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THE POPULAR SCIENCE MONTHLY. 



ments. If the performers tried to be true to these, the gist of the 

 story, as it were, would be lost in the telling. The performance, 

 therefore, may be psychologically true and yet mathematically false. 

 There is a natural tendency to raise notes under the influence of ex- 

 tremely excited, passionate phrases and to depress them in sinking to 

 a state of repose. But, irrespective of such changes, errors must cer- 

 tainly be made by singers and violinists of the nature of those now to 

 be defined, however they may be glossed over and hidden. 



Mr. Ellis illustrates his system of " Duodenes " by the first line of 

 the hymn " My country, 'tis of thee," which is better known to Ger- 

 man and Anglo-Saxon peoples as " God save the Queen." It will be 

 employed here, for it is not only familiar and very simple, but usually 

 has equally simple harmonies that may be conveniently illustrated 

 with low numbers and a few fractions. 



Whatever numbers may by chosen for convenience, the proper 

 proportions of the melodic notes are 15 : 16 : 18. Those of the chords 

 in columns 1, 4, 5, 6, are in the ratio of 4, 5, 6, and those in columns 

 2, 3 in the ratio of 10 : 12 : 15. It is understood that the doubling or 

 halving any of these chordal numbers will merely represent the note 

 in another octave, above or below, it will not change its character so 

 as to aifect the investigation. 



3. When the melodic proportions are true, the chords are untrue. 



At the third note of the melody the chord is out of proportion, for 

 the interval D : A, 27 : 40, should be D : A, 27 : 40^ to be in the 

 required ratio of 2 : 3 or 10 : 15. 



4. When the chords are true, the melody is temporarily out of 

 proportion. 



