De institutione musica II.8: Regulae quollibet continuas proportiones superparticulares inveniendi.
It often happens that one who discusses music inquires into three or four or any number of equal superparticular proportions. But lest, doing this by chance and ignorance, any error of difficulty impede, by this rule we shall draw any number of equal proportions from multiplicity. Each multiple, reckoned from the unit, precedes as many superparticular relations, of the name contrary to its own, as its own place is distant from the unit, in this way: the double precedes sesquialters, the triple sesquitertians, the quadruple sesquiquartans, and so on in this manner. Let the description below, then, be of double terms.
In the description above, the first multiple, two, has one number, three, to itself, which can make a sesquialter proportion. But three has none that can be sesquialter to it, because it lacks a half. Again, four is the second double. It precedes two sesquialters, six and nine; but nine lacks a half, and therefore none is compared to it in the sesquialter relation. And in the rest it is the same. The triples in the same manner create sesquitertians. Let a like description be made in the triple:
In the description above we see the sesquitertian proportions so born that the first triple precedes one sesquitertian, the second two, the third three, and the third part always closes in the last number as by a certain natural end. If you set the quadruple, you will find in the same way the sesquiquartans; if the quintuple, the sesquiquintans, and so on. Each multiple by name precedes as many superparticulars as the place at which it is distant from the unit. We shall set down the arrangement of the quadruple only, so that in it, as in the rest, the careful reader may exercise the keenness of his mind.
This speculation seems to have been invented for this use, that whenever anyone wishes to investigate four or any number of sesquialters or sesquitertians or sesquioctaves or any other proportions, he may not be led into any error; and that he may not seek to fit such proportions to the first number, which cannot precede and have after it as many as are proposed, but may rather set out the multiples and see how many superparticulars he requires, and look to that multiple which is at that place distant from the unit; as in the descriptions above, if perhaps he has sought three sesquialters, he should not begin the investigation from four (for this, since it is the second double, precedes only two, and he cannot fit a third to it), but should try to set the halves from eight. For this, since it is the third, will make the three sesquialter proportions he seeks. And in the rest in the same manner. There is also another way of increasing proportions, in this manner. The roots of proportions are called, in the same comparisons, the least proportions. Let the natural number, lacking the unit, be set out: 2, 3, 4, 5, 6, 7. The least proportions are thus the sesquialter 3 to 2, the sesquitertian 4 to 3, the sesquiquartan 5 to 4, and so on without limit, and whatever proportions have preceded by a unit. Let it be proposed to produce two sesquialter proportions by a continuous comparison. I take the sesquialter root and set it out: 2 and 3. I multiply two by two, making 4. Let three increase by two; there will be 6. Again we multiply three into itself; it makes 9. Let them be set out in this manner:
We shall find, then, the two proposed sesquialter proportions, 6 to 4 and 9 to 6. Let it now be proposed to find three. I set out the same numbers which I had set above in seeking two sesquialter relations, and the sesquialter proportions themselves. I multiply four by two, making 8; again six by two, making 12; again nine by two, making 18; again nine by three, making 27. Let them be set out in this manner:
And this manner will be the same in the rest. If you wish to extend sesquitertian proportions, set the roots of the sesquitertians, which are four and three compared with each other. And multiply in this manner. If you set out the roots of the sesquiquartans, and by the same multiplication you may extend any number of sesquiquartans. How much these considerations profit us, the following order will show.