POLYTECHXIC ASSOCIATION PROCEEDINGS. 773 



ment was here presented by the speaker, and illustrated on the 

 blackljoard.) It consists of a sheet or plate containmg a ch'cle 

 divided into spaces representing the intervals of the true as well 

 as the tempered system. A division of the circle into equal spaces, 

 so as to measure by multiples of one such space all intervals that 

 can be made by modulations, would be so minute as to be of no 

 practical service. I have substituted a near approximation to the 

 true intervals by dividing the circle into fifty-three equal parts 

 called commas. The comma used is the difference between the 

 first and the second intervals of the diatonic scale, expressed by 

 the ratio of |-J-. Taking the ratio of the air-waves producing each 

 sound of the true septave with those producing the tonic, and 

 deducing the ratio of the several sounds with each other, we have 

 the following ratios of intervals: |, ^^-, -i-|, |, ^, |, J|. The first 

 three ratios being nearly as the whole numbers 9, 8, 5, we may 

 measure the so-called major-tone, minor-tone, and semitone inter- 

 vals of the septave by 9, 8, 5, 9, 8, 9, 5=53, or the whole number 

 of units in the circle. 



The same circle is also divided into twelve equal parts repre- 

 senting an isotonic temperament of the chromatic scale, applicable, 

 however, to keyed instruments tuned by allotonic temperaments. 



Within this fixed circle is a movable circular disk, having 

 upon it the same divisions, representing the true and tempered 

 intervals. As the notes of the septave have the same fixed rela- 

 tion, it is evident that the position of the movable tonic at any 

 pitch shown on the fixed circle will determine the position of 

 every other note on the same circle. The notes of the septave 

 represented on the disk are distinguished by the common syllables 

 used in solfeggio, and the numbers by which they are known in 

 figured base. Within the movable circle, representing the inter- 

 vals, and in the major mode, is another series of signs, representing 

 the notes of the minor mode. Thus completing the symbols for 

 one key, by turning the tonic sign from C to G, D, A, E, B, and F 

 sharp respectively, we see at a glance the notes belonging to each 

 key in a modulation by sharps. Again, starting from C and pass- 

 ing to F, B flat, E flat, A flat, D flat, and G flat, we find in turn, 

 the sounds belonging to these several tonics. The employment of 

 numbers in the place of letters renders these changes in modulation 

 much more simple, and obviates the unwarrantable use of the terms 

 " flat" and " sharp." In a modulation by sharps, seven, added to 

 the last tonic, gives the next succeeding one; thus 12, 7, 2, 9, 4, 



