On the Institution of Music (De institutione musica)

...thereafter nothing is missing, and nothing is in excess. It is called either double, or triple, or quadruple, and in this order it proceeds into infinity. ¶ The second kind of inequality is that which is called superparticular Latin: superparticulare. A ratio where the larger number contains the smaller number plus one part (fraction) of it, such as 3:2 or 4:3.. That is, when the larger number has the smaller within itself in its entirety, plus one certain part of it. And if that part is a half, such as three to two, it is called a sesquialter proportion The ratio 3:2, which in music forms the interval of a Perfect Fifth.. Or if it is a third, such as four to three, it is called sesquitertian The ratio 4:3, which forms the interval of a Perfect Fourth.. And in this manner, even in later numbers, some part is contained by the larger over the smaller numbers. ¶ The third kind of inequality is whenever the larger number contains the whole smaller number within itself and some parts of it besides. And if it contains two parts above, it shall be called a superbipartient proportion, such as 5 to 3. If indeed it contains three parts above, it shall be called supertripartient, such as 7 to 4. And in other cases, the same similarity can exist. ¶ The fourth kind of inequality is that which is joined from the multiple and the superparticular. This is when the larger number has the smaller number within itself either twice or thrice or as many times as you like, plus one certain part of it. And if it has it twice plus its half part, it shall be called double sesquialter, such as 5 to 2. But if the smaller is contained twice plus its third part, it shall be called double sesquitertian, such as 7 to 3 The OCR text says 7 to 4, but mathematically 7:3 fits the definition of 2 + 1/3.. If it is contained three times plus its half part, it shall be called triple sesquialter, such as 7 to 2. And in the same way in other cases, the names of multiplicity and superparticularity are varied. ¶ The fifth is the kind of inequality called multiple superpartient, when the larger number has the smaller number within itself in its entirety more than once, and more than one certain part of it. And if the larger number contains the smaller twice, plus two parts of it besides, it shall be called double superbipartient, such as 8 to 3. And again, triple superbipartient, such as 11 to 3. And concerning these things, we now explain strictly and briefly, because we have written of them in the books on the Fundamentals of Arithmetic original: "Arithmetica institutione". This refers to Boethius's own earlier work, De institutione arithmetica., where we have unraveled them more diligently. Chapter V.

Which types of inequality are suited to consonants

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From these kinds of inequality, therefore, the last two are left aside because they are mixed from the ones above; but a study must be made of the three superior ones. The multiple e.g., 2:1, 3:1 seems to hold the greater power for consonances Musical intervals that sound "stable" or "pleasing" to the ear., and consequently the superparticular. The superpartient, however, is separated from the containment of harmony, as it seems to everyone except Ptolemy Claudius Ptolemy, the 2nd-century astronomer and music theorist..

Why multiplicity and superparticularity are assigned to consonances · Chapter VI.

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For those things are proven to agree in comparison which are simple by nature. And since unity and numbers consist in quantity, those things will especially seem to contain a sweet nature which are able to preserve the property of discrete quantity original: "discretae quantitatis". Quantity made of distinct, separate units (like integers), as opposed to continuous quantity (like a line).. For while one quantity is indeed discrete and another is continuous, that which is discrete is smallest in its beginning i.e., the number 1, but proceeds through larger things into infinity. For in this, the smallest is unity, and the same is finite. But the mode of plurality is increased into infinity by numbers which, since they begin from finite unity, have no end of growing. Conversely, that which is continuous is finite as a whole, but is diminished through the finite. For a line, which is continuous, is always diminished in infinite partition, whether its total is a foot long or any other defined measure. Wherefore number always grows together into infinity, but continuous quantity is diminished into infinity. Much...