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do together make-up Dia-pason; that is (as we now speak) a Fourth and Fifth do together make an Eighth or Octave: And, the Difference of those two (of a Fourth and Fifth) they agreed to call a Tone; which we now call a Whole note.
Such is that, (in our present Musick,) of La Mi, (or as it was wont to be called, Re Mi.) For La fa sol la, or Mi fa sol la is a perfect Fourth: And La fa sol la mi, or La mi fa sol la is a perfect Fifth: The Difference of which, is La mi which is, what the Greeks call, the Diazeuctick Tone which doth Dis-join two Fourths (on each side of it;) and, being added to either of them, doth make a Fifth. Which was, in their Musick, that from Mese to Paramese; that is in our Musick from A to B: supposing Mi to stand in B fa b mi, which is accounted its Natural position.
Now, in order to this, Aristoxenus and his Followers, did take, that of a Fourth, as a Known Interval, by the judgement of the Ear; and, that of a Fifth, likewise; And consequently, that of an Octave, as the Aggregate of both; and that of a Tone as the Difference of those Two.
And this of a Tone (as a known Interval) they took as a common Measure, by which they did estimate other Intervals. And accordingly they accounted a Fourth to contain Two Tones and an half; a Fifth to contain Three Tones and an half and consequently an Eighth to contain Six Tones or Five Tones and two Half-tones, And it is very near the matter, though not exexactly so.
And at this rate we commonly speak at this day; supposing an Octave to consist of Twelve Hemitones, or Half-notes. (Meaning thereby, somewhat near so many half-notes:) But, when we would speak more Nicely, we do not take those supposed Half-notes to be exactly Equal or each of them just the Half of a Full-note such as is that of La-mi.
Pythagoras and those who follow him, not taking the Ear alone to be a competent Judge in a case so nice; chose to distinguish these, not by Intervals, but by Proportions. And accordingly they accounted that of an Octave, to be, when the degree of Gravity or Acuteness of the one Sound to that of the other, is Double, or as 2 to 1; that of a Fifth, when it is Sesqui-alter, or as 3 to 2; that of a Fourth when Sesqui-tertian, or as 4 to 3 . Accounting That, the Sweetest proportion, which is expressed in the Smallest Numbers; and therefore (next to the Unisone) that
