



SCALE. 



km of notes to each other, would be necessary. 

 or to separate article the account of the systems 

 ,ve been proposed, it will b desirable here to lay 

 undatiou at_ the subject, which is easy enough to 

 Kurithms. 



SCALE. 



320 



Since all intervals are found by mull!j>licali'n> and iliriitiutt', it is 

 obvious that if for intervals we substitute the logarithms of intervals, 

 we form logarithms of new intervals by addition and sn/iir 

 Hitherto, we express a note which makes a vibrations while the fun- 



Q 



dainental note makes b vibrations, by j- ; let us now express it by 



log a log 4, the logarithm of the preceding. It only remains to see 

 what system of logarithms it will be most convenient to take. Having 

 mode the octave, or the interval from 1 to 2, consist of twelve semi- 

 tones (not equal indeed, but nearly so), let us take a new scale, to 

 which all others shall be referred, and which divides the octave into 

 1 2 . /iitil semitones. This is a tern 1% on the (theoretically) 



ximplext HVi-tem of temperament, and it is agreeable enough to the ear 

 in practice. Let 1 be made the logarithm of the interval of any one 

 of these mean Bemitones, then 12 is the logarithm of the interval of an 

 octave, or we must choose that system of logarithms in which log 2 = 12. 

 The preceding is such a table; to the mathematician it would lie 

 l>ed aa a system the base of which is '-\/2. But to the musician 

 it may be described as follows : it shows the number of mean semi- 

 tones contained in every H.VHJIO.MO of the fundamental note, r 

 first to the 250th inclusive. Thus, opposite to 21 we see written 52-71, 

 which means that the 21st port of a string sounds a note which is ;V2 

 mean semitones and '71 or ^ of a mean .- Mnitone above the funda- 

 mental note of the string ; or that there BT6 --71 mean semitones of 

 interval between two notes, one of which vibrates in a second 21 times 

 an often as the other. This into .1 mean semitone is one 



win h, repeated a hundred times, gives 71 mean semitones. All the 

 number* of the table must be understood to be approximate, within 

 the hundredth of a semitone or thereabout* ; which is more than 

 exact enough for practical purposes. The following rule is all that is 



ft 

 necessary : If a note make vibrations while the fundamental 



note makes 1, then that not* is log o - log * mean semitones above, or 



log ft -log a mean semitones below, the fundamental note, according as 

 a is greater than 4, or b greater than a. 



Example 1. What is the value of a mean semitones? 



Log 81 -log 80 is 76-08 -75-86= -2-2. and the answer is, that the 

 comma is 22-hundredths, or something less than a qimtcr, of a mean 

 semitone. Raising a note by a comma four times successively would 

 not raise it quite a semitone. 



Kxample 2. What is the enharmonic interval above obtained, in 

 mean semitones? Log 128 log 125 = 84-00 83-59 = -41, or about 

 Four-tenths of a mean semitone. This shows that an untcmpered 

 enharmonic scale, such as that proposed, if bearable when the sharps 

 and flats are only incidental deviations, would never do for any 

 key except the natural one. 



The following is the complete basis above given, of the enharmonic 

 scale, with all the intervals, measured from the fundamental note, 

 expressed in mean semitones ; it shows how much alteration a system 

 of mean temperament would require : it being rememberc* ! 

 although some few instruments have been made which give more than 

 twelve different notes in the octave, this is so unusual a circumstance 

 that it is not worth while to dwell upon it : 



The first column gives the name of the note ; the second, the ratio 

 of its number of vibrations per second to those of the fundamental 

 note ; the third, the interval from the fundamental note in mean semi- 

 tones ; the fourth, the interval between each consecutive pair of i 

 The small variations observable in the last column arise from impi T- 

 fection of the table (every table must be imperfect in its last figures) ; 

 and we see four intervals in it, namely, the old diatonic semitone 1-1-2, 

 the major and minor cliromnti<- as we will call them) : 



'71, and the enharmonic interval (or enharmonic diesis, as it ia called) 

 41. And the major tone is in every instance 



maj. chroin. sernit. + inin. do. -i- enh. hit. 

 while the minor tone is 



2 min. semit. + enh. int. 



If we confound the major and minor tone (and to distinguish them 

 is the ultima Tl nlr of temperament), we must take a mean vali 

 substitute it both for the major and minor chromatic semitone. The. 

 mean tune is 1-9, which is to the diatonic semitone nearly as 5 to 3. 

 The mean chromatic semitone is "81, about $ of the diatonic seinitunc, 

 and the enharmonic interval is its half. This ia a well-known system 

 of temperament (that of Huyghens) : the octave being divided into 31 

 equal parts, five of them are a tone, three a diatonic .-emit one. two a. 

 chromatic semitone, and one the enharmonic interval. In): 

 Kj and C^, which are omitied in the preceding, and we ha 

 following for a tempered enharmonic scale, upon which v.e ilouU i 

 any improvement being practicable, without attempting the distinction 

 between a major and minor tone : 



21 



1> V 

 11 



EJ> K 

 111 



> Ei F FS C' li ' A 8 1:1, r. ( , i; . r 



113 1121 21 I III 1 



This system, however, ia useless, inasmuch as instruments are required 

 to have only twelve notes in the octave; but we should recommend 

 the student to bear it in mind, as explaining those enharmonic el 

 which in piano-forte music are only fictions. This scale would enable 

 us to play with e<mal correctness in all keys up to seven Ihts among 

 the flats, and seven sharps among the sharps. Naming these keys by 

 their principal notes, they are the keys of 



(; l> A E 1) r: ( - 

 F B|> E|> A^ t>k 0^ C> 



Now suppose an incidental deviation into the key of Alt . Looking 

 into the preceding scale from A)( ascending, we find we can get a 

 whole tone at Bit , but the next whole tone is wanting, nor can 

 it except by interposing a note between D ^ and D, two tempered chro- 

 matic semitones above C, and therefore called C$1 or C x . On the 

 piano-forte we must be content with D for CJt t, and accordingly wo 

 have in like manner 



K for D;:, D for Ej>|>, D3 for F|>|, &c. 



In the preceding scale also, when enharmonic transitions are written, 

 they can most frequently be actually undo . on the piano-forte, though 

 written, they can only be made in imagination. The ear, knowing 

 coining, so soon as the enharmonic modulation is seen, pi i 

 for a change of key, and gives the chord in its possession to the mind 



