42 " Whether Music is necessary to the Orator, — 



I fear, sir, that I am tediously minute ; and yet I cannot dis- 

 miss the question without still further exposing its incongruity. 

 In a word then — If the fraction l-7th, l-9th, or any other dis- 

 cordant fraction whose numerator is a unit, were submitted for 

 examination bv this hitherto infallible rule, must it not prove an 

 equivalent, or nearlv an equivalent, to the octave ? 



So much for the vibratorv doctrine, which ever since the day of 

 Galileo has been implicitly acknowledged. That this doctrine, 

 however, is not only fallacious, in its application to musical in- 

 tervals, but also calculated to mislead — and that too for the so- 

 litarv purpose of exalting the fifth, in opposition to the unani- 

 mous declaration of ancient Greece, appears too evident indeed. 

 But what more eligible doctrine shall we substitute in its stead r 

 That of rational calculations — the pleasure of harmony as well as 

 of simple melody being obviously derivable from certain though 

 hitherto imperfectly defined proportions. 



For the ascertainment then of the relations between tone and 

 tone, 1 would thus proceed — considering the string like any other 

 integer as the root. 



First. I would lay down a series of numbers in duple pro- 

 gression, commencing with 7mif7j — as thus, 1.2. 4. S. 



Secondly. I would take the lowest terms of the series ; viz. 

 1 and 2 ; and representing the string by 2, I should be driven by 

 necessitv to acknowledge the unit as its most intimate relation. 

 Hence the base or unison =2, and its octave =1. 



Having thus obtained the numbers 2 and 1 as my extremes, I 

 cannot find an intermediate integer ; and must therefore resort 

 to the second and third numbers of the series ; viz. 2 and 4. 

 Nov.', taking the number 4 as the base, and 2 as the octave, the 

 third integer is wanted, which shall spring from these two con- 

 jointlv, as irom a common root. For the production of this new- 

 integer I am necessitated to add the numbers 4 and 2 together, 

 and divide their product by the lower term 2, generating by this 

 operation the required number 3, which is neither more nor less 

 than the mean *. 



Hence the Base = 4 



Fourth = 3 i. e. a I string. 

 Octave = 2. 



A fourth integer between the base and octave is next re- 

 quired ; which integer, and nothing but which, shall spring from 



* Not the common mean between two ordinary extremes, but that supe- 

 rior that primary mean which, springing from the purest geometrical source, 

 becomes the origin of arithmetical progression. 



Extend to intinity the series 1. 2. 4—3. G. 12, or any other equivalent 

 series, and the result is similar; for the addition of any two adjacent terms 

 divided by the less, she'll produce the original mean, viz. 3. Ex. gr. 6+12 = 

 18, which divide by G, and the product is 3, as required. 



the 



