PROCEEDINGS OF THE POLYTECHNIC ASSOCIATION. 521 



scale are fixed so that the rate of the vibrations producing tliem will har- 

 monize with each to a certain extent and be expressed by ratios. For 

 instance the waves producing the notes C and the first C above are as 1 to 

 2; the waves for the first C and for the first Gr above are as 2 to 3; for C 

 and E as 4 to 5; for C and F as 3 to 4; for and A as 5 to 6; for C and D 

 as 8 to 9, and for C and B as 8 to 15. 



Tlic consequence of 'this arrangement is that two classes of intervals are 

 formed, called tones and semi-tones, and in the diatonic scale there are five 

 tones and two semi-tones. In changing the pitch of the tonic, on all keyed 

 instruments, it is requisite to divide the tone-intervals into semi-tones, so 

 that this scale may be said to be made up of twelve semi-tones. As the 

 word semi-tone is a misnomer when applied to intervals, the speaker pro- 

 posed to call them grades; and as two grades are required for a tone-inter- 

 val, the intervals of the scale would be represented thus: 



Duograde, Duograde, Grade, Duograde, Duograde, Duograde, Grade, 



The true scale cannot be made for each key by any keyed instrument, 

 the sounds are therefore given as near as may be, by what is called tem- 

 pering. Instruments of the violin and trombone classes, as well as the 

 human voice, can make the sounds belonging to any key absolutely correct, 

 but as the tempered system is the only one in common use, what is now 

 said will be applicable to that only. 



Under the old system of counting, the intervals were measured along a 

 continuous right line, but another system is required to show the similarity 

 of sounds. The system invented by the speaker, is based on the spiral cr 

 helix, which measures by distance seven intervals, and returns nearly to 

 the place of beginning. Ten of these spirals in the same plane arouiid a 

 given point, would represent ten octaves or series of seven sounds. Each 

 spiral is divided into twelve equal parts, to represent the semitone or grade 

 intervals by means of twelve radial lines at equal distances apart. Seven 

 of these radial lines will mark the notes of the diatonic scale in any of the 

 twelve kej's, that is, when each of the twelve sounds of the series are in 

 turn used as the tonic or key note. 



These twelve radial lines are numbered like the dial of a watch in order 

 to readily remember their position. The number 12 is C, and is the tonic 

 in the natural key, major mode. If the first!*three notes of the diatonic 

 scale are on even numbers, the remaining four will be on odd numbers; or 

 if the first three are odd, the remaining four will be even numbers. Thus 

 12, 2, 4, 5, 7, 9, 11, represent C, D, E, F, G, A, B, the notes used in the key 

 of C, which aie made by the white keys of the piano. In the key of five 

 sharps 11, 1, 3, 4, 6, 8, 10 represent the order of the sounds used, distin- 

 guished in the old system, as B, C sharp, D sharp, E, F sharp, G sharp and 

 A sharp made by the five black keys and two white keys of the piano. So 

 the other ten keys might be given ; jet these two are sufficient to prove 

 the readiness with which the notes belonging to any key can be designated. 

 To ascertain what would be the order of tonics in a regular modulation by 

 sharps, the number seven must be added to the last tonic, thus we have 12, 

 7, 2, 9, 4, 11, C. For the tonics in a regular modulation by flats, add five 

 to the last tonic, thu^ we have 12, 5, 10, 3, 8, 1, 6. Both modulations end 



