Page:A Letter of Dr. John Wallis to Samuel Pepys Esquire, Relating to Some Supposed Imperfections in an Organ.djvu/3

 of an Octave, 2 to 1; then that of a Fifth, 3 to 2; and then that of a Fourth, 4 to 3.

And thus, that of a Fourth and Fifth, do together make an Eighth; For $4⁄3$ × $3⁄2$ = $4⁄2$ = $2⁄1$ = 2. That is, four thirds of three halves, is the same as four halves that is Two, Or (in other words to the same sense) the proportion of 4 to 3, compounded with that of 3 to 2, is the same with that of 4 to 2,m or 2 to 1. And, consequently, the Difference of those Two, which is that of a Tone or Full-Note, is that of 9 to 8. For $4⁄3$)$3⁄2$($9⁄8$ that is, three halves divided by four thirds is nine eights; or, if out of the proportion of 3 to 2, we take that of 4 to 3; the Result is that of 9 to 8.

Now, according to this Computation, it is manifest, That an Octave is somewhat less than Six Full-notes. For (as was first demonstrated by Euclide, and since by others) the Proportion of 9 to 8, being six times compounded, is somewhat more than that of 2 to 1. For $9⁄8$×$9⁄8$×$9⁄8$×$9⁄8$×$9⁄8$×$9⁄8$=$531441⁄262144$ is more than $524288⁄262144$=$2⁄1$.

This being the Case; they allowed (indisputably) to that of the Dia-zeuctick Tone (La mi) the full proportion of 9 to 8; as a thing not to be altered; being the Difference of Dia-pente and Dia-tessaron or the Fifth and Fourth.

All the Difficulty, was. How the remaining Fourth (Mi fa sol la) should debe [sic] divided into three parts, so as to answer (pretty near) the Aristoxenians Two Tones and an Half; and might, altogether, make up the proportion of 4 to 3; which is that of a Fourth or Dia-tessaron.

Many attempts were made to this purpose: And, according to those, they gave Names to the different Genera or Kinds of Musick, (the Diatonick, Chromatick, and Enarmonick Kinds,) with the several Species, or lesser Distinctions under those Generals. All which to enumerate, would be too large, and not necessary to our business.

The first was that Euclide (which did most generally obtain for many ages:) Which allows to Fa Sol, and to Sol la, the full proportion of 9 to 8; And therefore to Fa sol la (which we call the greater Third,) that of 81 to 64. (For $9⁄8$ × $9⁄8$ = $81⁄64$.) And, consequently, to that of Mi fa (which is the Remainder to a Fourth) that of 256 to 243. For $81⁄64$ ) $3⁄4$ ( $256⁄243$ that is, if out of the proportion of 4 to 3 we take that of 81 to 64, the Result is that of 256 to 243. To this they gave the name of Limma (λεῖμμα) Rh