The Positions in the System, or the Notes of the Musical Scale

Therefore, that between these 12 terms or motions of the 6 planets which revolve round the Sun there exist upwards, downwards, and in every direction proportions which are harmonic, or very close to such within an imperceptible fraction of the smallest melodic interval, has been proved so far by numbers which have been sought in the former case from astronomy and in the latter from harmony.

Book 3 Chapter 1 first extracted the individual harmonic proportions separately. Chapter 2 we assembled all that there were of them into one common system or musical scale.

• We divided one diapason of them which embraces the remaining ones in its dominion, through those remaining ones into steps or positions to produce a scale
• In this way, when we have found the harmonies which God embodied in the world, we next try to see whether the individual harmonies stand separately to answer:
  • Do they have no affinity with the rest?
  • Or do they all agree with each other?

These harmonies are fitted together so that they support each other mutually as if within a single structure.

• No single one clashes with another, inasmuch as we see that in such a many-sided comparison of their terms harmonies never fail to occur.

If all were not fitted to all to form a single scale, it could easily have come about (as has happened here and there, when necessity is so pressing) that several dissonances occurred.

Thus if anyone established a major sixth between the first and second term, and between the second and third a third, also major, without regard to the previous interval, in that case he would be admitting between the first and the third a dissonance and an unmelodic interval, 12-25.

Come now, let us see whether what we have already inferred by reasoning is in actual fact found to be so. However, let us preface this ___ with some words of caution, to avoid our being obstructed — while the inquiry is in progress. First, we should for the • present overlook those excesses, or deficiencies, which

are less than a semitone; for we shall see later what causes them. Next by repeated doubling, or on the contrary halving, of the motions, we shall bring them all within a system of a single octave, because of the identity of sound of every diapason.

Therefore, the numbers by which all the positions or notes of the system of the octave are expressed are set out in the table in Chapter 8 of Book 3, on page 197.

Those numbers refer to the length of the pairs of strings. In consequence, therefore, the speeds of the motions will be to each other inversely in those proportions.

Now let the motions be compared, in parts obtained by continuous division by two.

Now let the motion of Saturn, the slowest planet, at aphelion, that is the slowest motion, represent the lowest position in the system, G, numerically I minute 46 seconds. Then the motion at aphelion of the Earth will also represent the same, but five diapasons higher, because it is numerically I minute 47 seconds; and who would venture to argue about one second in the motion of Saturn at aphelion?

However, let it stand: the difference will not be greater than 106:107, which is less than a comma. Of this 1 minute 47 seconds, if you add a fourth part, 27 seconds, the total will be 2 minutes 14 seconds, whereas the motion of Saturn at perihelion comes to 2 minutes 15 seconds.

Similarly, for the motion of Jupiter at aphelion, but one diapason higher.

Therefore, these two motions represent the note or are very slightly higher. Take a third of 1 minute 47 seconds, 36 seconds —and add it to the whole: you will generate 2 minutes 23 seconds —standing for the note c\ and look, it is the motion of Mars at perihelion, of the same magnitude, but four diapasons higher.

To the same 1 minute 47 seconds add also a half, 54 seconds —the result will be 2 minutes 41 seconds - , standing for the note d’, and look, here to hand is the motion of Jupiter at perihelion, but one diapason higher: for it takes a number which is very close, that is 2 minutes 45 seconds.®^ If you add 2/3, that is 1 minute 11 seconds -i-, the sum is 2 minutes 58 seconds + . And look, the motion of Venus at aphelion is 2 minutes 58 seconds - .

Therefore, this represents the position or note e, but 5 diapasons higher; and the motion of Mercury at perihelion does not greatly exceed it, having 3 minutes 0 seconds, but seven intervals of a diapason higher.®® Lastly, divide twice 1 minute 47 seconds, that is 3 minutes 34 seconds, by nine, and subtract one ninth, 24 seconds, from the whole.

The remainder is 3 minutes 10 seconds + , standing for the note /,®® which is nearly represented by the motion of Mars at aphelion, 3 minutes 17 seconds, but three diapasons higher; but the actual number is a little greater than it should be, coming close to the n o t e F o r taking a sixteenth of 3 minutes 34 seconds, that is 13| seconds, from 3 minutes 34 seconds leaves 3 minutes 20^ seconds, to which 3 minutes 17 seconds is very close.

In music also fg is often used in place o f/ , as may be seen everywhere.

Therefore, all the notes in hard music within a single octave (except for the note A, which was not represented by the harmonic divisions in Book 3, Chapter 2, either), are represented by all the extreme motions of the planets, except for the motions at perihelion of Venus and the Earth, and the motion at aphelion of Mercury, for which the number is 2 minutes 34 seconds which is close to the note eg. For subtract from d, 2 minutes 41 seconds, a sixteenth, 10 seconds + : the remainder is 2 minutes 30 seconds, for the note eg.

Thus only the motions at perihelion of Venus and the Earth are outcasts from this scale, as you see in the figure.

On the other hand, if the motion of Saturn at perihelion, 2 minutes 15 seconds, is made the start of the scale, and it is directed that it should represent the note G; then the note A fits 2 minutes 32 seconds - , which is very close to the motion at aphelion of Mercury.

The note b fits 2 minutes 42 seconds, which is very nearly the motion at perihelion of Jupiter, by the equivalence of octaves.

The note c fits 3 minutes 0 seconds, the motion at perihelion of Mercury and Venus, very nearly.®^ The note d fits 3 minutes 23 seconds —and the motion of Mars at aphelion is not much lower, that is 3 minutes 18 seconds.

Thus this number is smaller than its note by almost the same amount as that by which previously it was in a similar way greater than its note.

The note dg fits 3 minutes 36 seconds, which the motion of the Earth at aphelion almost meets;®^ the note e fits 3 minutes 50 seconds and the motion of the Earth at perihelion is 3 minutes 49 seconds.

However, the motion of Jupiter at aphelion again occupies g.

On this basis all the notes within one octave of soft music, except for /, are expressed by most of the motions of the planets at aphelion and perihelion, especially those which had been left out previously, as you see in the figure.

On the other hand, if the motion of Saturn at perihelion, 2 minutes 15 seconds, is made the start of the scale, and it is directed that it should represent the note G; then the note A fits 2 minutes 32 seconds - , which is very close to the motion at aphelion of Mercury.

The note b fits 2 minutes 42 seconds, which is very nearly the motion at perihelion of Jupiter, by the equivalence of octaves.

The note c fits 3 minutes 0 seconds, the motion at perihelion of Mercury and Venus, very nearly.®^ The note d fits 3 minutes 23 seconds —and the motion of Mars at aphelion is not much lower, that is 3 minutes 18 seconds.

Thus this number is smaller than its note by almost the same amount as that by which previously it was in a similar way greater than its note.

The note dg fits 3 minutes 36 seconds, which the motion of the Earth at aphelion almost meets;®^ the note e fits 3 minutes 50 seconds and the motion of the Earth at perihelion is 3 minutes 49 seconds.®® However, the motion of Jupiter at aphelion again occupies g.

On this basis all the notes within one octave of soft music, except for /, are expressed by most of the motions of the planets at aphelion and perihelion, especially those which had been left out previously, as you see in the figure

Previously fg was represented, A was left out Now A is represented,/e is left out, for the harmonic divisions in Chapter II also left out the note

Therefore, there has been expressed in the heaven in a twofold way, and in two, so to speak, kinds of melody, the musical scale, or system of one octave, with all the positions by means of which natural melody is conveyed in music. The sole difference is in the fact thatin our harmonic divisions indeed both ways jointly start from one and the same term, whereas in the latter case in the motions of the planets what was previously now in the soft kind become G

For just as in Music the proportion is 2160:1800, or 6:5, so in the former system, which is expressed by the heaven, the ratio is 1728:1440, that is also as 6:5, and similarly for several other cases:

as:

2160 | :1800 | :1620 | :1440 | :1350 | :1080 as 1728 | :1440 | :1296 | :1152 | :1080 | :864

In fact there also remains another means by which we may grasp the double musical scale in the heaven, in which the system is indeed the same, but the tuning is conceived in twin ways, one according to the motion of Venus at aphelion, the other according to the motion at perihelion.

For the variation in the motions of this planet is of very small extent, inasmuch as it is contained within the size of a diesis, the smallest melodic interval. And the tuning at aphelion indeed, as above, has the motions at aphelion of Saturn, the Earth and Venus, and almost that of Jupiter, at G, e, and I) but the motion at perihelion of Mars, and almost that of Saturn, and as appears at first sight also that of Mercury, at c, e, and whereas on the other hand the tuning at perihelion gives a position also to the motions at aphelion of Mars and Mercury and almost that of Jupiter, but to the motions at perihelion of Jupiter, Venus, and almost that of Saturn, but also to a certain extent to that of Earth, and undoubtedly that of Mercury also.

For suppose that now it is the motion of Venus not at aphelion but at perihelion, 3 minutes 3 seconds, which occupies the position e.

The motion at perihelion of Mercury also approaches very close to it, over a double diapason, by the end of Chapter 4.

However, if of this motion of Venus at perihelion, 3 minutes 3 seconds, one tenth, 18 seconds, is subtracted, the remainder is 2 minutes 45 seconds, the motion at perihelion of Jupiter, which holds the position d\ and the addition of a fifteenth, 12 seconds, gives a sum of 3 minutes 15 seconds which is nearly the motion at aphelion of Mars, which holds the position f And similarly in the case of 1^, the result of almost the same tuning is also the motions of Saturn at perihelion and of Jupiter at aphelion.

But if an eighth part, 23 seconds, is taken five times, it yields 1 minute 55 seconds, which is the motion of the Earth at perihelion.

Although this does not square with the same scale as those which have been mentioned before, inasmuch as it does not include in order the interval 5:8 below e, nor 24:25 above G, yet if now the motion at perihelion of Venus, and similarly also the motion at aphelion of Mercury, take, out of order, the position dQ instead of e, then this motion of the Earth at perihelion will take the position G, and the motion of Mercury at aphelion will also agree.

For a third of 3 minutes 3 seconds, 1 minute 1 second, taken five times, becomes 5 minutes 5 seconds, half of which, 2 minutes 32 seconds + , approaches very closely the motion at aphelion of Mercury, which in this out of order arrangement will occupy the position

Then all these are related to each other in the same tuning.

However, the motion of Venus at perihelion divides the scale in one way, along with the 3 (or 5) earlier ones, in the same harmonic kind, that is to say as the same planet’s motion at aphelion, in its own tuning, meaning in the hard kind;“4 and the motion of the same Venus at perihelion divides the same scale in another way with the two later ones,“® meaning not into different melodic intervals but rather into a different order for the melodic intervals, that is to say the order proper to the soft kind.

The conflict in small details, will be made clear by the most lucid demonstrations in Chapter 9.