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Ficino, Marsilio - Commentary on Plato’s Timaeus

On harmony, ratios and proportions

It is said that when Pythagoras had observed in the smithies that harmony issued from the hammer-blows by a law of weights, he gathered the numbers which held the self-harmonising difference of the weights. Then he is said to have tautened strings by tying on the various weights which he had discovered in the hammers. And, as a result, he is said to have clearly thought that from a string which was tautened more than another according to the sesquioctaval ratio another tone was discerned by contrast, that is, a full, complete sound, as with the ratio of nine to eight.

Nor was he said to be able to divide the tone into two equal half-tones, since nine does not divide into two equal parts; but he discovered that one half-tone is slightly more than half while the other is slightly less than half; this he called diesis, while Plato called it limma.

How much the halftone was less than the full tone and away from the true half-tone, Plato shows in the difference between the numbers two hundred and forty-three and two hundred and fifty-six. For since an eighth of the smaller number is thirty and nearly a half and since the larger number is thirteen more than the smaller number, it produces neither a tone nor a full half-tone.

Pythagoras next observed that what I might call the full and complete breadth of the tone consists of two sounds and an interval, and thus he arrived at the first elements of harmony.

The first of these harmonies, the diapason, is based on the first or double ratio, where the first string had twice as much tension as the second because of the double weight on it; when both were struck, the first resumed its rectilinearity twice as energetically and twice as quickly and sounded a far higher note than the lower one, yet one that was so friendly to the other that it appeared as a single sound, more restricted in one respect but fuller in another. By regulating and measuring the first sound when it happened to be higher, he ascertained that it is positioned at the eighth step above the note that is accounted low and that, as a rule, it consists of eight notes, seven intervals, and six tones.

Moreover, when the ratio of tension and slackness between the two strings was that of one to one and a third, he found the diatesseron harmony, comprising two tones, a minor half-tone, four notes, and three intervals.

In the ratio of one to one and a half he found the diapente, and in this he noticed three tones and a smaller half-tone, five notes, and four intervals.

Then he observed that the diapason consisted of the diatesseron and the diapente,, for the double ratio which produces the diapason, where the mean occurs, is composed of the sesquitertial and the sesquialteral, of which these consist. For the first double having a mean is of two in relation to four, composed of the sesquialteral between three and two and of the sesquitertial between four and three.

But from its triple appearance he considered the diapason diapente, having twelve notes and eleven intervals. For the diapason is based on the double, while the diapente is based on the sesquialteral. But the unbroken sesquialteral of the double produces the triple. For six is the double of three. Consider nine, from which is born the sesquialteral of six, and you immediately have the triple of three.

However, it is not a triple if you add a sesquitertial to the double: for example, if you compare eight to six; for the sesquioctaval is missing, that is, the relation of nine to eight.

And since the sesquitertial, when added to the double, is superpartient, as in the ratio of eight to three (for here there is a double superbipartient relation through the bisesquitertial), Pythagoras forbade any continuation beyond the double, that is, the diapason through the sesquitertial (diatesseron), although Ptolemy sometimes admits this addition.

But the sesquitertial was agreeably added to the triple, for hence comes a quadruple proportion, and through the quadruple the disdiapason harmony, having fifteen notes and fourteen intervals; for example, if a triple is placed alongside nine and three, you will reach twelve, which is the sesquitertial of nine, and you will at once obtain the quadruple born of twelve in relation to three.

But for the sake of melody he forbids any progression beyond the quadruple, not only because violence invades the senses from the more vehement movement and from the broken sound, but also because as soon as the quintuple is born between five and four beyond the first quadruple, there immediately arises a superbipartient between five and three which produces dissonance.

But so that we may not go beyond the quadruple, he also prohibits a frequent descent below the sesquitertial, for the sake of avoiding heaviness. He forbids the frequent continuation of two sesquitertials, for two reasons: they are generally displeasing, and they do not complete a double or a diapason.

If you take nine, twelve, and sixteen, you have two sesquitertials, but you certainly do not have a double, for there is no sesquioctaval – that is, the ratio of eighteen to sixteen - so that the double is born from the ratio of eighteen to nine.

Nor does he like to continue a pair of sesquialterals, for when they are continued they exceed the double by the sesquioctaval ratio. This you will be able to observe if you write down the numbers four, six, and nine.

For here there are two sesquialterals, from four to nine. Nine exceeds eight, which is the double of four, by one-eighth. From this it is clear that the power of the sesquialteral is greater than half the double.

It is also clear that the power of the sesquitertial is less than half by just the same amount, since the double is created from the two of them together. Indeed, the sesquialteral exceeds the sesquitertial by just one-eighth. For eight to six gives you a sesquitertial, while nine to six gives you a sesquialteral. But nine to eight gives a sesquioctaval, and you can see that the sesquialteral exceeds the sesquitertial by this one-eighth part.

However, it should be remembered that the diatesseron - the arrangement of four notes rising to the fourth step heard through itself - is not approved, but is happily taken from the one which, with the addition of a tone, easily becomes the diapente, that extremely pleasing harmony of the fifth note. Again, if it is connected to the diapente, it produces the diapason - the most perfect harmony of all.