"TEMPERAMENT"

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. Equal-tempered fifths, for example, are closer to pure (3:2) than the fifths used in mean-tone 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 equal-tempered 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." . . . [pp24-25]

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, Christiaan--who was proficient on the lute, flute, and harpsichord--used logarithms to calculate the division of the octave into thirty-one 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]

Common Equal Tempered Scales

The Argument For Equal Temperament

Equal Temperament

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

When . . . music, conceived on the piano, is . . . played by an orchestra . . . a new difficulty appears: that of notation. The chords of the piano have introduced into music new intervals that accommodate themselves in some way or other to temperament. But if those chords are to be played on untempered instruments just as they are written, the result may be completely discordant. This is the case with Wagner, for example, whose music, conceived on the piano, was much more dissonant in his time, when the musicians of the orchestra made some distinction between sharps and flats, than it is nowadays when practically all musicians play in the tempered scale. p134

Defects of Equal Temperaments Bar Chart

Definition of Tuning Terms

History of Tuning & Temperaments

Tuning Page

The Scales Page

36 TONE EQUAL TEMPERAMENT

More Complex Scales

We 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 stopping-places 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 12-note scale in this respect is the 53-note 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 12-note 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 53-note 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 complexities--especially 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 perfect--as with the unaccompanied violin of to-day? [Sir James Jeans, Science & Music, Dover Publications, Inc., New York, 1968 (reprint from 1937, Cambridge University Press,) pp188-190]