of Musical Intervals. 443 



in p. 364 of your last nuir.ber : Wherein, after assigning' the 

 symbol >J«to the Interval 83l + 2f+7m, I have mentioned, that 

 the same is always the value of a double sharp ( *), or a doubln 

 flat (bb) ; — which last, although true in all the instances of aSharp 

 added to a Sharp, or a Flat to a Flat, as C to C**, and Cb to 

 Cb^ (or three times flattened), &c. yet it fails in a very few othei 

 cases of Mr. Liston's Scale : because, from an extension of his 

 Euharmonic Scale of Intervals, so as to produce a Note corre- 

 sponding to every one of the 612 Artificial Commas, into whici? 

 the S's of my Notation divides the Octave* (which extensioii 

 I have effected, since the paper referred to was written) I have 

 lately found, tliat the following Intervals, viz. between C* and 

 Cb, between C* and C b, betv/een C* and C b, between C'^* (or 

 three times acuted) and Cb, and between C'-^* and C'^b ; also 

 between C« and Cb, between Q\ and Cb, between C'^* (oi 

 three times graved) and Cb, between C-'* and Cb, and be- 

 tween C'* and Cb, are each of the value 2l, or 23 = 94S-f-2f 

 •4- 8m; respectively; but in all other instances, of this very ex- 

 tended new Scale, the values of double Sharps, or double Flats, 

 or of the sums of single Flats and single Sharps, of the same 

 Notes (as from G* to Gb, D* to Db, &c.) is the Tone minimum, 

 or 832 -}-2f + 7m, as has been stated, in the page referred to, but 

 without these qualifications, of its general application, here given. 

 The happy discovery, bv Mr. Liston, of a Tuning process, 

 which, by help oi perfect major Thirds, Fifths, and Octaves only, 

 is found capable of giving any required extension to the Scale, 

 still preserving throughout the same, a true harmonic relation to 

 the note first assumed, or C (as mentioned in p. 421 of your 39th 

 volume, and which the late Kir. Maxwell failed in effecting) has 

 led me to the arrangement, of the extended scale here spoken of, 

 in a Table, composed of numerous small squares, which has the 

 effect of exhibiting its many curious properties, in a very striking 

 manner. The bulk of this table is too considerably extended, by the 

 many squares in the opposite corners of the sheet, which are nn- 

 fiUed, for it to seem probable, that you could give thisTal)le a place 

 in your Magazine ; otherwise, for the sake of the insight which it 



* It appears, however, and the Student mubt alw ays l)Car ii in nii.nd, that 

 these divisions, altliough extremely near to it, arc not exactly equut amongst 

 theniselve.-., a thiiii; impossible to see clVcctod, with )et;ai<l to tii;> aliquot, 

 division of any prime musical Interval whatever; but of the 612 i;itcrval» 

 or diffeifuces which occur herein, between 0, and 615i 2 + 12 f + 

 53m, 441 of these, have exactly the value of 2 0; 118 of these 

 differences are of tlie value 2 4-f, and have the ratio, ,3"'_f.2'''-' x 5', or 

 1'1496009G X 2 : and the rcniainini; 53 of them liavc the value 2 — 2f+m, 

 and ratio 3* x 5'''-h2''", or ■7085I04B x 2; — or, near enough for every prac- 

 ticable purpose, my 612 artificial commas in the octave, always have dif- 

 fcreuces between them, cither ^ ' ihs, j ggths, or \ ^gths of a Schixma. 



affords. 



