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...has the same type of sesquialtera A 3:2 ratio, which in music creates the interval of a Perfect Fifth., $12/8$. which in a certain sense can be called composite types; the same should be understood regarding other types of this class. Here are the examples:
The sesquialtera proportion The 3:2 ratio. consisting of its root terms.
3.
2.
The same proportion outside of root terms These are multiples of the root 3:2 ratio..
6. 12. 24. etc.
4. 8. 16.
The sesquitertia proportion The 4:3 ratio, producing a Perfect Fourth. consisting of root terms.
4.
3.
The same proportion outside of root terms.
8. 16. 32. etc.
6. 12. 24.
The root sesquiquarta proportion The 5:4 ratio, producing a Major Third..
5.
4.
The same proportion outside of root terms.
10. 20. 40. etc.
8. 16. 32.
The sesquiquinta proportion The 6:5 ratio, producing a Minor Third. in root terms.
6.
5.
The same proportion outside of root terms.
12. 24. 48. etc.
10. 20. 40.
This is to be understood for infinite other types, by observing the same method of ascending mentioned above.
CHAPTER VI.
On the third Type of Proportion.
The third type of proportions is called superpartient superpartient: A ratio where the larger number contains the smaller number once, plus more than one "part" of the smaller number (for example, 5:3, where 5 contains 3 once plus two-thirds of 3)., whose larger term contains the smaller once, and additionally some parts of it that are not aliquot parts, but "aliquants," or collective parts. original: "non aliquotas, sed aliquantas, seu aggregativas." An aliquot part (like 2 in 6) divides the whole exactly; an aliquant part (like 4 in 6) does not.
According to the diversity of the middle differential term which can intervene between proportions of this kind, various types and names also arise. If the middle term of the differ-
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