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)
