178 



lntervU. M e not alternately and J, or if and S, but are both 

 * V*' alike, or S ; and this sum is, 942+2 f+ 8 m ; a cu- 

 rious anomaly of the scale, which Mr. Farey has point- 

 ed out ; and, in a note to p. 445, in his paper already 

 qi'oVd, has ascribed it to the practice of turning both 

 ways from C, upwards to G and downwards to F, in 

 forming the scale. 



\Ve shall now be enabled to give rules of very useful 

 Application, by which, any note being given, either' by 

 its literal designation, or in Mr. Farcy's notation, or by 

 its ratio of vibrating strings, either of the other two ex- 

 pn>sions for the same note may be quite correctly ob- 

 tained, more than the artificial commas can show, 

 which are approximate numbers. 



1st, When the Literal designation of a Note is given. 

 It will be necessary to premise the following Table of 



the values of the literals, in the other two modes of no- 

 tation, with the addition of the. numerals in the Vast 

 column, viz. , 



INTERVALS. 



takeout the literal designation answering thereto, as Interv 

 that sought ; then from the rule above given, calculate """""Y" 

 the numbers of 2, f, ajitl m, which should agree with 

 those given, unless the given note falls without the 

 limits of Mr. Farey's tuning table, Vientioiied above ; 

 in which case, this rule fails. At a future time, we 

 may, perhaps! be able to supply one which does not. 

 The ratios will be found by the rule already given. 

 For example, if 306'2+5f+27m be given, on search, 

 ing through the Table, we find, in the first line of its 

 7th division, that the note should be D%.'. In order 

 to try whether this be correct, on applying the first 

 rule, we have 1042 + 2 f + 9m, -f 166'2 + 4f+l4m, 

 + 362+f+3m, = 3062 + 7f+26>n, which happens 

 not to be the note given ; but Bj; , as wijl be seen from 

 the last note in this division, and from the following 



It will also be necessary to bear in mind the following 

 value of the signatures, viz. 



Tb 



bb 



or 

 * 



} 



+ 11+0 + 1 

 _ iio l 



=:83dt2= 



44 1 

 _ 4 + 11 



7=t3=fc:l 



c 

 c 



J 



8 



* 



Rule. To the given literal note, or its proper oc- 

 tave, apply, as the signs indicate, the values of the 

 given number of graves or of acutes, and of pairs of 

 sharps or of flats (*) ; and if there be an odd one in 

 such, apply S for it, and the required result will be 

 had; either true, or a major comma (c) too great or 

 too small, which the artificial commas in the great 

 Table will show and detect. 



For example, if B-V be given, then, for 2,f and m, 

 we have 5552 + 11 f+48m, + 5(ll2 + m), 832 

 2 f 7 m, 47S f 4 m, equal to 480S + 8 f+42jm. 

 For the indices of the primes in its ratio, we have 

 31 1, +20 20+5, 10+2+3, 7+3+1, 



equal to 6 16 8, or -, the interval above C. 



Again, if B$ 3 be given, then, as B must be taken an 

 octave lower, we have 572 f 5 m, + 832 + 2 f + 

 7m, + 472+f+4m, equal to 732+2f+6m; but 62 

 appearing to be the artificial comma, answering to this 

 note in the large Table, we have 622 + 2 f + 5 m for 

 the interval required above C ; and + 4 1 1, 

 + 10 23, +3 + 1 2 (or cf), is equal 17 2 

 6 - 2 ' 7 - !31072 

 1 3*5 6 ~ 140625' 



2d, When farcy's Notation of a Note is given __ 

 Rule. Search through the great table for the artificial 

 comma, which answers to the number of SO given, and 



process, viz. 5552+11 f+48m ) 2492 6 f 21m, 

 = 3062 + 5 f + 27 m, as given. For the ratio of the 

 last interval, we' have 3 1 1, 30 + 6 + 9, = 

 27 + 5 + 8. 



3d, When the Ratio of a Note is given. Rule. If the 

 ratio is given in numbers, find their component primes 

 by division, by 5, 2, and 3, successively ; then, from 

 the Table and rule, in p. 275 of vol. ix. find the num- 

 bers of 2, f, and m, and apply the 2d and 1st rules 

 above, for finding and proving the literal designation. 



After what has been done above, farther examples 

 are, perhaps, unnecessary. 



From the ratio of an interval above C of the tenor 

 cliff line, it is easy to deduce correctly its number of 

 vibrations ; from the circumstance of 240, the vibra- 

 tions per 1" of this C (see our article CONCERT Pitcli,) 

 happening to be composed of the musical primes 2, S, 

 and 5, viz. 2< x 3 x 5, or 4 + 1 + 1. 



Rule. From the constant indices of primes, 4+1 + 1, 

 deduct those of the given interval (or add them with 

 contrary signs,) and the primes composing the number 

 of vibrations will be had. 



If, for example, the fourth of Mr. Farey's Tempe- 

 rament, in p. 273 of vol. ix. be given, viz. 357 2 + 

 7 f + SI m, whose ratio is 14 + 7 + 1 ; we have 



4+1 + 1, +14-7-1, = 15-6-0, or ^g* = 



359.5940, the vibrations per 1", which were required. 



Inlenals have been considered in our work, or are 

 in the course of being so treated, under various classes, 

 as follows, viz. 



Acute, or such as are raised a major comma, and are 

 marked ('). 



Close. See that article. 



Commensurable. Ditto. 



Composed, according to Euler, are greater than VIII. 



Compound. See that article. 



Concinnous. See that article. 



Concordant. See our article CONCORD. 



Defective. See that article. ' . 



Deficient. Ditto. 



Diminished. Ditto. 



Discordant. See our article DISCORD. 



Double. Besides those intervals, which are a multiple 

 of some other, by 2, and properly called double : some 

 practical musicians, as Holden, Calcott, &-c. on some 

 occasions denominate those intervals double which have 

 a major eighth added to them, and fall between VIII. 

 and XV. Euler calls these Composed Intervals. 



Excessive. See that article. 



Extreme. See that article. 



Flattened. See our article FLAT. 



Grave. See that article. 



