A Question in Musick Lately Proposed to Dr. Wallis, Concerning the Division of the Monochord, or Section of the Musical Canon: With His Answer to It

which was placed on the far Side of the Monstrous Head, the other as usual in other Calfs. It breathed equally at both Mouths, and had Communication with the same Throat, but took its Nourishment only at the perfect Mouth: The under Jaw of the other being so weak, that the Mouth always stood open and drivell'd. It appeared on the Left side to be a perfect Calf, and look'd very lively, and was, at Three Days Old, as large and strong as other Calfs usually are at Ten Days or a Forth night. IV. A Question in Musick lately proposed to Dr. Wallis, concerning the Division of the Monochord, or Section of the Musical Canon: With his Answer to it. Question. Take a String of any Musical Instrument, and divide the same into two equal Parts, and stop the String there; it shall be an Eighth, which consists of twelve Semi-tones. Hence it appears, that the Frets are nearer to one another toward the Bridge, and wider toward the Nut or Head of a Viol. And that they decrease or proceed in a Geometrical Proportion. Quære, How is it possible, from the foresaid Hypothesis, to divide the other 11 Semi-tones, in their due Proportion, and to demonstrate the same. And whether the other Distances assigned by Simpson in his Compendium of Musick (and Chapter of Greater and lesser Semi-tones) are demonstrable from the said Hypothesis. Answer. WHAT Method is used by Simson (in the Book mentioned) to divide a String or Chord, I know not: Nor have I the Book at Hand to consult. That a String open (or at its full Length) will sound (what we call) an Octave (or Diapason) to that of the same String stopped in the Middle (or at half its Length) is very true. And hence it is that we commonly assign, to an Octave, the Duple Proportion (or that of 2 to 1:) because such is the Proportion of Lengths (taken in the same String) which give those Sounds. And (upon a like Account) we assign to a Fifth (or Diapente) the Sesqui-alter Proportion (or that of 3 to 2:) And, to a Fourth (or Diatesseron) the Sesquiterian (or that of 4 to 3:) And to a Tone (which is the Difference of a Fourth and Fifth) The Sesqui-octave (or that of 9 to 8:) Because Lengths (taken in the same String) in these Proportions, do give such Sounds. And (universally) whatever Proportion of Lengths (taken in the same String, equally stretched) do give such and such sounds; such Proportions (of Gravity) we assign to the Sounds so given. But when an Eighth (or Octave) is said (in common Speech) to consist of 12 Hemitones, or 6 Tones; this is not to be understood according to the utmost Rigour of Mathematical Exactness, (of such 6 Tones, as what they call the Diazenetic Tone, or that of la mi, which is the Difference of a Fourth and Fifth;) but, as exact enough for common Use. For Six such Tones (that is, the Proportion of 9 to 8 Six Times repeated) is somewhat more than that of an Octave (or the Proportion of 2 to 1:) And, consequently, such an Hemitone, is somewhat more than the Twelfth Part of an Eighth, or Octave, or Diapason. But the Difference is so little, that the Ear can hardly distinguish it: And therefore (in common Speech) it is usual so to speak. And, accordingly, when we are directed to take the Lengths (for what are called the 12 Hemitones) in Geometrical Proportion: it is to be understood (not, to be so in the utmost Strictness, but) to be accurate enough for common Use; for placing placing the Frets on the Neck of a Viol, or other Musical Instrument; wherein a greater Exactness is thought not necessary. And this is very convenient, because (thus) the Change of the Key (upon altering the Seat of mi) gives no new Trouble, for this doth indifferently serve any Key; and the Difference is so small, as not to offend the Ear. But those who choose to treat of it with more exactness, go this way to work. Presupposing the Proportion for an Octave (or Dia-pason) to be that of 2 to 1; they divide this into Two Proportions; not just equal (for that would fall upon surd Numbers, as of $\sqrt{2}$ to 1;) but near-equal (so as to be expressed in small Numbers.) In order to which, instead of taking 2 to 1, they take (the double of these Numbers) 4 to 2; (which is the same Proportion as before;) and interpose the middle Number 3. And, of these Three Numbers, 4, 3, 2, that of 4 to 3, is the Proportion for a Fourth (or Dia-tesseron.) And that of 3 to 2, the Proportion for a Fifth (or Dia-pente.) And these Two put together, make up that of an Octave (or Diapason,) that of 4 to 2, (or 2 to 1.) And the Difference of those Two, that of a Tone or 9 to 8. As will plainly appear in the ordinary Method of Multiplying and Dividing Fractions. That is, $$\frac{4}{2} \times \frac{2}{1} = \frac{4}{1}.$$ And $$\frac{3}{2} \times \frac{2}{1} = \frac{3}{1}.$$ And $$\frac{3}{2} \times \frac{2}{1} = \frac{3}{1}.$$ Thus, in the common Scale (or Gam-ut) taking an Octave, in these Notes, la, fa sol la, mi, fa sol la; suppose, from E to e, (placing mi, in b fa b mi;) which is called the Natural Scale;) the Lengths for the Extremes la la, an Octave, are as 2 to 1, or 12 to 6. Those for la la (in la fa sol la) or mi la (in mi fa sol la) a Fourth, as 4 to 3, or 12 to 9, or 8 to 6. Those for la mi (in la fa sol la mi) or la la (in la mi fa sol la) a Fifth, as 3 to 2, or 12 to 8, or 9 to 6. Those for la mi, the Diazeuctick-Tone (or difference of a Fourth and Fifth,) as 9 to 8. So have we for those Four Notes la la mi la, their Proportionate Length in the Numbers 12 9 8 6. Then Then, if we proceed in like manner, to divide a Fifth (or Dia-pente) la fa sol la mi, or la mi fa sol la, or the Proportion of 3 to 2, into near-equals, (taking double Numbers in the same Proportion, 6 4; and interposing the middle Number 5;) of these Three Numbers, 6 5 4; that of 6 to 5, is the Proportion of a lesser Third (called a Tri-hemitone, or Tone and half,) as la fa (in la mi fa.) And that of 5 to 4, is the Proportion of the Greater Third (commonly called a Ditone, or Two Tones,) as fa la (in fa sol la) which Two put together, make a Fifth, as 3 to 2; That is $\frac{6}{5} \times \frac{5}{4} = \frac{3}{2}$. And their difference is as 25 to 24: That is $\frac{5}{4} \times \frac{1}{24}$. So have we for these 3 Notes la fa la, their proportionate Lengths in Numbers, as 6 5 4. In like manner, if we divide a Ditone (or Greater Third) as fa la (in fa sol la) whose Proportion is as 5 to 4 (or 10 to 8) into Two near-equals (by help of a middle Number 9;) then have we (in these Three Numbers 10 9 8) that of 10 to 9, for (what they call) the Lesser Tone: And that of 9 to 8, for (what they call) the Greater Tone. But, whether fa sol shall be made the Lesser (as 10 to 9) and sol la the Greater, as 9 to 8;) or, This the Lesser (as 10 to 9) and That the Greater (as 9 to 8) or sometime This and sometime That, as there is occasion, (to avoid what they call a Schism;) is somewhat indifferent: For, either way, the Compound will be, as 5 to 4; and the Difference (which they call a Comma) as 81 to 80. That is, $\frac{5}{4} \times \frac{1}{80} = \frac{1}{8} \times \frac{1}{8} = \frac{1}{8}$. And $\frac{1}{5} \times \frac{1}{8}$. Lastly, It from that of the Tri-hemitone. (or Lesser Third) la mi fa; whose Proportion is as 6 to 5; we take that of the Tone la mi (which is the Difference of a Fourth and Fifth) as 9 to 8; There remains for the Hemi-tone mi fa (or la fa) that of 16 to 15. That is $\frac{2}{3} \times \frac{1}{15} = \frac{1}{15}$. Or, the Tri-hemitone (or Lesser Third) whose Proportion is as 6 to 5; may be divided into Three Near-equals, (by taking Triple Numbers, in the same Proportion, 18 15; and interposing the Two Intermediates, 17 16;) which will therefore be as 18 to 17, and as 17 to 16, and as 16 to 15; That is, $\frac{18}{17} \times \frac{17}{16} \times \frac{16}{15} = \frac{18}{15}$. Where also the Greater Tone, who's Proportion is as 9 to 8, or 18 to 16, is divided into its Two Near-equals (commonly called Hemitones,) that of 18 to 17, and that of 17 to 16: That is, $\frac{18}{17} \times \frac{17}{16} = \frac{18}{16}$. And And the Lesser Tone, that of 10 to 9, or 20 to 18, may be in like manner divided into that of 20 to 19, and that of 19 to 18: That is, \( \frac{2}{3} \times \frac{19}{18} = \frac{19}{18} = \frac{19}{18} \). Which Divisions of the Greater and Lesser Tone, answer to what is wont to be designed by Flats and Sharps. So that (by this Computation,) of these Eight Notes, la, fa sol la, mi, fa sol la; their Proportions stand thus; that of la fa (or mi fa) is as 16 to 15. That of fa sol as 10 to 9, and that of sol la as 9 to 8: (or else that of fa sol as 9 to 8, and that of sol la as 10 to 9.) That of la mi as 9 to 8. And if either of the Tones (Greater or Lesser) chance to be divided (by Flats or Sharps) into (what they call) Hemi-tones, their Proportions are to be such as is already mentioned. There may be a like Division of a Fourth (or Dia-tessaron) into Two Near equals: And of some others of these, into Three Near-equals. Which might be of use for (what they were wont to call) the Chromatick and Enarmonick Musick. But those Sorts of Musick, having been long since laid aside, there is now no need of these Divisions, as to the Musick now in use. V. Part of a Letter from Mr. Ray, F. R. S. to Dr. Sloane, giving an Account of the Poisonous Qualities of Hemlock-Water-Drop-Wort. I shall now communicate to you, a Story or Two of the direful Effects of Oenanthe aquatica, Cicuta facie succo Viroso of Lobel, which we may English Hemlock-Water-Dropwort, upon several Persons that eat of the Roots of it, sent me not long since in a Letter from Dr. Francis Vaughan, a Learned Physician in Ireland, living at Clonmell, in the County of Tipperary. This Gentleman observing me, notwithstanding what Dr. Johnson in his Gerardus emaculatus, and Lobel in his Adversaria