772 Transactions of the American Institute. 



Twelve similar right lines, at equal distances around the cone, 

 would represent the intervals of the chromatic scale, as made by 

 keyed instruments; and seen from the top of the cone, they would 

 appear as shown in Fig. 2, which measures all the intervals made 

 on the laro^est sized oro^an. 



The true spiral projection, representing a single septave, is shown 

 in Fig. 4. The line CO being the length of a string or pipe giving 

 the sound C; and c O a similar string or pipe, half as long as the 

 first, which gives a sound an octave higher than the first sound. 

 The perfect concord of these two sounds arises from the fact that 

 every wave of air producing the lower sound strikes the ear simul- 

 taneously with every other wave causing the octave. 



The intermediate sounds of the diatonic scale would be measured 

 on the septave by lines proportionate to the length of strings or 

 pipes producing such sounds. Only three are shown at E, F and 

 G. The ratio of the vibrations producing these sounds respectively 

 with those producing C being as 4:5, 3:4, and 2:3. The generation 

 of a spiral on this plan, to embrace every septave, requires that 

 the length of the radius vector should be doubled with every revo- 

 lution of the generatrix, and would, therefore, on account of its 

 size, be quite inconvenient. For general use I substitute a true 

 circle for one ring of the spiral, and divide it into twelve equal 

 parts or grades to represent the tempered intervals of one septave, 

 as made by keyed instruments. It resembles the ordinary watch 

 dial, 12 being the starting point or the tonic of the natural key. 

 The position of the figures on a watch face being so early learned 

 and so familiar to every one, there is but little difficulty in adapting 

 them to musical intervals. Inverting the circle of figures, its appli- 

 cation to the ke^'S of a piano, or an organ, are shown in the annexed 

 illustration. Fig. 5. 



Tvro septaves of kej^s are shown in Fig, 3. The numbers 12, 2, 

 4, 5, 7, 9, 11 correspond with the letters designating the notes of 

 the natural key of C, while the numbers 1, 3, 6, 8, 10 designate 

 notes used in other keys. This application of numbers to the 

 twelve sounds embraced in a septave obviates the necessity of 

 sharps and flats which do not properly belong to the tempered 

 system employed on keyed instruments. 



The position of the numbers used in the key of C, on the stafi', as 

 fi:xed by the base and treble clefs, are shown in Figs. 6 and 7. 



These preliminary statements will make plain the use of my tono- 

 meter, which measures to the eye all musical intervals. (The instru- 



