FAREY'S EQUAL TEMPERAMENT OF THE MUSICAL SCALE. The apparent simplicity of the ISOTONIC SCALE, wherein the octave is supposed to be divided into 12 exactly equal semitones, and its agreement with the vulgar notions of our musical notation, have occasioned an unusual number of theoretical writers to give their opinions in favour of this system : The names of more than 20 such authors are before us, many of whose recommendations of it are most unqualified : yet few of them have been at the least pains to inquire as to the harmoniousness of its concords, or otherwise ; leaving these to be inferred from its semitones, which are discords! - the length of string calculated for sounding each note, having usually been thought fully sufficient information to the musical student ;* at the same time it has been, and may fairly yet be questioned, whether this system has, in one single instance, been actually tuned on an instrument, with sufficient accuracy for judging fairly of its practical effects in performance.

It was for removing this defect of information, as to a scale so simple and elementary as the Isotonic, that the gentleman whose name appears at the head of this article was induced to pay considerable attention to a system, which he was convinced was at bes but a most violent and unnatural simplifiction of a subject vastly more profound and extended, as Mr. Liston has since proved, in his admirable "Essay on Perfect Intonation," and it was in the course of this investigation, that Mr. Farey discoved a new regularly tempered system, having its fifth G#Eb, less than each of its other eleven equal fifths, by what Mr. Overent has denominated the most minute (m), the last interval that he or any other person has yet dicovered, and being insensible, perhaps, in the nicest experiments in harmonics. None of its semitones differ from each other more than this very small quantity m, and the whole of them have finite or determinate ratos, expressible by help of the primes 2, 3, and 5 ; and, lastly, each of its notes may actually be tuned (on an organ having a sufficient number of pipes, like the Euharmonic organ of Liston), by means of untempered or perfect concords only. That is to say, if on C (on a spare range of pipes) the five successive perfect fourths CF, FBb, BbEb, EbAb ad AbDb, be very correctly tuned upwards, and from the highest of these notes, desceding again by two perfect fifths, DbGb, and GbF_*#, and a major third F_*#G_* (which notes, in Farey's artificial commas, are, 0, 254, 308, 762, 1016, 1270, 912, 534, and 357, respectively), the last of these sounds, G_* , or G according to the common notation, is the proper fifth above C, in this system. After this, new G has been transferred to the range of pipes intended to be tuned, a new beginning is to be made at this note, and 5-4ths up, and 2-Vths and a IIId down, are to be carefully tuned for obtaining D, a proper new fifth, (357 as before) to G. From this D, A is to be tuned in like manner, and likewise E, F#, C# and G# in succession. This process is then to be discontinued, and a new beginning made from C, by tuning upwards the three fifths CG, GD, and DA', and the major third A'C#, and thence downwards, the four minor fourths C#G'#, C'#D'#, D'#A'#, and A'#F*, which last note, being the same with Mr. Liston's E'#) is the proper fourth of this system, i.e. in Art Com. 358, 716, 1074, 1271, 1017, 763, 509, and 255. Upon F*, two other schisma-excessive fourths are to be tuned in succession, as above, for Bb and Eb ; and thus eleven fifths will be obtained, each equal 5-4ths - 2 V - III, and a resulting one, G#Eb, equal to 29V + 11 III - 48-4ths. Its fourth CF*, and complementary fifth F*C, being found ready tuned on Liston's organ ; and whereon there are 14 other pairs of notes, at the exact distance apart of this fourth, and 14 fifths (their complements), which agree exactly in their quantity therewith, but not with their places in the scale, as Mr. Farey has observed in the Phil. Mag. vol.xxxix, p.472, note.

* Even Dr. Robert Smith has bestowed too little attention on this much recommended system, contenting himself with mentioning, (Harmonics, 2d edit. p.156), that its harmony "is extremely coarse and disagreeable ;" and has, at p.167, erroneously state the teperaments of its V, VI, and III, to be 1-10th, 7-10th, and 6-10ths of a comma, instead of 1-11th, 8-11ths, and 7-11ths, respectively, which temperaments may nevertheless be very neatly obtained, from his col. 1 of Table 11, in p. 158, viz. 7-76ths, 56-76ths, and 40-76ths, instead of 7-77ths, 56-77ths, and 49-77ths, as above.
The asterisk reversed, or at the bottom of the line _* is here, and will hereafter be used, to denote the fall of the interval schisma or Σ, (to be read "fall schisma,") and the asterisk in its usual position *, to denote the reise of a schisma, (to be read "rise schisma") [] the asterisk and grave united, *\, will denote the fall of a minor comma or [C], and */ the rise of the same interval : and, in like []/ will denote the rise of the diaschisma or [d], and []\ the fall of the same interval, attached either to the literal or the numerical [] of intervals. See NOTATION of Musical intervals.

In order to fully exemplify this system, and shew clearly its almost perfect agreement with the Isotonic or perfectly 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 IIIds, respectively, are to be tuned upwards, and the numbers with - affixed, of the same intervales, 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, (calculated by our 4th Theorem in the article BEATS), shew the number of beats in a second of time, made by each several concord, 3d, IIId, 4th, Vth, 6th, and VIth, above the note on the same like 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 3ds from C to G#, and the remaining beats from C to D are the halves of those of their complementary 3ds ; also the beats of the 6ths below E, are the same as those of the IIIds from C to G#, [] the remainder are double of those of their comple[]tary IIIds ; and the beats of the 4ths from C to F[]the same as their complementary Vths from F to [] the remainder are double of their complementary [] respectively.

In order to shew, in different ways, the extr[] near agreement of the above system with the Is[] (which agree in col. 2), it may be proper to stat[] in lengths of strings, the greatest difference (on [] but .000037, and the mean of all the difference[] but .000017 ; that in the vibrations, the greatest diff[] (on G#) is .0022, and the mean difference only [] With respect to the beats, which offer by far the [] accurate mode of judging the practical effects [] two systems, it may be proper to compare togeth[] sums of the beats of each concord, in each syst[] 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 6th or Bb. (g)

Luke Hebert, "Piano-forte" (1836)

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