388 Notices respecting New Books. 



called the seven-fifth and five-fifth seuiitones respectively — the 

 former consisting (approximately) of 5, and the latter of 4 intervals; 

 and it is easily shown that five of the former and seven of the 

 latter make up an exact octave*. 



The facts now recited serve as a basis for a notation which will 

 indicate all the 53 intervals of the system, by an extension of the 

 ordinary notation for the seven notes of the octave C, D, E, . . . . 

 In fact the intervals C to D, D to E, E to Gr, &c. consist of a five- 

 fifth and a seven-fifth semitone, which together make nine intervals 

 in the system of 53 ; while E to E and B to c are each a five-fifth 

 semitone or four intervals, and of course 5 x 9 -h 2 x 4=53. Also any 

 note is sharpened by raising it a five-fifth semitone, and flattened 

 by depressing it a seven-fifth semitone. If now we denote the 

 successive notes in the system of 53 bv the numbers 1, 2, 3, . . . 53, 

 54, . . . and call C, 4, then C# or J)\) i's 8, D is 13, . . . A^f or ^^ is 

 48, B is 53, c is 57, &c. Thus we have twelve notes correspond- 

 ing exactly to the seven white and five black keys of an octave on 

 the piano, except that the intervals from C to C# and from C# to 

 D are in the ratio of 4 to 5 instead of being equal. The notation 

 for the other notes is now easily stated. A mark of elvation (/) 

 or a mark of depression (\) put before a letter signifies a note an 

 interval higher or a note an interval lower than that denominated 

 by the letter ; two marks signify a note higher or lower by two 

 such intervals ; and so on. The whole of the fifty-three notes of 

 the octave can now be indicated ; e. y. nos. 3, 4, 5 are \C, C, /C 

 Nos. 7, 8, 9 are \C$f, CJf, /C:jf, and so on. Under certain circum- 

 stances the same note could be indicated by either of two nota- 

 tions ; e. g No. 7 could be indicated by ///C or \C#, indifferently. 

 It is to be observed that all these notes can be easily written on the 

 musical staff. 



These notes all admit of being played on a harmonium furnished 

 with a key-board (called a generalized key-board) resembling that 

 of a piano, but in which each key of the piano is replaced by seven 

 keys one above another — an octave of the whole key-board consist- 

 ing of seven tiers of keys, 84 in all. In the arrangement of the 

 keys, every note determined by an exact fifth is placed a quarter of 

 an inch further back and a twelfth of an inch higher that that which 

 precedes it in the series of fifths — e.g. G is set a quarter of an inch 

 further back and a twelfth of an inch higher than C. " The most 

 important practical point about the key-board arises from its sym- 

 metry — that is to say, from the fact that every key is surrounded 

 by the same definite arrangement of keys, and that a pair of keys 

 in a given relative position corresponds always to the same interval. 

 Erom this it follows that any passage, chord, or combination of any 

 kind has exactly the same form under the fingers in whatever key 

 it is played" (p. 20). 



This statement, it will be observed, is an answer to what we may 

 call Dr. Stainer's challenge. We may just add that any system 



* In fact 5(7.r - 4y)4-7( — bx-\-^ii)=y ; the above statement is tlierefore 

 true whether the fifths be exact or approximate. 



