Page:A Letter of Dr. John Wallis to Samuel Pepys Esquire, Relating to Some Supposed Imperfections in an Organ.djvu/4

 that is, the Remainder (to wit, over and above Two Tones.) But, in common disourse (when we do not pretend to speak nicely, nor intend to be so understood) it is usual to call it an Hemitone or Half-Note (as being very near it ) and, the other, two Whole-Notes. And this is what Ptolemy calls Diatonum Ditonum, (of the Diatonick kind with Two full Tones.)

Against this, it is objected (as not the most convenient Division,) that the Numbers of 81 to 64, are too great for that of a Ditone or Greater Third: Which is not Harsh to the Ear; but is rather Sweeter than that of a single Tone who's proportion is 9 to 8. And in that of 256 to 243 the Numbers are yet much greater. Whereas there are many proportions (as $5⁄4$,$6⁄5$,$7⁄6$,$8⁄7$,) in smaller numbers than that of 9 to 8; of which, in this division, there is no notice taken.

To rectify this, there is another Division thought more convenient; which is Ptolemy's Diatonum Intensum (of the Diatonick Kind. more Intense or Acute than, that other..) Which, instead of Two Full tones for Fa sol la; assignes (what we now call) a Greater and a Lesser Tone; (which, by the more nice Musicians of this and the last Age, seems to be more embraced;) Assigning to Fa sol, that of 9 to 8 (which they call the Greater Tone:) and to Sol la, that of 10 to 9, (which they call the Lesser Tone:) And therefore to Fa la (the Ditone or Greater Third) that of 5 to 4. (For $10⁄9$=$9⁄8$ =$5⁄4$.) And consequently, to Mi fa (which is remaining of the Fourth) that of 16 to 15. For $5⁄4$) $4⁄3$ ($16⁄15$. That is; if out of that of 4 to 3, we take that of 5 to 4, there remains that of 16 to 15.

Many other waies there are(with which I shall not trouble you at present) of dividing the Fourth or Dia-tessaron or the proportion of 4 to 3, into three parts, answering to what (in a looser way of Expression) we call an Half note, and two Whole-notes. But this of $16⁄15$ × $9⁄8$ × $10⁄9$=$4⁄3$ is that which is now received as the most proper. To which therefore I shall apply my discourse. Where $16⁄15$ is (what we call) the Hemitone or Half note, in Mi fa: $9⁄8$ that of the Greater-Tone, in Fa sol and $10⁄9$ the Lesser-Tone, in Sol la.

Onely with this addition; That each of those Tones, is (upon occasion) by Flats and Sharps (as we now speak) divided into two Hemitones or Half notes: Which answers to what by the Greeks was called Mutatio quoad Modos (the change of Mood;) and what is now done by removing Mi to another Key. Namely $9⁄8$=$18⁄16$=$18⁄17$×$17⁄16$; and $10⁄9$=$20⁄18$=$20⁄19$×$19⁄18$.

Thus,