Page:A Letter of Dr. John Wallis to Samuel Pepys Esquire, Relating to Some Supposed Imperfections in an Organ.djvu/7

 many) be in continual proportion; that is, each to the next subsequent in the same Proportion.

For it hath been long since Demonstrated, that there is no such thing as a just Hemitone practicable in Musick, (and the like for the division of a Tone into any number of Equal parts; three, four, or more.) For, supposing the Proportion of a Tone or Full-note to be  (or, as 9 to 8;) that of the Half-note must be as $$\surd 9$$ to $$\surd 8$$ (as the Square-root of 9 to the Square-root of 8; that is, as 3 to $$\surd 8$$, or 3 to $$2\surd 2$$,) which  are Incommensurable quantities. And that of a Quarter-note, as $$\surd qq \, 9$$ to $$\surd qq \, 8,$$ (as the Biquadrate root of 9, to the Biquadrate root of 8,) which is yet more Incormnensurate. And the like for any other number of Equal parts. Which will therefore never fall-in with the Proportions of Number to Number.

So that this can never be perfectly adjusted for all Keys (without somewhat of Bearing) by multiplying of Pipes; unless we would for every Key (or every different Seat of Mi) have a different Set of Pipes, of which this or that is to be used, according as (in the Composition ) Mi is supposed to stand in this or that Seat. Which vast number of Pipes (for every Octave) would vastly increase the Charge. And (when all is done) make the whole impracticable.

These are my present thoughts, of the Question proposed to me, and upon these grounds.

You will please to excuse me for the trouble I give you of so long a Letter.

I thought it necessary, to give a little intimation of the Ancient Greek Musick compared with what is now in practise; which is more the same than most men are aware of: though the Language be very different. But I was not to be large in it, Those who desire to know more of it; may see my thoughts more at large, in that Appendix which I have added at the end of my Edition of Ptolemy's Harmonicks in Greek and Latin.

The two Eminent Sects amongst them, the Aristioxenian and the Pythagorian, differ much at the same rate as doth the Language of our ordinary practical Musicians, and that of those who treat of it in a more Speculative way.

Our Practical Musicians talk of Notes and Half-notes, just as the Aristoxenians did; as if the Whole Notes were all Equal; and the Half notes likewise each the just Half of a Whole Note. And