Columns 5, 6, 7, 8, and 9, are intended to explain the Equal-beating System of his Lordship (see vol. xxv. pp. 301, 302; xxvii. p. 203; and xxviii. p. 150): the 5th contains the complete vibrations (which Earl S. would call Beats) made in one second of time; wherein 240 is assumed as the pitch of C on the Tenor Cliff line (a ledger line below the Treble, or the same above the Bass stave in music), on the authorities quoted in the article Concert pitch, in vol. ix. part I. of Dr. Rees's New Cyclopædia, lately published.

I am aware that Earl Stanhope, (vol. xxv. p. 303,) refers, in his Tuning Table, to the Octave below my C instead of above it, at least for tuning of some of his notes, but I have preferred this Octave, and added to my calculation, vol. xxvii. p. 203; extending the same to the equally beating THIRDS in this his Lordship's System.

If the Third EbA in his Lordship's Table be tempered. sharp 1.066 commas, and the Third bAc, also sharp ⚫843 parts of a comma, both of these will beat 10.00 times per second; and his three successive Fifths GD, DA and Ae, if tempered flat 4721, 3163 and 2116 parts of a comma respectively, will each be found to beat 3.158 times per second nearly. The half of the number of Vibrations in columns 4 or 5, or of Beats in this Octave, will answer to the first Bass Octave, and twice these numbers to the first Treble Octave respectively; and the half or double of these again, will express the next descending or ascending Octave respectively, and so on, throughout the whole scale.

Columns 6 and 7 contain the logarithms and lengths of strings in this System, for comparison with the notes in his Lordship's Monochord System, vol. xxvii. p. 195; as column 8 is intended, to compare with vol. xxviii. p. 141; in which column, I have preserved the terms f and m, the same, and thrown all the differences between this and the monochord system, into the term E, as the same are expressed in column 9: from whence it appears, that half the notes differ more than 14 Schismas from each other respectively, in these two Stanhopian Systems.

A Table

The Equal-Beating System of Earl Stanhope, expressed in

Mr. Hawkes's System, in

New Notation.-See vol.

Since it appears, (vol. xxviii. p. 304,) that Mr. William Hawkes asserts the superiority of his System (vol. xxvi. p. 171,) over that of Earl Stanhope, as being "the best principle for tempering our present scale" with 12 notes in an octave, I have devoted a column of my present Table to his System, and whereby a comparison of the same may be made, with either of Earl Stanhope's Systems; and where the differences in schismas and minutes, between any of the respective notes in each, may be obtained by a very easy subtraction.

From such a comparison it will appear, that Earl Stanhope's notes some of them differ materially from Mr. Hawkes's corresponding ones, and that both of them differ much, comparatively, from the Equal Temperament, a mode of tuning, which Earl Stanhope says, (vol. xxv. p. 291,) was approved by one half of the most eminent musicians in England, whom his Lordship particularly consulted on the subject although Mr. H. and his Lordship so cordially agree, (vol. xxviii. p. 305,) and (vol. xxv. p. 305,) in condemning the same.

In my last, I omitted to mention, respecting the Isotonic system in column 4, (vol. xxix. p. 347,) that if cach note therein, all of which contain fractional parts of the very small interval m, be increased to the next whole number, as 48 to 49 m, 46 to 47 m, &c., these trifling alterations will reduce it to my Equal Temperament, mentioned vol. xxviii. p. 65, and the same can then be practically tuned, by the help of perfect conchords only! We may therefore now hope, that "Equal Temperament," which so many have commended, and others condemned, probably without having ever heard any music performed in it, will be brought to the test of experience, and practice sufficiently extensive, to get over the prejudice which will naturally arise, on the hearing of any new system whatever.

Many, I know, have imagined, that the system which was aimed at by the professional Tuners before Earl Stanhope wrote, was an Equal Temperament, to be effected by the judgment of the ear, and Mr. Hawkes thinks that their practical results agrce very nearly with his system: both of these opinions I have much reason to doubt, and cannot

refrain

refrain from again endeavouring to call the attention of some gentleman, possessed of good Instruments and the necessary knowledge and experience in making experiments and calculations in harmonics, and requesting him to employ the best professional Tuners to tune his instruments, without any intimation to the Tuners, of his intentions or object; and before such instruments are put out of tune by use or standing, to ascertain exactly, by the beats of the different conchords, by a monochord; and by other methods also for further satisfaction, the exact value of every interval in an Octave, using single strings only: these experiments varied and repeated, on Organs and Piano-Fortes, tuned by as many good tuners as possible, would, by their results, enable us to say, how far any one system whatever has been adopted, or can be accomplished, by the method of tuning in use, and within what limits the different tuners, or the same persons at different times, do in practice fix each note. Such an one would doubtless perform the most valuable piece of service to the practical musician, and perhaps enable him to profit from the labours of theorists in harmonics, by enabling him with more certainty and facility, to accomplish the "tuning," with which by practice his auditors are become acquainted, and wherewith most of them are satisfied*, if the same did not lead to an amelioration of the system. With such information before them, the musical public would perhaps be enabled to judge, of the pretensions of the many musical quacks, who are almost every year bringing forth some new and fanciful system of temperament, (of which an almost inexhaustible fund yet lay behind,) and crying up the same with a confidence, equalled only by that with which rival empirics condemn them: until at length the Science of Harmonics, and the valuable discoveries of Dr. Robert Smith on the nature of imperfect

* Let it always be recollected, that performances on perfect Instruments or by voices, are free from defects in harmony if skill and good ears but direct them, and that the present inquiry is limited to the use of Instruments with 12 strings or pipes in an octave, where, or even with double that number of fixed sounds, temperaments, or errors in harmony are impossible to be avoided. consonances,

A 4

consonances, and of Mr. Maxwell on the system of perfect consonancy, are in danger of falling into utter contempt.

I beg here to mention, respecting the new notation for musical intervals, which I have explained vol. xxviii. p. 140, that the Octave, happening to contain just 12 of the lesser fractions f, and one of these to fall near each note of the equal temperament; in almost all calculations respecting Douzeaves, the temperaments or results, are free of f, and two only of the three independent or prime terms, of which every accurate notation must consist, are in general found at last; while the smallness of the most minute, m, it being less than the th part of the Schisma, E, which is itself but a very trifle more than th part of a Comma, c, (or TT + TT m) render it allowable in most practical cases to neglect m, and to consider the Σ s as elevenths of a comma, in the results; although I would advise the previous calculations to be always carried on strictly, in £, f and m, especially, as the number of fs will generally point out, to what finger-key or number of half notes, any step in the process answers.

I am, sir, your obedient servant,

12, Upper Crown-Street, Westminster,

JOHN FAREY.

February 1, 1808.

II. Essay upon Machines in General. By M. CARNOT, Member of the French Institute, &c. &c.*

Preface.

ALTHOUGH the theory to be discussed be applicable to

every subject which concerns the communication of motion, I have given to this work the title of Essay upon Machines in General ;-in the first place, because it is principally machines I purpose to treat of, as being the most important

For a Translation of Carnot's "Reflections on the Theory of the Infinitesimal Calculus," see Phil. Mag. vol. viii. p. 222, and 335 ; and vol. ix. p. 39. branch