700 THE POPULAR SCIENCE MONTHLY. 



pound tones of the organ, sustained with equal energy, caused the 

 imperfections of the concords to be markedly apparent. They could 

 not be ignored, as when occurring with the evanescent tone of a 

 piano-forte. The super-harmonics, or summation-tones, disagreeing 

 with each other, developed, in common with difference-tones, a mass 

 of attendant dissonances, which even the employment of a double 

 foundation (sixteen-feet manual stops) and powerful pedal-organ, etc., 

 could barely hide. 



The music of the Church was then as calm and free from passion 

 as the sculptures of Phidias; no exciting dissonances held the listener 

 spellbound until they were resolved. The organ-builders, accord- 

 ingly, produced instruments that were solemn, sedate, serious and 

 grave, forcible and powerful, without betraying passion or the flut- 

 terings of private, personal emotion. To use this temperament would 

 be to destroy these features, which the piano-forte-makers did not 

 desire to retain. They sought brilliancy and contrast for secular 

 requirements, and were at once manifestly gainers by the system ; 

 while the organ-builders were compelled to reflect very seriously on 

 matters of ritual and their own reputations. 



To make this clear : Let the length of a stretched string be repre- 

 sented by unity, and |- or -^^-^ the octave. Now, three major-thirds 

 are less than an octave ; for f X f X f = -f^. Again, let | or J^Ve 

 represent the octave, and it will be seen that four minor-thirds are 

 greater than an octave ; for | X f X f X f = i^V 



The contraction of the minor-thirds, therefore, by one-fourth of the 

 ratio 625 to 648, caused a further depression of the significant note in 

 the minor-chord, by which it became more gloomy and depressed ; and 

 the expansion of the eloquent major-third by one-third of the diesis, or 

 ratio of 125 to 128, caused the expression of joy in the major-chord to 

 be exaggerated or intensified, or even over-excited, when compared 

 with the peaceful, contented, restful character of the pure harmony. 



These violent contrasts affected the quality of the tones, not indi- 

 vidually, but when heard in combination. The general character of 

 the tone-tint was thus greatly modified. 



That the ear is much more sensitive to imperfections of intervals 

 with simpler ratios is proved by the uneasiness experienced while a 

 violinist tunes his strings to the interval of a fifth (|). Twelve such 

 fifths exceed the octave by an interval having the ratio of 524.288 to 

 531.444. Each successive fifth must, therefore, be contracted one- 

 twelfth of this interval (or diascMsma) , that the series may be bound- 

 ed by a perfect octave ; and in this manner the tonal system of keys, 

 which is in reality a slowly-ascending infinite spiral, practically takes 

 the form of a circle. This fact must be borne in mind if one would 

 satisfactorily account for many laws of art. The ideal purity which 

 enraptures the creative artist is ncA'er realized ; for all performances 

 are marked by the imperfections attending human efforts. 



