244 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1705. 



deric and Mr. Christian Stefkins; when it appeared, that the theory was certain, 

 since all the stops were owned by them to be perfect. And, that they might 

 be proved agreeable to what the best ear and the best hand performs in modern 

 practice, Signer Gasperini played another sonata on the violin in concert with 

 them, in which the most complete harmony was heard. 



The full knowledge and proof of this experiment may be found in the two 

 following schemes, in which music is exhibited, first arithmetically, and then 

 geometrically ; the mathematician may, by casting up the proportions, be satis- 

 fied that the five sorts of half notes, here set down, exactly constitute all those 

 intervals of which our music consists. And afterwards he may see them exhi- 

 bited on a monochord, where the measure of all the notes and half notes comes 

 exactly to the middle of the string. The learned will find that these are the very 

 proportions which the old Greek authors have left us in their writings ; and 

 the practical musician will testify, that these are the best notes he ever heard. 



:91 



Rgure the 1st, 

 containing the 

 proportions set 

 out arithmeti- 

 cally. 



An eighth 



A seventh 



A sixth 



A fifth 



A fourth 



A greater third 4 

 5 



8 9^ 



9 10 

 tone major : tone minor 



2 

 3 



8 



9 . 

 tone major 



9 



10 

 tone minor 



Is 



8 



9 

 tone major 



15 



16 

 bemitone 



ar; 



An octave with 



* 

 17 ^ 16 



r •' r -n^^n^^T? ^^ f ^7 „ lb" 19^ 18 17 . lb 15 

 a greater third. ^ Ti ^ I7 ^ 20 ^ l9^ TS ^ Ti ^ I^ ^ 25 ^ I9 ^ Ti l7 ^ l6 ^- 



* * 



17 t,l6^ 19^ 18 

 17 



* 

 17 . 16 



en< 



« ^''a ^^Ti^^r ^^r '^n^^n^^ 15 p »7 p l6p 19^ 18 . An octave with a 

 bi r8^T7 16^18^17^20^19 l6^Ti^T7^^ 19 lesser third ^. 



The Explication of the first Figure. — Between the two lowest lines, you 

 have the series of all the 12 half notes in an octave, from A-re to A-la-mi-re ; 

 which added together make an octave, or exact duple proportion : the several 

 parts also added together make all those intervals of which it is constituted. ' 

 As for example, the two half notes from a to A ♦ -j-r? ^nd from a ♦ to B^-f 

 make a major tone \ ; to which if a hemitone, from b to c w, be added, you 

 have a lesser third -f. 



In like manner between the two next lines, you have the series of all the 12 

 half notes, in an octave from c fa-ut to c sol-fa-ut : the first two tones added 

 together make a greater third : and so you may add a tone or hemitone, till 

 yop arrive at every interval in the octave, which is so called because, eight 



