De institutione musica IV.5: Monochordi regularis partitio in genere diatonico.
V. But it is now time to come to the division of the regular monochord. Concerning this it must first be said that, whether the division to be described is established in the measure of the string or in numbers and their proportion, the greater space of the chord and the greater multitude of number will produce the deeper sounds. But if the length of the string is shorter and the multitude of numbers not great, higher notes must be sounded. And from this comparison, as each is either longer or marked by more numbers, and another is shorter or marked by fewer numbers, so much the deeper or higher is it found. Let it not trouble the reader that in raising we have often marked the spaces of the proportions with the greater number, but in lowering with the smaller, since raising makes the high note, lowering the deep. For there we were marking only the spaces of the proportions, taking no trouble about the property of depth or height, and therefore we often raised to the high with greater numbers and lowered to the deep with smaller. But here, where we shall measure the spaces of the strings and the sounds, we must follow the nature of things and give to the greater length of the strings, from which depth arises, the larger numbers, and to the smaller, from which the height of the voice is born, the shorter. Let the stretched string be AB. Let a rule be equal to it, to be divided by the proposed divisions, so that, the rule being laid beside the string, the same divisions may be marked on the length of the string which we had earlier marked on the rule. We now divide as if we were dividing the string itself and not the rule. Let AB be divided into four parts by three points, which are C, D, E. The whole AB will then be double of DB and of AD, and AD and DB severally are double of AC, CD, DE, EB. AB will therefore be the deepest, that is, proslambanomenos, and DB the mese. For it is half of the whole, and as AB is double of DB in space, so DB is double of AB in height. For, as was said above, the order of space and of height is always inverse. For a chord is so much the greater in height as it is smaller in space. Therefore EB also will be nete hyperboleon, since EB is the half of DB in quantity and the double in height. Again, since the same EB is the fourth part of AB in space, it will be the quadruple of it in height. As has been said, then, nete hyperboleon will be double in height of mese, and mese double in height of proslambanomenos. And nete hyperboleon is quadruple in height of proslambanomenos. Proslambanomenos will therefore sound the octave with mese, mese the octave with nete hyperboleon, proslambanomenos the double octave with nete hyperboleon. Again, since AC, CD, DE, EB are equal parts, and AB is four of those parts, CB three, AB is the sesquitertian of CB. Again, since CB is three equal parts, and DB two, CB will be the sesquialter of DB. Again, since CB is three equal parts, of which EB is one, CB is the triple of EB. CB will therefore be lichanos hypaton diatonos, and proslambanomenos will sound with lichanos hypaton diatonos the consonance of the fourth, and the same lichanos hypaton diatonos will sound with mese the consonance of the fifth, and the same lichanos diatonos will sound with nete hyperboleon the octave and fifth. Again, if from the whole AB I take away a ninth part, which is AF, the parts FB will be eight. FB will be hypate hypaton, to which AB, that is proslambanomenos, holds the sesquioctave proportion, in music a tone.
The figure above, containing the signs below, belongs to this description, where we have set marks to the strings, since it would have been long to write out their names.
Likewise, if we divide AB by three cuts, the third part will be AG; the two parts of it will therefore be GB. AB, proslambanomenos, will therefore sound with GB, which is hypate meson, the consonance of the fifth, constituted in the sesquialter proportion; CB to GB will be sesquioctave and will contain a tone, and this falls in order. For lichanos hypaton diatonos, that is CB, contains a tone with hypate meson, that is GB. Again, AB, proslambanomenos, to CB, lichanos hypaton diatonos, has the consonance of the fourth, and AB, proslambanomenos, to GB, hypate meson, has the consonance of the fifth. Likewise CB to DB, that is lichanos hypaton diatonos to mese, has the consonance of the fifth. GB to DB, that is hypate meson to mese, has the consonance of the fourth. Lichanos hypaton, that is CB, compared with hypate meson, that is GB, will be distant by a tone. If I take the fourth part of CB, there will be CK. CB therefore will hold to KB the sesquitertian proportion. KB will be distant from DB by the sesquioctave proportion. KB will therefore be diatonos meson, and CB, that is lichanos hypaton diatonos, to KB, that is diatonos meson, will contain the consonance of the fourth. Again, if I take the ninth part of DB, I shall have DL. LB will therefore be paramese. If I take the fourth part of DB, there will be DM. MB will therefore be nete synemmenon. If I take the third part of DB, there will be DN. NB will be nete diezeugmenon. If KB is divided into two equal parts, there will be KX, and XB will be paranete hyperboleon.