Theorems on Musical Temperament, 39 



free space inside; the increase of expense on this account 

 would not, I suppose, much exceed twelve or fifteen thou- 

 sand pounds, in addition to that before stated. 

 I am, sir, 



Your obedient humble servant, 



Bath, June 1810. W. Ca; LFIELD LENNOX. 



VI. Six Theorems, containing the chief Properties of all 

 Regular Douzeave Systems of IvJusic ; with Twelve Co- 

 rollaries thence deduced, showing others of their Relations ; 

 and Thirteen Scholia, containing the Temperaments of as 

 many Systems, calculated thereby. With Remarks. By 

 Mr. John Farey. 



To Mr. Tilloch. 



Sir, A am much pleased to observe, that at length a be- 

 ginning has been made, at publishing tables of the Beats 

 in 15"; made by the 72 concords in different systems of 

 Musical Temperament, by your new correspondent Mr. 

 C.J. Smyth, of Norwich, in your last Number, who will, 

 I hope, persevere, and give us tables of many other sy steins, 

 accompanied by such critical remarks, on their comparative 

 merits and detects in praciice, as he appears well qualified 

 to make, either m your Magazine, or in the separate work 

 which he has announced on the subject. 



Some time ago, I had thoughts of preparing a work on 

 Harmonics, perhaps as a kind of supplement to Dr. Robert 

 Smith's justly celebrated work ; but the prospect being now 

 rather distant, of my being able to find leisure to complete 

 this design, I am induced by the above paper of Mr.Smvth's, 

 and the publication of a small work on Harmonics, by Mr. 

 J. Marsh of Chichester, to transcribe from my papers some 

 Theorems, showing the properties of regular douztaves, or 

 of such systems pr twelve notes in the octave, as have all 

 their fifths alike tempered, except, that between *G and 

 bE, when there is a bearing fifth or quint wolr": but first I 

 beg to make a few remarks. 



In douzeaves, or systems of twelve notes, there are ge T 

 nerally 16 wolves or tempered concords, differing* from 



* Wolves, taken in their general sense, are not always larger than the 

 temperaments, but may be equal to them, as happens throughout the iso- 

 tonic or equal temperament scale, and may even be le.-s ;han their re- 

 spective temperaments, in some cases, as in scholia 1 and 7 ; they are, in 

 fact, the places in the douzeave or other defective scales, where the resulting 

 intervals or unavoidable inequalities fall, and, as such, are very important 

 to be known and attended to by the compoiers of music, to be performed in 

 fuch scales. 



C 4 the 



