal Temperaments Bar Chart History of Tuning & Temperaments


DEFINITION OF TUNING TERMS: "EQUAL TEMPERAMENT" ••19 TONE EQUAL TEMPERAMENT / NEIL HAVERSTICK (VIRTUAL CHAUTAUQUA.COM) ••MY 31TONE GUITAR BY JOHN S. ALLEN INTERVALS IN 31 NOTE EQUAL TEMPERAMENT ABOUT 31 TONE EQUAL TEMPERAMENT BY PAUL RAPOPORT CHRISTIAAN HUYGENS (16291695) •


More Complex ScalesWe have already seen that the complexities of the present scale centre in the fact that 12 fifths are not exactly equal to 7 octaves. Let us first examine whether we can replace the numbers 12 and 7 in this statement by others which will reduce the degree of inexactness. Using the method of continued fractions, we find that the following are increasingly good approximations to the ratio of the intervals of a fifth and an octave.
Each of these approximations is the best that can be obtained without extending the scale beyond the number of notes it contemplates, so that if the only problem was that of reducing the comma of Pythagoras to a minimum, the logical stoppingplaces would be at 12, 41, 53 and 306 notes to the octave. This is, however, far from being the whole problem: we want a scale which is rich in 5:4 consonances (major thirds) as well as in 3:2 consonances. Now in the various scales just mentioned, the number of notes which constitute the exact 5:4 consonance are found to be 3•86, 13•20, 17•06 and 98•51 respectively. The only scale which is even as good as the present 12note scale in this respect is the 53note scale. On this the present "fifths" are replaced by intervals of 31 notes, the tuning being almost perfect, while the present "major thirds" are replaced by intervals of 17 notes, these being flat by only a seventieth part of a present semitone. . . . We have already seen that the present 12note scale has its roots embedded very deeply in the unalterable properties of numbers; we now find that music will have to go very far before finding a better scale. But a 53note scale would give far purer harmonics than the present scale, and we can imagine future ages finding it worthy of adoption, in spite of all its added complexitiesespecially if mechanical devices replace human fingers in the performance of music. For, in the last resort, our limited scales have their origin in the limitation of our hands. Yet, if ever music becomes independent of the human hand, may not the race then elect to use a continuous scale in which every interval can be made perfectas with the unaccompanied violin of today? [Sir James Jeans, Science & Music, Dover Publications, Inc., New York, 1968 (reprint from 1937, Cambridge University Press,) pp188190] 



JUNE
12, 2003
