The Harmonies of the World by Johannes Kepler

p. 1028

All these are simple intervals. When one or more octaves are added to any simple intervals the resultant interval is a "compound" interval.

1 : 3 equals ½ × ⅔—an octave and a perfect fifth

3 : 32 equals (½)³ × ¾—three octaves and a perfect fourth

1 : 20 equals (½)⁴ × (16/20)—four octaves and a major third

Concords: All intervals from diesis downward on above list.

Consonances: Minor and major thirds and sixths, perfect fourth, fifth, and octave.

"Adulterine" consonances: sub-minor third, ditone, lesser imperfect fourth and fifth, greater imperfect fourth and fifth, imperfect minor sixth, greater major sixth.

Dissonances: All other intervals.

Throughout this work Kepler, after the fashion of the theorists of his time, uses the ratios of string lengths rather than the ratios of vibrations as is usually done today. String lengths are, of course, inversely proportionate to the vibrations. That is, string lengths 4 : 5 are expressed in vibrations as 5 :4. This accounts for the descending order of the scale, which follows the increasing numerical order. It is an interesting fact that Kepler's minor and major scales are inversions of each other and hence, when expressed in ratios of vibrations, are in the opposite order from those in ratios of string lengths:

An arbitrary pitch G is chosen to situate these ratios. This g or "gamma" was usually the lowest tone of the sixteenth-century musical gamut.

ELLIOTT CARTER, JR.

1032:1 cf. Footnote to Intervals Compared with Harmonic Ratios, p. 1026.

1034:1 The choral music of the Greeks was monolinear, everyone singing the same melody together.—E. C., Jr.

1034:2 In plain-song all the time values of the notes were approximately equal, while in "figured song" time values of different lengths were indicated by the notes, which gave composers an opportunity both to regulate the way different contrapuntal parts joined together and to produce many expressive effects. Practically all melodies since this time are in "figured song" style.—E. C., Jr.