413 



Mr, A. J. Ellis on the Temperament [June 16, 



C. Combined Systems of Equal Temperament. 



No. 30 (4). The combined harmonic and melodic errors are a minimum. 



By combining the equations of No. 1 5 and No. 26, we have (539 + 106) Ic 

 = (1998 + 420>,or^=^4-5gZ;=-001444, log 746439. 



No. 31 (32). The tones are a mean between those of No. 1 and No. 2. 



Here (sum of the two values of x in No. 1 and No. 2) = '0006744, 

 logt?=-1754169. 



This is proposed by Drobisch (Nachtrage, § 9). 



No. 32 (42). The errors occasioned by using the tempered c, c?, /,/, g, 

 Sbb, c^b, for the just c, dy e, /, are a minimum. 



Using 5 for '0004901, and forming the values of these errors by 

 Table XII., we have4.2;'^ + (« — 8^)^ -\-x^-\-{s—^xf-\-{s— 7xy = a minimum, 

 x=^^ 5=-000059084, log v = -1760322. 



This is proposed by Drobisch as a system of temperament adapted to 

 bowed instruments (Mus. Tonbestim. § 57), allowing them to use a system 

 of perfect fifths, and yet play the perfect scale very nearly by substitution. 

 Sach a system would be more complicated than the just scale for any in- 

 strument, and would require many more than 27 tones. It is, therefore, un- 

 necessary for the violin, and impossible on instruments with fixed tones. 



D. Cyclic Systems of Equal Temperament. 



When it was supposed that the number of just tones required would be 

 infinite, importance was attached to cycles of tones which by a limited 

 number expressed all possible tones. Hence Huyghens's celebrated Cyclus 

 Harmonicus, which he proposed to employ for an instrument with 31 

 strings, struck by levers and acted upon by a moveable finger-board 

 {abacus mohilis), acting like a shifting piano or harmonium. The condi- 

 tion of forming a cycle is not properly harmonic or melodic ; it is rather 

 arithmetic. If log v : log 2 be converted into a continued fraction for any 

 of the preceding values of log v, and y: zhe. any of the convergents, then, 

 putting log 2=2 .h, we shall have log v=^yh, which is commensurable 

 with log 2, and consequently the logarithms of all the intervals will be 

 multiples of A, and therefore commensurable with log 2. A cycle of z 

 tones to the octave will thus be formed. If z is less than 27, the number 

 of tones otherwise necessary, the cycle may be useful, otherwise it can only 

 be judged by its merits as an equal temperament. As an historical interest 

 attaches to several of these cycles, I subjoin a new method for deducing 

 them all, without reference to previous calculations of log v. 



Since log v=^y . h, and log 1=7 log 2—12 log v={7z~l2y) . h, we 

 have only to put 7z—\2y= ... — 2, — 1, 0, 1,2,... and find all the positive 

 integral solutions of the resulting equations. This gives for 



7._12^-_2 y-^^ 27 41 55, 69 



7. vzy- J-22' 46' 70' 94 lT8" • • 



