In order fully to exemplify this system, and shew clear ly its almost perfect agreement with the Isotonic or per fectly equal temperament, we subjoin the following Table, consisting of 11 columns, numbered at top, and entitled at bottom, because intended to be read upwards, according to the practice of musicians.
The 1st, 2d, 4th, and 5th, of the above columns, seem to require no explanation. The numbers in the 1st, 2d, and 3d range of col. 3, shew how many Vths, 4ths, or 1IIds, respectively, are to be tuned upwards, and the numbers with — affixed, of the same intervals, are to be tuned downwards respectively from tenor cliff C, in order to produce each several note.
The numbers in the 6th to the 11th columns, (calcu lated by our 4th Theorem in the article BEATS), shew the number of beats in a second of time, made by each several concord, 3d, IIIcj, 4th, Vth, 6th and VIth, above the note on the same line in col. 1, either flat or sharp beatings, as is marked at the bottom of the columns : and hereon it may be proper to remark, that the beats on all the VIths above D are repetitions of those of their complementary ads from C to Gil, and the remaining beats from C to D are the halves of those of their complementary ads; also the beats of the 6ths below E, are the same as those of the Inds from C to G14, and the remainder are double of those of their complementary Inds; and the beats of the 4ths from C to F4 are the same as their complementary Vths from F to c, and the remainder are double of their complementary Vtlis respectively.
In order to shew, in different ways, the extremely near agreement of the above system with the Isotonic, (which agree in col. 2), it may be proper to state, that in lengths of strings, the greatest difference (on G14) is but .000037, and the mean of all the differences only .000017 ; that in the vibrations, the greatest difference (on G) is .0022, and the mean difference only .0006. With respect to the beats, which offer by far the most accurate mode of judg ing the practical effects of any two systems, it may be proper to compare together the sums of the beats of each concord, in each system, as follows, viz.
by which, the very insensible differences of these two equal temperaments will sufficiently appear ; the beats differing but 1 in 50, in the most extreme case, the Gth or B17. (e) •
FAREY's NOTATION OF MUSICAL INTERVALS. This new mode of expressing the magnitudes of intervals ha ving been adopted in our work, it may be proper to say a few words in this place on the discovery of this nota tion. The late Mr Marmaduke Overend, organist of Isle worth Church, near London, and author of "a brief ac count of, and an introduction to, Eight Lectures in the Science of Music, (intended) to be read," &c. 4to. pp. 20. Payne & Son, 1781, bestowed inconceivable pains and la bour on the calculations and comparison of musical inter vals, by actual involutions, multiplications, &c. of the terms of their ratios, never using logarithms, and but rarely re sorting to the indices of the component primes, for short ening his work: he adopted a consistent nomenclature throughout, according to which he named the several in tervals, and fixed on a symbol or character as we have done in our 30th Plate in Vol. II. and throughout our work.
At the conclusion of each of his arithmetical calcu lotions, as above, Mr Overend was careful to express his results in form of equations, by means of his sym bols, and to transcribe the whole neatly into thick quarto volumes. After Mr Overend's death, Dr John Wall Callcott, one of the most able but unfortunate of musi cians, having purchased all Mr Overend's manuscripts from his family, kindly offered the use of the above men tioned quarto volumes to Mr Farey, with permission to make all such extracts from, and use of their contents, as he might be able. In perusing 14 of these volumes, in June 1807, Mr Farey found an immense number of in tervals correctly deduced by Mr Overend, and expressed in his symbols, which no previous writers had noticed, the whole of which he was desirous of preserving, in an arranged Table; but, previous to this, it became necessary to reduce them all to some one notation ; a thing which never seemed to have occurred to Mr Overend, but who had in each instance left his intervals expressed in the symbols of those particular intervals, which he happened to have used in calculating their ratios.