( 6?8 )

.(Zeeoscl *5 Noce in the Ta-T First ( c , .

The ^ Third ble, differs from S>Second «rby S»f >°‘ 1 5 tr,n & -

CFourth, &c.j the J Third, (sic. C j 4 J ^ C '

The Divißon 'osthe Mono-chord ; ly D r,Z. Waliis,n. 238, v. 80.

III. Any String or Chord oF a Afusical Instrument Ofen (oratit’s fulllength)will Sound (what we call) an Oft ave (or Diapason ) to that of the fame'.String stopt in the Middle, or at half it’? Length. Hencc it .is. that we com-monly affign, to .an O clave, the Duple Proportion ( or that of 2. to 1 ) be-cause such is the Proportion of Lengths ( taken in the fime String ) whichgive thole Sounds. And (upon a like Account) we affign to a Fiftb ( orDiapente ) the Sespui-alter Proportion ( or that of.3 to 2. ) And to a Fourtb( or Dia-teffcron ) the Sesqui-tertian for that of 4 to 3.) And to a Tone(which is the Difference of a Fourtb and Fiftb. ) The Sesqui-oftave ( or thatof 9 to 8: ) Because Lengths ( taken. in the fima. String ) in these Propor-tion?, do give such Sounds.

And ( univerfally ) whatcver Proportion of Lengths ( taken in the fameString equally stretched) do give such and such Sounds; Pich Proportion s, of Gravity ) we affign. to the Sounds fo given. .

But when an Eight {oy Oft ave) isfidd (in common Speech) to consillof12 Hemi-tones, or 6 Toties this. is not to be understood according to the Ut-most Rigour of Mathematical Exactneß, (of (uch 6 Toties, as what theycallthe Dia^enticl{ Tone, or that of Ia, mi, which is' the-Difference 'of~-a Fourtb and-VFiftb ; ) but, as exact enough for common Ufe. For 6 such Tones ( that is,the Proportion of 9 to 8, 6 times repeated ) is lbmewhat more .than that of .an Oftave ( or the Proportion of 2 to 1.) : And, confequently, such an He-mi-tone, is lbmewhat more than. the Twelftli-part of an Eigbt, or Oftave, or vDiapason. But the-Difference is lb little, that the Far can hardly distinguistiit : And therefore (in common Speech.) it is usual so to lpeak.

And, accordingly, when we are directed to take the Lengths ( for what are .called the 12 Hemi-tones ) in Geometrical Proportion it is to be understood( not, to be lb iri the utmost Strictnels, but ) to be accurate enough forcommon Ufe ; for placing the Frets on the Neck of a Viol, or other MusicalInstrument ; wherein a greater Exactneß is thought not neceffary. .And this.isvery convenient, becaufe (thus ) the Changc of the Isiy ( upon alteringthe .Seat of »») gives no new Trouble,. for this doth indifterently ferve anyTfy • and the Difference is so -sinall, as not to offend theEar.

But thofe who choose .to treat of it with more Exactneß, gothis way towork.

Prduppofmg the Proportion for an -Octave. ( or Dia-pafon ) to be that of 2.to 1 ; they divide this into two Proportions ; not justEqual (for that wouldfall upon Surd Numbers, as V' 2 to t ; ) but near eu.ua! ( so as to be ex-preffed. in finali Numbers.) In order to which, instead .of taking. 2 to 1, theytake, (the Double of rhese Numbers) 4 to 2; (which is the fame.Proportion as befbre; ), and interpole, the Middle Number 3. And of thesechree Numbeta, 4, 3., a* that of 4.10.3,- is the. Proportion of a Fourtb (-or*JDia-teJseron. ) And that of 3 to 2, the Proportion for a.. Fiftb (or L»ch-'-

pente. ) ^