Hoc in volumine haec opera continentur

2144, but b is 4312; but c is 1876; d, however, is 3248; but f is 9304; f is 2392; g is 1441; it is that g is greater than double [of] a.

A decorative drop cap 'D' is depicted. The diapason interval is multiple; for let the nete hyperbolaeon be a, the mese be b, [and] the proslambanomenos be c; therefore, the a c interval existing [as] a disdiapason double octave is consonant; therefore, it is either superparticular or multiple; but it is not sesqualter, for no proportional mean falls into a superparticular interval; therefore, it is multiple. Since, therefore, two intervals composed from b c make a whole multiple, therefore a b is also multiple. The diatessaron interval and the diapente fifth [are] each superparticular; for let the nete synemmenon be a, the mese be b, [and] the hypate meson be c; therefore, the a c interval existing [as] a disdiatessaron double fourth is dissonant; therefore, it is not multiple. Because, therefore, many equal intervals a b, b c composed do not make a whole multiple, therefore a b is not multiple; but it is a consonant superparticular; therefore the same demonstration [applies] also to the diapente. The diapason interval is double; for we have demonstrated it to be multiple, therefore it is double, and is greater than double; but because we have demonstrated that a double interval consists of two maximum superparticulars, if therefore the diapason is greater than double, it will not consist of only two superparticulars, but of more. It consists, however, of two consonant intervals, of the diapente and the diatessaron; therefore the diapason will not be greater than double; [it is] double; but because the diapason is double, and consists of the two maximum superparticulars, therefore the diapason consists of the sesqualter and the sesquitertius; for these are the maximum. They consist, namely, of the diapente and of the diatessaron, which are superparticular; therefore, because the diapente is greater [it] will be sesqualter, [and] the diatessaron will be sesquitertius; there is no doubt that the diapente and diapason is triple; for we have demonstrated that from a double interval and a sesqualter, a triple interval is made; likewise also the diapason and diapente [is] triple; but the disdiapason is quadruple; it is demonstrated, therefore, that any of the consonants has notes containing [intervals] in some ratios among themselves. It remains to discuss the toniaeo tonal interval, as it is sesquioctavum; for we have learned that if you take away a sesquitertius interval from a sesqualter interval, the remainder will be sesquioctavum. If the diatessaron is taken away from the diapente, the remainder will be the tonizum tonal interval; therefore, the tonizum interval is sesquioctavum. The diapason is less than six tones; for it is demonstrated that the diapason is double, but the tone is sesquioctavus, but six sesquioctava intervals are greater than a double interval; therefore, the diapason is less than six tones. The diatessaron is less than two tones and a half; let the nete diezeugmenon be b, the paramese be c, the mese be d, the hypate meson be f; therefore, the c d interval is a tone, but the b f, since it is a diapason, is less than six tones. The remainder, therefore, b c and d f, since they are equal, are less than five tones. Therefore, b c is less than two tones and a hemitonium, which is the diatessaron; but b d is less than three tones and a hemitonium, which is the diapentetonus; it will not be divided into two equal [parts] nor more; for it is demonstrated [to be] superparticular; but a superparticular interval has no mean, neither many nor one.