Steps to Parnassus (Gradus ad Parnassum)

...an aliquot part original: "partem aliquotam"; a part that divides into the whole perfectly, like 2 into 4.. For example, in this proportion: 3/2. In the sesquialtera The 3:2 ratio, which in music produces the interval of a Perfect Fifth., the larger number contains the smaller once, and at the same time contains one aliquot part of it; which is clearly evident through the division of the larger number by the smaller: $^3/_2. | 1 \text{ \textonehalf}$. Here, the number three contains the number two, and additionally its half part, which when taken twice makes exactly two, its whole.

Infinite species of this kind can exist, which also receive different names according to the diversity of the numbers: such as Sesquialtera, sesquitertia, sesquiquarta, sesquiquinta, sesquisexta, etc.

Before I prepare myself for the method of forming this kind, it is necessary to know what the little word Sesqui means: namely, that this word signifies the half of some whole. From this it is clear that this particle can be applied to no other species of this kind except the sesquialtera, in which alone the larger number includes the smaller once, and additionally its half part; for in the sesquitertia proportion $^4/_3.$ The 4:3 ratio, which in music produces the interval of a Perfect Fourth., the larger number contains the smaller once indeed, but not a half, but only a third part of it: $^4/_3. | 1 \text{ \textonefrac13}$. In this proportion $^5/_4.$ The 5:4 ratio, known as a Major Third., the part remaining from division is not a half, but a fourth part: $^5/_4. | 1 \text{ \textonefrac14}$. , and so in all other species. However, because it is already accepted by use—or rather by misuse—that this particle is prefixed to all species of this kind, I have judged it more advisable to change nothing to avoid confusion.

This kind is formed in this easy way. Let two numbers be taken, leaving aside unity, in the order they follow each other in the Pythagorean table The standard multiplication table used since antiquity., placing the larger above and the smaller below, as: $^3._2, ^4._3, ^5._4, ^6._5, ^7._6, ^9._8$ etc. Behold six ratios, or proportions, differing in species; the first of which is called sesquialtera, the second sesquitertia, the third sesquiquarta, the fourth sesquiquinta, the fifth sesquisexta, the sixth sesquioctava The 9:8 ratio, a Whole Tone., and each of these takes its name from its denominator, or the smaller term placed below.

All ratios consisting of their root terms original: "terminis radicalibus"; the simplest form of a ratio (e.g., 3:2 is the root of 6:4). can also exist outside of root terms, which is achieved by doubling the terms of the ratio. For example: let the root terms of the sesquialtera ratio be doubled (unless one prefers to say sesquialterae A grammatical correction from the masculine to feminine form., neglecting the laws of Grammarians as we are accustomed to do with technical terms) $^3/_2.$ in this way $^6/_4.$, which ratio is of the same sesquialtera kind outside its root terms. This ratio doubled again exhi-