THE EAR. 503 



of vibrations of the component sounds, and the notes are 

 more harmonious as the mathematical ratio of their numbers 

 of vibrations becomes simpler and simpler. Kvidently the 

 simplest mathematical ratio is 1 to 1, but this, of course, is 

 unison. The next simplest mathematical ratio is 1 to 2. 

 This is the ratio of a note and its octave \ that is to say, the 

 octave above a note has always just twice as many vibra- 

 tions as the note itself. If middle C, therefore, has 256, the 

 C just above in order to be a harmonious octave must have 

 512. The next simplest ratio is 2 to 3. This ratio is found 

 to exist between a note and the perfect fifth of that note. 

 Using the key of C in all of this, it is the combination of C 

 and G. In other words, for every two vibrations in C, G 

 has three, or, if C has 256 G has 384 per second. The next 

 simplest ratio possible is 3 to 4, and this proportion in their 

 vibrations exists between C and F. The ratio of 3 to 4 is 

 thus found in \\\o. perfect fourth. The ratio of 4 to 5 is the 

 ratio of the interval between C and E, or the major third. 

 The ratio of 5 to 6, 6 to 7, 7 to 8, 8 to 9, and so on, are 

 becoming too complicated to appear harmonious to the ear, 

 and we feel these ratios as dissonances. But there are other 

 simple ratios; 3 to 5 is a simple ratio, and this is a ratio 

 which exists between C and A, the major sixth. The ratio 

 of 1 to 3 exists between C and G in the next octave, while 

 the ratio of 1 to 4 exists between C and the upper C of the 

 next octave. 



All these ratios are simple ratios, and it is this simplic- 

 ity of ratio, which is simply the rhythm of their vibrations 

 expressed mathematically, that produces the effect of har- 

 mony upon the ear, or rather, upon the mind. 



Complex ratios become more and more dissonant, while * 

 very complex ratios finally lose all harmonious qualities and 

 tend to become mere noise. If the question be asked why 

 notes having a simple ratio to each other appear pleasant, 

 or harmonious, the answer would have to be referred out- 

 side of the domain of physiology. It would probably be 

 that the mind naturally and for inexplicable reasons likes 



