414 



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



the division into degrees and sixteenths adopted in my previous paper 

 Proceedings, vol. xiii. p. 96. 



No. 45 (37). Cycle of 45; /i=-006689, log «=26y^=- 1738940. 



No. 46 (25). Cycle of 43; ^='0070007, log z?=25/z=-1750175. 



This is Sauveur's cycle, defended in Me'm. de I'Acad. for 1701, 1702, 

 1707, and 1711. 



No. 47(10). Cycle of 31; ^='009711, log tJ= 18^=-1747900. 



This is Huyghens's Cyclus Harmonicus, \vhich nearly represents No. 2 

 (2). It was adopted, apparently without acknowledgment, by Galin (De- 

 lezenne, loc. cit. p. 19). 



No. 48 (44). Cycle of 26; ^='011578, log 15A=-1736700. 



No. 49 (25). Cycle ofl9; ^='0158437, log z;=l U = -l 742807. 



This is the cycle adopted by Mr. Woolhouse (Essay on Beats, p. 50) as 

 most convenient for organs and pianos. It may therefore go by his name, 

 although it is frequently mentioned by older writers. It is almost exactly 

 the same as No. 3 (23). 



No. 50 (35). Cycle of 12; ^='0250858, log ?;=7A=-1756008. 



As this is a cycle of twelve equal semitones, it may be termed the Hemi- 

 tonic temperament. It is the one most advocated at the present day, and 

 generally spoken of as "equal temperament" without any qualification, 

 as if there were no other. It was consequently referred to by that name 

 only in my former paper (Proceedings, vol. xiii. p. 95). For its harmonic 

 character see No. 53. 



E. Defective Systems of Equal Temperament. 



It has been from the earliest times customary to have only twelve fixed 

 tones to the octave, on the organ, harpsichord, piano, &c., and to play the 

 other fifteen by substitution, as shown below, where the tones tuned, ar- 

 ranged in dominative order, occupy the middle line, and the tones for which 

 they are used as substitutes are placed in the outer lines, and are bracketed. 



[^bb, ^c>b, jBbb, Jl?, cb, Bb, ^b] 



Eb, J5b, F, C, D, A, B, F% Qf, 



[D#, ^, m Fx, ax, Gx]. 



The consequence was, that while the Vths in the middle line were uni- 

 form, the Vths and 4ths produced in passing from one line to the other 

 (as Git-Et> for ^b^E'b or GftDj) were strikingly different. Similar errors 

 arose in the other concordant intervals. It is evident that the interval 

 error thus produced must be the usual interval error of the system increased 

 or diminished by the logarithm of the diesis, where log 3=log^b— log/ii= 



