( 7 °« )

Now, concerning this, there were amongst the Ancicnt Greeky, Two (themoft considerable ) Sects of Afustcans the Arißoxenians, and Pythagoriam.

They Both agreed thus sar; that Dia-tcffcron and Dia-fentc, dp togethcrmalte up Dia-pafon that' is ( as we now /peak) a Feurth and Fifth ) do to-gether make an Eighth or Oftavc .- And, rhe Disterence of thole two of aFourth and Fifth , they agreed to call a Tone , which \ve now call a WaoleKote.

Such is that, ( in our prefent Musicis ) of la nn , ( or as it was wont to becalled rc, mi. ) For la, fa, sol, la, or mi, fa, sol, la, is a perfect Fourth AndIa, fa, sol, la, mi, or la, mi, fa, sol, la, is a perfect Fifth The Disterenceof which is la, mi, Which is, what the Greckj call, the Dia-^euticf Tone ;which doth Disjoin two Fcurrbt (on each fide of it;) and being added to ei-ther of them, doth make a Fifth-, Which was, in th^ir Mustek,, that fromMese to Paramese, that is in our Mustek., from A to B: (üpposing mi to standin B-fa-b-mi, which is accounted its Naturas Position,

Now in order to this, Aristoxenus and bis Followe^s, did takc, that of a<Feurth, as a Known Jntcrval, by the Judgment of the Ear; and that of aFifth, likewife; And consequcntly that of an Öitavi, as the Aggregate ofboth; and that of a Tone, as the I/istcrence of thole Two.

And this of a Tone ' as a Known Interval) they cook as a common Mca-lüre, by which they did Ellimate other Intcrvah. And acccordingly theyaccounted a Fourth to contain two Tones and an Half; a Fifth to contain ThresTones and an Half, and conlcquently an Eigth to contain Six Tones ; or Fiv-e •Toms and two Half Toms.

And at this Rate our PraElical Mußeians talk of Kotes and Half Kotes asthis Day; lupposing an Oclavc to consist of Twelve Hcmi-tones , or HalfNotes,

But Pythagoras and thole who follow him, not taking the Ear afone ro be»a Competent Judge in a Cafe lc> nice; cho/eto distingiiilh thefe, not by EquatIntel vals, but by Due Proportions : And this is followed by farline, Kopier,

C. artes and others, who treat of Speculative Mustek, in this.and tlve last Age,-Accordingly they accounted that of an Oclavc, to be* when the Degree ofGravity or Acutcness of the one Sound to that of the other, is Double, oror as i to i; that of a Fifth, when it is Scs.jui-alter, or as 5 to i; that of'a Fourth when Scfqui-tcrtian, or as 4 to 3. Accounrrng that the SweetefHProportion, which is exprest in the Smallcst Numbers, and therefore (next^to the Unljon ) that of an Oclavc, 2. to 1, then that of a Fifth, 3 to-2,and than that of a Fourth, 4 to 3.

And thus that of a Fourth and Fifth, . do togethcr make an Eighth ; For-y-x ~ — = 2, or the Proportion of 4 to 3, compounded i

with that of 3 to 2, is the fame wirft that 0f4.ro 2, or 2 to 1, And T con-fequcrttly, ehe Disterence of thole two, which is that d a Tone, or Füll Not«;.,

L that of y to 8. For —) — (-T ; or, ifout of the Proportion of 3 to z 3 -3 - 0

we take that of 4 to 3 ; the Reluit is that of ci to 8