( 7°i )

Now, äccording to this Computation, it is manifest, That an OBave isfömewhat iels than Six Füll Notes. For (as was Erst demonstrattd by Eu-aik-, and fince by othcrs ( the Proportion of 9 to 8, being fix Times com-

poundcd, is lornewhat morc than that of 2 to 1.

For

v 9_ 9_ „T3M4I

8 8 262144.’

is more than

524288

262144

2

I

This being the Cafe • they aliuwed (indilputably ) to that of thc Din-4 cuticlTone (In »>i ,) the füll Proportion of 9 to 8, as a thing not to be al-tcred; being the Difference of Dia-peute und Din-teJJhon, Or th e Fiftb andFourtb.

All the Difficulty, was, How the remaining Fourtb (mi,fn, jol, In,) shonldbe divided into three part.;, so as to anfwer ( pretty near ) the ArifloxeninusTwo Tones and a halt: and tnight, altogether malte up the Proportion of4 to 5 5 which is that of a Fourtb or Din-tcfferon.

Many Attempts were made to this purpole: And according to thofc, theygave Names to the Disterent Genera or Kinds of Mustek., (the Dintonick,Chromnticke, and Ertnrmonus Kinds,) with the fevtral Species , or leiser Distin-ctions under thole Generals.

The first was that of Euciide ( which did nrost Generally obtain for manyAges:) Which allows to fa,ß!, and to ßl, In, the füll Proportion of 9 to 8 ,And therefore to fa,foI, In, (which we call the Grcnter Third) that ot 81 to

64- (kor

8i_

64

And, consequently, to that of Mi, fn, (which

« the Remainder to a Fourtb ) that of 256 to 24z.

For

8jt_ _4 2.56

64 3 2.43

that is, if out of the Proportion of 4 to z, we talee that of 8 t to 64, theRefult is that of 256 to 24z. To this they gave the Name of Limmn( u* ) that is, the Remainder (to wit, over and above two Tones. )But, in common Difcourse ( when we do not pretend to fpeak nicely, ncrintend to be sö underlfood. it is uliial to call it an Hemi-tone , or Ilnlf-Note, (asbeing very near it ) and the other, Two iVhole Notes. And this is what Pto-lemy calls Dintonum Ditonum, (of the Dintomck. kind with Toto Füll Iones.)

Against this, it is Objected ( as not the most convenient Division ) thatthe Numbers of 81 to 64, are too great for that of a Ditonc, or GrcnterThird-, which is not Harsli to the Kar; but is rathet Sweeter than that of aSingle Tone, whofe Proportion is 9 to 8. And in that of 256 to 24;, theNumbers are yet much greater. Whereas there are many Proportions ( a 3

—, —, —, —,) in smaller Numbers than that of 9 to 8 j of which,

4567

in this Division, there is noNotice taken.

To