a ae eae ee ae ee ee eee eee 
— 4 eS ee ee ae ee ee ee ee ee en ee ee 
Fe ee EE ee eT ee ee SI LP ee eS ee ee ee a — ee ” 
On Musical 7 emperament. 21 
ous in their kinds with these concords respectively 5 according 
to the corollary of Prop. I. | 
Hence we have only to find those temperaments of the Vths, 
IIIds, and $dsg, in the compass of one octave, which will render 
them all, as nearly as possible, equal! y harmonious. The tem- 
peraments of the different concords of the same name ought 
evidently to be rendered equal ; since, otherwise, their har- 
mony cannot be equal. This can be effected only by render- 
ing the major and minor tones equal, and preserving the equali- 
ty of the two semitones. If this is done, the temperament of 
all the TI Ids will be-equal, since they will each be the sum of 
two equal tones. For a similar reason the Sds, and consequent- 
ly the Vths, formed by the addition of ILIds, and 3ds, will 
be equally tempered. Sere 
In order to reduce the octave to five equal and variable tones, 
and two equal and variable semitones, we will suppose the 
intervals of the untempered octave to be represented by 
3c—5x 3e—5x 
Cae 5 ae: a [pel ge hg Eee By 
c OS ee ee eee oe ee 
parts CD, DE, &c, of the line Ce, Denoting the comma by 
e, we will suppose the tone DE, which is naturally minor, to 
be increased by any variable quantity, 2 ; then, by the fore- 
going observations, the other minor tone, GA, must be increas- 
ed by the same quantity. As the major tones must be render- 
ed equal to the minor, their increment will be —c. As the 
octave is to be perfect, the variation of the two semitones must 
be the same with that of the five tones, with the contrary sign ; 
and as they are to be equally varied, the decrement of each 
will be 2% —5¢3 or, what amounts to the same thing, the 
@ | 
increment of each will be — . 
The several concords of the same name in this octave are 
now affected with equal and variable temperaments. The 
common increment of the [IIds will be 22— c ; that of the Sds 
4-¢— Sx; and consequently that of the Vths 4. 2—c. 
