Degrees, or Tones Musicall

For 2 causes chiefly are Degrees required in Musick;

1 That by their assistance a Transition may be made from one Consonance to another, which cannot, so conveniently, be effected by Consonances themselves with Variety, the most gratefull thing in Musick:

2 That all that space, which the sound runs over, may be so divided into certain intervals, as that the Tune may alwayes passe through them more commodiously than through Consonances.

If we consider them in the first capacity; there can be only Four kinds of Degrees, and no more: For then they ought to be desumed from the inequality, found between Consonances, and all Consonances are distant each from other $\frac{1}{9}$ part, or $\frac{1}{10}$, or $\frac{1}{16}$, or finally $\frac{1}{25}$ [36];

Besides the intervals which make other Consonances: therefore all Degrees consist in those numbers, the two first Tones whereof are called Major and Minor, and the two last are called Semitones, Major and Minor.

But we are to prove that Degrees, considered in this capacity, are generated from the inequality of Consonances; which is thus done. So often as there is a transition made from one Consonance to another, either one Term is moved single, or both together; and by neither of these two ways can any such transition be made, unless by those intervals, which design the inequality betwixt Consonances: Therefore. The first part of the Minor is thus demonstrated.

Let from $A$ to $B$, be a Fifth; and from $A$ to $C$, be a Sixth Minor; and, of necessity, from $B$ to $C$ will be that difference, which is betwixt a Fifth and a Sixth Minor, viz. $\frac{1}{16}$, as is evident [38]:

but that the Posterior part of the Minor may be proved, wee are to observe; that wee are not, in sounds, to regard only the proportion while they are emitted together, but also while they are emitted successively, so that, as much as possible, the sound of one Voyce ought to keepe Consonance with the immediately præcedent sound of the other voyce; which can never bee effected, if the Degrees did not arise from the inequality of Consonances. For Example, let $DE$ be a Fifth, and let each Term be moved by contrary motions, so that a Third Minor may be created; if $DF$ be an intervall, which doth not arise from the inequality of a Fourth to a Fifth, $F$ cannot, by relation, be consonant to $E$; but if yea, then it can: and so likewise in the rest, as may soon be experimented. Here observe, that as concerning that Relation, we sayd it ought to be consonant so much as possible: for alwayes it cannot be, as will appeare in the succeeding Discourse.

But if we consider them in the second Capacity; namely, how these Degrees may, and ought to bee ordained in the whole intervall of sounds, that by them one solitary voyce may be immediately elevated, or depressed; then, among the Tones already found out, those Degrees shall only be accounted Legitimate, into which the Consonances are immediately divided. To the manifestation of this, wee are to advert, that every intervall of sounds is divided into Eighths, whereof one can by no means differ from another, and therefore that it is sufficient, if the space of one Eighth be so divided as that all the Degrees may be obtained: as also, that that Eighth is already divided into a Ditone, a Third minor, and a Fourth [39], all which evidently follow from what wee have sayd concerning the last Figure of the Superior Tractate.

Hence also is it manifest, that Degrees cannot divide a whole Eighth, unless they divide a Ditone, a Third minor, and a Fourth; which is thus done. A Ditone is divided into a Tone major, and a Tone minor [40]; a Third minor is divided into a Tone major, and a Semitone majus [41]; a Fourth, into a Third minor, and also a Tone minor [42], which Third is again divided into a Tone