Attempt at a New Theory of Music

...relations are reported, which can be perceived with equal ease: thus, to the first degree belongs the single, simplest ratio of all—equality—which, wherever it is present, is most easily noticed immediately; two equal sounds constitute this ratio. Following this is the second degree, to which likewise no more than one ratio may be assigned: the double ratio original: "ratio dupla"; in music, this is the 2:1 ratio. This is perceived more easily than all others except for the ratio of equality, and among musical sounds, it encompasses the interval which is called the diapason or the octave The term "diapason" is the Greek name for the octave, literally meaning "through all [the notes]". To the third degree, however, it seemed appropriate to assign two ratios, namely the triple and the quadruple, since these two ratios are perceived with equal facility. In this manner, we have followed the remaining degrees in order, assigning to each those ratios that are equally easy to perceive. We call these the "degrees of sweetness" original: "gradus suauitatis"; this refers to the sensory pleasure or consonance of the sound, because from them it may be understood how much sweetness each consonance...