I will now give some account how the real scale of Musick is indeed generated from the proportion which the diameter and radius of a circle bear respectively to the circumference; but as towards which, this great, or secret discovery, it is as in the first place, to be understood, that it was after I had made several strict experiments of divers, or diverse divisions of the octave, and they as from or by such necessary, or proper preparations, or aparatusses to the purpose, as from my other business I was enabled to make: yea, I may boldly say as thence, from far more correct, or natural qualifications to the purpose, than any before me were ever able to make or have, [nay, and still are - as yet the same], and that, as so at last, I found to my great surprize, or admiration [viz. as from the same strictness of trial of the result of the properties of a circle, as here above specified, and as with such, the same apparatusses to the purpose] the real foundation of the matter to exist, or to be, as thence, by the hand of providence established: and the which (as in brief) I shall explain as followeth.
Let the ratio of the octave, or, as even here, as well as below to the purpose, the octave itself, be represented by the logarithm of 2 [viz. .30103]: and let that same number be also taken or supposed as the circumference of a circle -
And then, [as in the margin] (.30103 x 2 = .60206 + .09582 = .69788) let the space or quantity of two octaves and a sharp 3rd be taken, or be as chiefly, or rather as primarily to the purpose notified, viz. when [as according to my discovery] the said sharp 3rd is in its most strictly musical proportion, and that is as when, with respect to the octave the same is taken as the diameter of the circle [viz. here, as .09582]: For the proportion which the circumference of the circle bears to the diameter (and as true enough to this purpose, as well as to others) is about as 3.1416 to 1: So, as 3.1416 is to 1, so is .30103 to .09582.
And then, as five larger notes [but not with tones major and minor, as hath been imagined, and that from of old] and as together with two of the lesser notes [as all along foolishly styled half notes major] are, or must be, exactly contained in the octave: so therefore, as in taking half the diameter for the larger note, viz. .04791, as I from strict, or proper experience, found it to be -- as an interval of melody, right truly pleasant [although, as barely in itself, as well as the lesser note, nothing to do with harmony], and that four 5ths, thence as below to be generated [viz. of each containing .17447], (margin note .17447 x 4 = .69788) and as when, as I am proof sure, to be then near in their most strictly musical proportion, will, as according to nature, be equal to the two octaves and sharp 3rd, and at the same time, as already intimated, each one of the four 5ths will also be as without any infringement in any case [viz. as with respect to the product of nature] so generated by subtracting five times the radius from the circumference, where will be left such a quantity or space, as the two lesser notes must, with equal shares, take up; and that will be .06148, so the half of which, viz. .03074 must be the lesser note; and the lesser note subtracted from the greater will leave .01717, properly to be called a flat or a sharp [or the difference of the notes], and not nonsensically the half-note minor; the lesser note having withal the same authority to be called a whole note as what the other has; but they may respectively or properly be styled tone major and tone minor, viz. without meaning the fictitious nonsense as of old: and (as well understood) a 5th must contain three of the larger notes and one of the lesser [viz. as in the case or cases here .17447].
But as not withstanding, that from what is here above, are indeed the real steps or intervals of tune, or of natural melody, exactly pointed out, or are to be thence truly generated [viz. accordingly as they are taken by the voice or by voices]; so there must, as in consequence thereof, be also the real consonances, or chords of natural harmony, truly limited or described; nay as so, in both respects [viz. as touching both melody and harmony] I found to my great surprize, to be confirmed upon strict instrumental musick, as I have shewn above.
But still (and as just has been intimated) that though from what is shewn above, the true steps of melody, as also the true consonances of natural harmony, are, as touching them all, or each of them, exactly to be defined, yet, as from thence, no ratios at all can be said to be (that of the octave to be excepted), so the said chords &c. must be denominated as they have all along been: and, in the logarithm way, as here to the purpose the best way, as the ratio of any chord is to be had by subtracting the logarithm of the lesser number from that of the greater, so therefore, and as only proper, viz. as in what is here, as first above - may differ from such ratios, so each chord, or interval, must to its properness. or sweetness of relish, in tunes or lessons of musick, be said to have respectively such and such flatness or sharpness of latitude: as the 5ths to have, .00162 flat latitude, the 4th (its complement to the octave) as much sharp; the sharp 3rd to have .00109 flat latitude, the flat 6th as much sharp; the flat 3rd to have .0053 flat latitude, and here I may notify, that the 3rds will bear their flat latitudes better than the 6th will bear their sharp; nay the 5th will bear its flat latitude of .00162 as well or better than the sharp 6th its sharp latitude of .00053: But to bear have I said, as touching them all! whenas, as when in that, their exactly right degrees, they are only as so rendered perfect! I speak from strictly due experience [viz. from such as no man before me could ever make, nay, and are as still the same]; and therefore, as each interval respectively so results from the properties of a circle, as I have shewn, they cannot each one, or any one, as by proof from thence, be said to have a defect of any part or parts of a foolishly feigned nonsensical comma; no, for this, as here otherwise shewn, is certainly the true essence of all that can be said of the matter, whatever nonsense any book, as heretofore in the world may consist of. Now whether my style of writing in this affair, be right proper to the purpose or not, I thought it must be better than that the contents of this book should be in danger of sleeping in oblivion; yea, notwithstanding what I had -- as verbally communicated to the world.
FINIS

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