94 



Mr. A. J. Ellis on a Perfect Musical Scale 



[Jan. 21, 



i?i eadem consonantia neque in duaimm consonantiarum successione misceri 

 possuntj error etiam ab acutissimo auditu percipi non poterit." It will 

 appear in the sequel that these assertions, when tested by experiments on 

 instruments with fixed tones, are all incorrect. 



The musicaVscale has formed the subject of many recent investigations* ; 

 but I have been unable to find a complete account of the necessary con- 

 ditions to be fulfilled by a perfect scale, the least number of fixed tones 

 required, and the practical means of producing them uncurtailed without 

 inconvenience to the performer, although instruments which produce a 

 limited number of just tones have been practically used by Gen. Perronet 

 Thompson, Mr. Poole, Prof. Helmholtz, Prof. Wheatstone, myself, and 

 others. This is therefore the subject of the present paper. 



The following notation is employed. I have introduced it for the purpose 

 of supplying a want which has been greatly felt by all writers on the theory 

 of music. It is founded on the principle of retaining the whole of the 

 usual notation unaltered, but restricting its signification so as to prevent 

 ambiguity, and introducing the smallest possible number of additional signs 

 to express the required shades of sound with mathematical accuracy, 

 selecting such signs as are convenient for the printer, and harmonize with 

 the ordinary notation of accidentals on the staff. 



A letter, as C, called a notey will represent both a certain tone and its 

 pitch, defined to be the number of double vibrations in one second, to 

 which the tone is due. The letters i>, E, G, A, B represent other 

 tones and pitches, so that 



8Z) = 9C; 4^=5C, 3i^=4C; 2 6^ = 3(7, ^A = hC, 8^=15C. 



Small and numbered letters will be so used that 



and similarly for other letters. The pitch of c is that of the " tenor or 

 middle c," usually written on the leger line between the treble and bass 

 staves ; and the other letters are noted on the staff as usual in the scale of 

 C major. 



* Gen. T. Perronet Thompson, F.R.S., Instructions to my daughter for plajdng 

 on the Enharmonic Guitar, 1829 ; Just Intonation, 6tli ed. 1862. H. W. Poole, 

 On a perfect musical Intonation, Silliman's American Journal of Science, 2nd ser. 

 vol. ix. pp. 68 and 199. W. S. B. WoolJiouse, Essay on Musical Intervals, 1835. 

 Prof. A. De Morgan, Cambridge Philosophical Transactions, vol. x. p. 129. M. 

 Hauptmann, Die Natm' der Harmonik, Leipzig, 1853. M. W. Drohisch, Ab- 

 handlungen der Fiirstlich Jablonowskisclien Gesellschaft, 1846; Poggendorff's 

 Annalen, vol. xc. C. B. Naumann, Ueber die verscliiedenen Bestimmungen der 

 Tonverhaltnisse, Leipzig, 1858. Prof. H. Helmholtz, Lebre von den Ton- 

 empfindimgen, Braunschweig, 1863. To this last writer we owe the first satis- 

 factory theory of consonance and dissonance. 



2 C = 4 = 8 C3 = . . ., 



