Notes from Charles Lucy written Sat. 8th July 1995

Searching for something else through a battered metal trunk from Nigeria this afternoon, I discover the following notes which I had transcribed from Dr. Robert Smith's Harmonics in the British Library I am posting it here for those interested in the history of tuning in eighteenth century England. (Harrison's era)

(You may remember John Harrison wrote of how he felt that Smith had exploited his ideas. The mention of geometry and senses by Smith is interesting, although of course he is advocating JI logic.)

excerpt from: Dr. Robert Smith Harmonics MDCCXLIX (1749)
Preface pages xi - xv

He told me he took a thin ruler equal in length to the smallest string of his base viol. and divided it as a monochord, by taking the interval of the major IIId, to that of the VIIIth, as the diameter of a circle, to its circumference. The by the divisions on the ruler applied to that string, he adjusted the frets upon the neck of the viol. and found the harmony of the consonances so extremely fine that after a very small and gradual lengthening of the other strings at the nut, by reason of their greater stiffness he acquiesced in that manner the placing of the frets.

It follows from Mr. Harrison's assumption that his IIId major is tempered flat by a full comma. My IIId determined by theory upon the principle of making all the concords within the extent of every three octaves as equally harmonious as possible, is tempered flat by one ninth of a comma; or almost one eighth, when no more concords are taken into the calculation that what are contained within one octave.

That theory is therefore supported on one hand by Harrison's experiment, and on the other by the common practice of musicians, who make the major IIId either perfect, or generally sharper than perfect, with a design I suppose, to improve the false concords, though to the manifest detriment of all the rest. We may gather from the construction of the base viol, that Mr Harrison attended chiefly, if not solely to the harmony of the consonances contained within the octave; in which case the difference between his and my temperaments of the Major IId, VIth and Vth and their several dependents, are respectively no greater than 4, 3 and 1 fiftieth parts of a comma. And considering that any assigned differences in temperaments of a system, will have the least affect in altering the harmony of the whole when at the best, I think a nearer agreement of that experiment with the theory could not be reasonably expected.

Upon asking him why he took the interval of the major IIId to that of the VIIIth as the diameter to the circumference of a circle, he answered that a gentleman lately deceased had told him it would bring out the best division of a monochord whoever was the author of that hypothesis for so it must be called, he took the hint, no doubt, from observing that as the octave, consisting of five meantones and two limmas is a little bigger than six such tones, or three perfect major IIIds, so the circumference of a circle is a little bigger than three of its diameters. When the monochord was divided upon the principle of making the major IIId perfect, or but very little sharper, as in Mr Huygen's system resulting from the octave divided into 31 equal intervals, Mr. Harrison told me that the major VIths were very bad and much worse than the Vths and VIths major when equally tempered should differ so in their harmony, after various attempts I satisfied my curiosity; and this gave me the first insight into the theory of imperfect consonances. With a view to some other inquiries I will conclude with the following observation. That, as almost all sorts of substances are perpetually subject to very minute vibrating motions, and all our senses and faculties, seem chiefly to depend upon such motions excited in the proper organs, either by outward objects or the power of the will, there is reason to expect that the theory of vibrations here given will not prove useless in promoting the philosophy of other things besides musical sounds. Such readers as can only dip into this treatise must remember, that by the word vibration so often repeated I mean the time of a single vibration, which I notified once for all in sect I art. 8 31/12/1748.

Section I art 8.8 Harmonics 7

If two musical strings have the same thickness, density and tension and differ in length only, (which for the future I shall always suppose) mathematicians have demonstrated that the times of their vibrations are proportional to their lengths (f). 8. Hence if a string of a musical instrument is stopt in the middle, and the sound of the half be compared to the sound of the whole, we may acquire the idea of the interval of two sounds, whole single vibrations (always meaning the times) are in the ratio of 1 to 2; and by comparing the sounds of 2/3, 3/4, 3/5, 4/5, 5/6. 8/9, 9/10, etc of the string with the sound of the whole, we may acquire the ideas of the intervals of the two sounds, whole single vibrations are in the ratio of 2 to 3, 3 to 4, 3 to 5, 4 to 5, 5 to 6, 8 to 9, 9 to 10, etc.

Footnote: (f) As a clear and exact demonstration of this curious theorem depends upon one or two more of no small use in harmonics, and requires a little of the finer sort of geometry, which cannot well be applied in few words, I have therefore reserved it to the last section of this treatise.

Harmonics or the philosophy of musical sounds, by Robert Smith DD FRS Cambridge MDCCXLIV (1744)