TEMPERAMENT. 



concords, yet, in other refpects, it fpoils and falufics both 

 perfetl and impcifeft concords, and renders difcords more 

 liarfh than they would otherwifc be, if the intervals were 

 julUy taken. To explain this, we inull confider that all the 

 intervals are founded on the primary proportions ariling from 

 the numbers 2, 3, and 5, that is, if we do not exceed the 

 compafs of an oftave, -], 4, and 4. Sec Intekval. 



The nearer we come in praftice to the true intervals, the 

 more perfeft the melody and harmony will be ; ajid it is cer- 

 tain, that the human voice, and fome inilrumcnts, as violins, 

 &c. which have no flops nor frets, will execute mufic to a 

 great degree of exaftnefs ; but the cafe is not the fame with 

 fixed or fretted inftruments, as harpfichords, organs, lutes, 

 viols, &c. Accuracy is here impoffible, unlefs we would 

 content ourfelves with always playing in the fame key, with- 

 out any tranfition or tranfpofition whatfocvcr. In this cafe, 

 indeed, the harpfichord or organ might vie with the accu- 

 racy of the voice or violin. For inftance : if we were to 

 compofe or play in the key of C, then we miglit make the 

 feveral intervals of that key to be in the following true pro- 

 portions, I, I, -^, -14, ?, -',^, i, 1 1, that is, in whole 



numbers, -{ , of- and the 



' I24 27 30 32 36 40 45 483 



mftrument tuned in this manner, would perform any piece 

 of mulic in C, juftly compofed, with great beauty and ex- 

 aftnefs ; taking for granted, that every key, fundamental 

 note, or found, ought to have its true fifth and fourth, and 

 that thefe ought alfo to have their true fifths and thirds. 



Now this being preniifed, it will prefently appear, that in 

 making any tranfpofition or tranfition from C, we fliall find 

 fome falfe concord. Thus, for inflance, if we proceed 

 to G, and confider it as a key, or fundamental found, we 

 fliall have the following feries of numbers for the o&ave 



Ot G, "VIZ. < ^ o iC /r ^ 



13.6 40 45 48 54 60 64 72 J 

 But here the interval between 40 and 54 is falfo, being 

 a comma too much, for the fecond of a key mufl make 

 a true fifth with the fifth of the fame key. In like man- 

 ner, if we were to proceed fi-om C to A, as a new key, 

 we fhould find the following feries for the oftave of A, 



fABCDEFGa-} , ,, •, , 

 -i n ^ t: of where the mterval 



{^40 45 48 54 60 04 72 80 J 



between A 40 and its fourth D 54 is falfe, being too great 

 by a comma. If any other tranfition were examined, we 

 ihail always find fome note falfe.; as in F, the fixth would 

 be redundant by a comma ; and in D, the fifth would be 

 deficient by a comma. All which (hews the impoffibility 

 of truth and exaftnefs of mufic on fixed inftruments. Yet 

 as thefe inftruments have their ule and convenience in fome 

 refpefts, it was proper to endeavour to find out a method of 

 making them tolerable. It has been obferved under the 

 article Interval, that the tone major exceeds the tone 

 minor by a comma. Their difference is ncccffary for the 

 truth and perfeftion of mufic ; but yet if thefe tones were 

 rendered equal, the ear would not be offended. And this 

 has fuggefted the means of tempering fixed inftruments. If 

 we were to make all tones equal to the tone major, as fome 

 imagine the ancients did, then we ftiould find the ditonus, or 

 third, exceeding a true third major by one comma, which 

 would be intolerable. In like manner, if all tones were to be 

 minor, we fhould have thirds major defedlive by a comma, 

 which would alfo be intolerable, not to mention other falfe in- 

 tervals that muft neceffarily arife from fuch a fuppofition. 



Suppofing then one tone increafed, and the others dinii- 

 nifhed by half a comma, we fhould have our thirds major 

 remain perfeft. But ttill it would be neceffary to examine 

 what fifths this fuppofition would give. Now it is evident 

 that a tone major added to an oftave, makes jufl two fifths, 



thus i X -J = 4 = .; X }. But the tone here added is a 

 tone major, and the tone we have afTumrd is a ttniporate 

 tone deficient from the tone major by half a comma ; hence 

 the fum of the twu fifths, on this fuppofition, will fall fhort 

 of the truth by half a comma, and confiquently one fifth 

 will be deficient a quarter of a comma. Which difTi-rcnce, 

 although it be fenfiblc, yet experience (hews, that fifths fo 

 diminithed are tolerable. 



This temperament is what is called the common or vulgar 

 temperament, and confifts, as lias been faid, in diminifhing 

 the fifth by a quarter of a comma, in preferving the third 

 major perfed, and dividing it into two equal tones. Which 

 being fuppofed, it follows that the fourth muft exceed the 

 truth by a quarter of a comma ; that the third minor will be 

 deficient by the fame quantity ; that the fixth minor will be 

 perfeft, and the fixth major redundant by a quarter of a 

 comma ; and laftly, that the feinitonc major will exceed the 

 truth by a quarter of a comma. If we introduce chromatic 

 notes, or flats and ftiarps, the femitone minor will alfo ex- 

 ceed the truth by a quarter of a comma, and confcqucntly 

 the difference between the two femitones, or the diefis en- 

 harmonica, will be preferved. 



If then we had a harpfichord or organ, with each feint 

 or half note divided, we ihould have the foHowing notes or 

 founds, viz. CC», Db, D, D», Eb, E, F*, Gb, 

 G, G.«, A b, A, A*, B b, B, c, in the compafs of an 

 oftavc. Yet this fyflem of notes, numerous as thev feem, 

 would not be fufficient for all tranfitions aid tranfpofitions. 

 For though a piece of mufic tranfpofed to any pf the na- 

 •tural keys C, D, E, F, G, A, B, and to the flats, as 

 E b and B b, and fome others, would do well ; yet, in 

 tranfpofing to fharps, as to C«, we (liould not find a true 

 third major, uidcfs we introduced E *. And even in flats, 

 as A b and Eb> we fhould 'not find a true third major in 

 defcending, or a fixth minor in afccnding, unlefs we intro- 

 duced Fb and Cb. And in like manner, tranfpofitions to 

 G« and Eb would oblige us to introduce B* and Cb. 

 Nor would even this fuffice, for if neceffity required a tranf- 

 pofition from the key of C to that of D *t, wo fhould not 

 find a true third major without introducing F** and r. 

 So that at laft we fhall come to a temperate fyftem, where, 

 in afcending, the notes C, D, F, G, A, would each have 

 its fliarp and double fharp, and the notes B and E each a 

 fingle Iharp. In defcending, the notes E, D, B, A, G, 

 would each have their flat and double flat, and the notes 

 F and C each a fingle ilat. And tlius the oftavc would 

 be divided into 3 1 intervals, whofo defignations are 

 C Dbb C* Db C«« D Ebb D« Eb D»« 



I 234 56 789 10 

 E Fb E» F Gbb F* Gb F*« G 



II 12 13 14 15 16 17 iS 19 

 Abb G* Abb G«f« A Bbb A* Bb 



20 21 22 2^ 24 25 26 27 



A«;* B C B* C. where the letters C, D, E, 



28 29 30 31 

 F, G, A, B, fignify the common diatonic notes : thofe 

 marked with a fingle * or b are the chromatic ; and thofe 

 marked « ith a double * * or b b arc enharmonic notes ; fo 

 called, becaufe the interval between them and the next dia- 

 tonic note is an enharmonic diefis ; for which rcafon, the 

 notes E*, Fb, and B «, Cb» arc alfo enharmonic. 



But even in this divifion of the oftave, all the notes would 

 not have a third major in afcending and defcending : thus, for 

 inftance, D** has no third major ; for this would be F***, 

 which is not in the fcale, nor can any number of additional 

 notes fuffice in all cafes. But this inconvenience is cafily reme- 

 died, and the fyftem confiderably improved, by making all the 

 thirty -one intervals equal. We have already obferved, that in 



the 



