De institutione musica I.4: De speciebus inaequalitatis.
Things that are unequal keep among themselves the moments of inequality in five ways. For either one is exceeded by the other by multiplicity, or by single parts, or by several, or by multiplicity and a part, or by multiplicity and parts. And the first kind of inequality is called multiple. It is multiple where the greater number contains the smaller number in itself entire, either twice or three times or four times and so on, and nothing is lacking, nothing exceeds. It is called double or triple or quadruple, and proceeds in this order to infinity. The second kind of inequality is that called superparticular, that is when the greater number contains the smaller number entire and one some part of it, either the half, as three of two, and it is called the sesquialter proportion, or the third, as four to three, and it is called sesquitertian, and in this manner also in the later numbers some part is contained by the greater over the smaller numbers. The third kind of inequality is whenever the greater number contains within itself the smaller entire and moreover some parts of it. If it contains two parts above, it will be called the superbipartient proportion, as are five to three; if it contains three above, it will be called supertripartient, as are seven to four, and in the others the same likeness can hold. The fourth kind of inequality is that which is joined of the multiple and superparticular, when the greater number has in itself the smaller number either twice or three times or however often and one part of it; and if it has it twice and its half part, it will be called double supersesquialter, as are five to two; but if the smaller is contained twice and its third part, it will be called double supersesquitertian, as are seven to three. If it is contained three times and its half part, it will be called triple supersesquialter, as are seven to two, and in the same way in the others the names of multiplicity and superparticularity are varied. The fifth kind of inequality is called multiple superpartient, when the greater number has in itself the smaller number entire more than once and more than one part of it. And if the greater number contains the smaller twice and two parts of it besides, it will be called double superbipartient, as are eight to three, and again triple superbipartient, as are 11 to three. We explain these things briefly now for the reason that we have set them out more carefully in the books which we wrote on the instruction in arithmetic.