The origin of third intervals

The origin of the third intervals are not exactly melodic, yet are useful in melody.

They serve the purpose of melodic intervals.

They arise from the subtraction or comparison of melodic intervals (in the same way as melodic intervals from those of consonances).

For between melodic intervals or seconds are the following third intervals:

8/9 and 9/10 ..... 80/81

8/9 and 15/16 ..... 128/135

9/10 and 15/16 ..... 24/25

Which is composed of 24:25 and 80:81 and is a very little less than 15:16. 24 25

To these can be added twice the interval 15:16, or in other words 225:226. This is greater than the interval 8:9 by a little less than 15:16 is greater than the interval 128:135.

The first three indeed arise from the mutual division of various melodic intervals; but this last from the addition of two melodic intervals which are equal, but in a less common way.

Hence springs to notice a very splendid Arithmetical corollary in numbers and in the following diagram: 35 15 1 16 4 ) K 8 i i 36 5 6 25 24 II } k 63 i i 64 1 8 7 9 49 48 . Ji 81 10 j 1 80 That is to say, the square of every number below ten, together with the rectangle having the two numbers which are its closest neighbors

as sides, makes up either a consonance or a melodic interval or a third interval, with the exception of 49, the square of the sevenfold, and its two rectangles 35 and 63. But here the melodic interval 9:10 is ban­ ished, and most of the consonances, except 3:4. It is therefore fortu­ itous, depending on the order of the numbers and the structure of this diagram.

In vain will the arithmetician seek causes from this direction, in vain will the Pythagorean be obsessed by his fascination with the sevenfold, as a counting number: the matter must be pursued more deeply in geometry, and in the counted and figured numbers, that is to say in the inconstructible figures themselves, of which the first is the hepta­ gon.

For what prevents the possibility of the diagram’s being continued beyond ten, with the nature of the melodic interval following, is now no longer the sevenfold, but the other numbers belonging to incon­ structible figures, 9 and 11, which make the rectangle 99, which with 100, the square of the tenfold, makes up an interval which is totally abhorrent to the nature of melody. So great a difference is there be­ tween axioms of conjecture and axioms of knowledge.

The order of the melodic intervals in perfection, and their names.

What are the differences between the intervals which are less than consonances?

• What are their names?
• We could not keep them absolutely the same as the ancients since we differ from them on these matters and on their causes.

Therefore it is in agreement with what has been said above, especially Axiom II, that of those intervals which belong to the nature of the melodic, each one retains the nature of the consonances from which it is established. Therefore, since of the consonances which are smaller than the double interval, the most perfect are 2:3 and 3:4, on account of the nobility of the figures from which they take their origin, their Joint progeny among the melodic intervals also, that is to say 8:9, must be elevated above the others. We shall therefore give this interval, in common with the ancients, the name WHOLE, and on account of this preeminence, we shall call it a perfect tone, but on account of the size of the proportion, a major tone.

On the other hand if you compare the greater perfect interval 2:3 with the greater imperfect interval 3:5 on the higher side, or the smaller of the perfect intervals 3:4 with the smaller of the imperfect intervals 5:6 on the lower side, there will be born from this marriage the rather imperfect melodic interval 9:10. It is smaller than 8:9; and since that interval in ancient music before Ptolemy was generally not mentioned, inasmuch as the theoreticians demon­ strated all intervals through the full tones previously defined, we shall give it the name of minor or little tone, so as to mark its imperfection.

Here let the reader take note, in a word, that some have given that name to another interval, so that if he happens by chance to read them he may not be caught off his guard and confused.

Yet if you link the greater perfect interval 2.‘3 with the smaller imperfect interval 5-8 on the higher side, or the smaller perfect interval 3:4 with the larger imperfect interval 4:5 on the lower side, the melodic interval arising from the comparison, that is to say 15:16, again brings in ____ ______ . . _________ _______ an element of im- perfection from i s 2 5 :^ the fact that this is its origin, and will be called a semitone,^® the same term as is used for this interval in normal present day music, because it is a little larger than half a major tone. Some have maintained that it is a minor tone; but the reader should be wary of them, so as not to be confused.

These three, therefore, arising from perfect intervals by com parison of them either with each other or with imperfect intervals, have acquired the property of being melodic in their own right and always.

On the other hand if you compare with each other the imperfect intervals arising from the pentagon or decagon, either on the higher side 3:5 with 5:8, or on the lower side 4:5 with 5:6, the interval arising from it, that is to say 24:25, is of such imperfection that it almost ceases ^ ———- a slackening of the string. Nor am I at pains to propose by this term the same size of interval as the ancients: and again let it suffice to give notice of that.’’

There are three causes of its imperfection: origin, small size (since it does not equal a third part of a perfect tone), and because it is also listed among the third intervals above, that is to say among those which are serviceable for making tuneful kinds of melody.

For it also arises from the comparison of the minor tone and the semitone. Now this interval is not melodic in its own right nor always; for the human voice does not usually pass over this interval in one and the same dYCoyfj, “ap­ proach,” as it does the other intervals, but it leaves it out and over­ shoots it, with the sole exception of a modulation in the melody, to

dd flavor.

Then it becomes extraordinarily melodic, but in such a way that it begins, so to speak, a new kind of melody; and it requires art and no little toil to achieve that with the human voice without an instrument. Thus this interval only marks the difference between kinds of melodic intervals, and is serviceable for them on that basis.

We have begun to speak of third intervals; for the same interval which is the first of them, 24:25 or a diesis, was also the last of the melodic intervals. There now follow the designations of the remaining ones.

For 128:135 which arises from 15:16 and 8:9 _____ can be designated a major and irregular diesis. As stated above it is a very little (that is to say by the amount of 2025:2048) smaller than the melodic interval of a semitone and is scarcely distinguished from it.

Under this title it is among the melodic intervals, because it plays the part of a legitimate diesis, particularly at a modulation in the melody. For its genesis is both natural and necessary in practice, so that semitones and dieses are available in all directions, for the sake of various flavors of melody. For that reason, when a legitimate semi­ tone is split off from a major tone, and this interval remains, we can designate it too by a Greek name, limma or remainder. Finally we can call the difference, 80:81, intervening between 8:9 and 9:10, a comma, in Latin a segment or cut. For the ancients cut their diesis into four parts, and hence called them commas, believing that this was the common element of all consonances. Now this inter­ val is a little larger than a fourth part of our diesis, and smaller than a third. For 24:25 is 72:75 or 96:100.

Therefore a third part would be 74:75, and a fourth 96:97, about; and 80:81 is between the two. We could define a comma by a closer number as an eighth part of a major tone, that is to say 8:9.

That is also clear as follows. The major tone 8:9 is divided into a diesis, 72:75, a semitone, 75:80, and a 24 comma, 80:81. Now a comma was just found to be $. 7 2 . about a third of a diesis. Therefore about four commas are equal to half a tone, and eight to a whole tone, l 6- approximately indeed, not absolutely. Therefore this interval is plainly not among the melodic ones which P* 8i« are sung in succession, because their small size is scarcely perceptible by the hearing, still less expressible in human melody on their own independently, by two notes in succession.

But it does not cease to be melodic, like 11:12 and similar intervals, because we are also comparing things which are separated in space and time.

However a double semitone must be established because in the division of tones which succeed each other in order, two semitones are sometimes placed in succession; and people occasionally use them combined together as a tone when they are aiming at variety and novelty, to express grave disturbances of the mind.

Note that between the semitone 15:16 and the diesis 24:25 there is 125:128, roughly 42:43 or a double comma. If you add to that a comma 0^81, the result is 625:648, roughly 27:28 or a triple comma.

However the same 80:81 subtracted from a diesis, 24:25, leaves 243:250, which is as nearly as possible 35:36. The same comma subtracted from a semitone 15:16 leaves a Platonic limma, 243:256, which is roughly 19:20; but added to 15:16 it makes 25:27 which is between 12:13 and 13:14.

Thus two major tones, 8:9 combined together make 64:81; and it was by dividing that interval into 3:4 that Plato established his limma.

However on subtracting 243:256 as a limma from a major tone Plato had as a remainder 2048:2187, which he named an apotome;^*^ and it is larger by one comma, 80:81, than our limma, 128:135, and exceeds a semitone, 15:16 very little.

Although these are abnormal intervals, nevertheless mention will be made of some of them below in Book V.