INTRODUCTION 5 



Helmholtz the influence of phase is inappreciable. This has 

 been contested by some writers, but there can be no doubt that 

 in most cases the difference of quality is a question of relative 

 amplitudes alone. 



Comparisons of loudness can only be made strictly between 

 sounds of the same quality and about the same pitch. 



It follows from the preceding that, so far as Ohm's law is 

 valid, the sensation of a musical note must be complex, and made 

 up of the simpler sensations, or tones, which correspond to the 

 various simple-harmonic elements in the vibration-form. This 

 doctrine has to contend with strong and to some extent 

 instinctive prepossessions to the contrary, and some preliminary 

 training is usually necessary before it is accepted as a fact of 

 personal experience. We shall return to this question later; at 

 present we merely record that that element in the sensation 

 which corresponds to the gravest simple-harmonic constituent 

 is called the " fundamental tone," and that the others are termed 

 its " overtones " or " harmonics." 



3. Musical Intervals. Diatonic Scale. 



There are certain special relations, familiar to trained ears, 

 in which two notes or two simple tones may stand to one 

 another. These are the various consonant and other "intervals." 

 Physically they are marked by the property that the frequencies 

 corresponding to the respective pitches are in a definite 

 numerical ratio, which can be expressed by means of two small 

 integers. The names of the more important consonant intervals, 

 with the respective ratios, are as follows : 



Unison 1 : 1 Octave 1 : 2 



Fifth 2 : 3 Fourth 3 : 4 



Major Third 4 : 5 Minor Sixth 5 : 8 



Minor Third 5 : 6 Major Sixth 3 : 5. 



The ear has of course no appreciation of the numerical 

 relations themselves ; but each interval is more or less sharply 

 " defined," in the sense that a slight mistuning of either note is 

 at once detected by the beats, and consequent sensation of 

 roughness, which are produced. The explanation of these latter 

 peculiarities must be deferred for the present. 



