The Origin of the Melodic Intervals Smaller than Consonancesby Johannes Kepler
All the proportions which we have shown to be consonant, with the sole exception of equality, must be taken to represent the same number of intervals which are m like manner consonant; whereas the proportions which we have said to be dissonant represent a like number of dissonant intervals.
However, there is a great difference between the dissonant intervals, so that not only are the consonant intervals taught to us by Nature and approved by hearing at her prompting, but other smaller intervals are also established by the same sense which although they J are dissonant are yet suitable for conveying melody.
Harmony, following Nature, attaches to them the name of melodic, and distinguishes them from the unmelodic, which have no place in the flow of any ordered melody. In Greek they are called in melody” and 8KM,eX,f|, “outside melody.”
When the ancients saw this ingenuity of Nature in distinguishing between the melodic and unmelodic, they therefore thought they should try to find what was the smallest element common to the melodic and the consonant, by taking some number of which any consonance or melodic interval could be made up.
For it seemed necessary that some such smallest interval should exist, as simple, and prior in origin to the consonances themselves, which seemed to be made up of such a smallest element, inasmuch as some intervals were larger than others. Yet the reality is far different, as can be learnt from many examples.
For if in all species the individuals, which differ in size, are made up of one common smallest element, therefore there will be some single smallest quantity of the human species; and from some definite number of striplings of that kind, as if from elements, any man you like may be composed, a lofty one from many, a dwarf from few.
For in harmony the quality known as consonance shapes the proportion of the strings, or the interval of the notes, just as much as the shape of a man shapes that mass of matter which is surrounded by a man’s skin.
Why did they forget geometry, in which there are a great many examples of every kind of incommensurable quantities, which are defined as sharing no common measure whatever, which belongs to quantities of the same kind, as a definite quantity of some element of their composition
Therefore we must accept that consonant intervals (except in cases here One is a multiple of the other) are, like the actual proportions, incommensurable, in such a way indeed that although their differences may be expressed in numbers, which in simple numbers is a sign of commensurability, yet these differences, not of course being simple numbers but fractions, are not an aliquot part or aliquot parts of the differing terms, in relation to any number. For instance the two proportions 1:2 and 1:4 are to each other as the number 1 to the number 2. They are, then, commensurable, for 1:4 is twice T2.
There is room for this in the series of continuous doubling alone. For in the series of triples, and the other multiples, two consonant proportions do not occur.
Thus 1:9 is indeed triple 1:3; but only 1:3 is among the consonances, and T9 is among the dissonances, by Axiom 111. We can see the same thing in non-multiples.
Thus in the case of the sesqui-alterate proportion, 2:3, a consonance, its multiple certainly occurs and is thus commensurable; for 4:9 is to 2:3 as the number 2 is to 1, but 4:9 is not among the consonances.
On the contrary two others are consonant, as in the series of continuous doubles, like T4 and 2:3.
These two proportions are not commensurable with each other, that is, they are not as number to number; for the excess of T 4 over 2:3, 3:8, is not measurable by either T4 or 2:3.
Therefore the consonant intervals are by nature prior to the smaller intervals which we name melodic; and they are not composed of melodic intervals as if of elements, or of some smaller quantity, but on the contrary the melodic intervals arise from the consonances, as if from causes.
At this point we must take note that the word “composition” is ambiguous. Sometimes it denotes the natural origin of a thing, some times however the quantitative division of a thing, which is not an origin, but rather a destruction, as when we say that a circle is made up of 3 thirds, first mentally dividing the circle into three, or when we say that the human body is composed of members, not because the members existed before the body, and the body was assembled and constructed from them, as a house is from stones and wood, but because the body in virtue of its bulk is divisible into these members, which separately and independently are no longer a functional body.
In the former sense we must say that consonant intervals are composed neither of other consonances nor of melodic intervals.
In the latter sense the consonant intervals, which are larger, certainly do consist, and are thus in a sense composed (as we ourselves have assumed previously) of the smaller consonances, and the smallest consonances of melodic intervals, and so on, because they are analyzed into these elements, so to speak; but the various intervals among themselves do not consist of some larger number of intervals of a single very small kind, common to each other, and cannot be analyzed into any such.
However, the consonant intervals also have related causes, yet they do not all have the same cause, but each its own special cause, distinct from the others’ causes, as has been explained previously.
For consonance is a property of the actual intervals, not according to their quantity directly, nor directly according to their relationships, but according to their relationships qualitatively (that is, in a sense, as figured).
Thus to seek to establish a smallest interval which is common to them is inappropriate, since smallest and greatest are observed not in qualities but in bare quantities and in their proportions, whereas to divide consonances, as consonances, is to destroy a kind of consonance, and in its place to establish either other kinds of consonance, or dissonant melodic intervals, or even downright unmelodic intervals.
An interval therefore does not take the causes or elements of its consonance from parts as if they were basic principles, in the same way as commensurable quantities are built up by the multiplication of a common measure, and along with that measure belong to one and the same type.
On the contrary, what the ancients took to be the basic principles of consonances (tones, I mean, and semitones and dieses”^) originate from the consonances as their true basic principles.
For although the consonances do consist of these melodic inter- The type of a vals which are not consonant (if not from a single one in common, consonance as such does not at least from several combined with each other in various ways), yet arise from that must not be attributed to the actual consonance of the interval, number of melodic intervals
For if the melodic intervals imparted to a larger interval, which was composed of them, its own consonance, that would always occur in any multiple of melodic intervals, and the more melodic intervals there were in it, the better would be the consonance. However that is false, for as we shall hear below, two tones combined make a consonance, three combined make up a dissonant interval.
Nevertheless the fact that a consonance can be analyzed into dissonant melodic intervals, as will follow, is clearly accidental to that consonance considered on its own, and occurs only insofar as several consonances are compared with each other, each originating from its own basic principles.