CHAP, m.] STRINGED INSTRUMENTS. 163 



of the strings in order to produce the series of notes of the diatonic 

 and chromatic scale. This is generally done by means of comparison 

 from one fifth to another, and requires great delicacy of ear, and a 

 certain amount of skill, as temperament must be taken into account. 



The necessity for temperament l in keyed instruments may be 

 regarded as springing from the fact that the exact concords, viz., the 

 octave, fifth, and major third, are intervals incommensurable in magni- 

 tude. In rigorously just intonation the constituent notes of a certain 

 number of exact concords are provided, arid consequently the propor- 

 tion of available concords to the number of notes is small. Mr. Ellis's 

 system of Duodenes is a method for dealing with just intonation of 

 this type. 



Temperaments of different kinds are systematic processes, in which 

 these intervals are altered by small quantities so as to make them 

 commensurable ; let us employ an old rule as an illustration . The 

 interval of the major third is to the octave as the diameter of the 

 circle to the circumference, very nearly. (Smith's Harmonics, Preface). 

 But on ordinary keyed instruments three tempered major thirds 

 make an octave exactly; consequently all the thirds are too large, 

 just as three diameters would have to be stretched to make the 

 circumference of a circle. 



Again, the fifths of the ordinary key-boar~d are too small by about 

 -/y of a semitone ; tuners learn to estimate this by the ear with 

 varying accuracy. The system thus obtained is called the equal tern- 

 perament : it is now universally used ; in it the octave is divided into 

 12 equal semitones. Four of these constitute a tempered major third, 

 which is -137 of a semitone sharp ; seven equal semitones constitute a 

 tempered fifth, about -^ of a semitone flat. 



There are many temperaments other than the equal temperament 

 which possess historical and other interest. In all of these commen- 

 surable relations exist between the fifths, thirds, and octaves; but 

 temperaments may be divided into two principal classes : non-cyclical, 

 in which neither the fifths nor the thirds taken alone are commensur- 

 able with the octave ; and cyclical, in which they are so. Of non- 

 cyclical systems we may enumerate : (1) the Pythagorean system, in 

 which everything is tuned by exact fifths, the thirds being sacrificed ; 

 (2) the mean tone system, in which the fifths are sacrificed to the 

 thirds this was Handel's system ; (3) a system known as Helmholtz's 

 1 For these remarks on temperament we are indebted to Mr. Bosanquet. [ED.] 



M 2 



