Unit II — Five Species of Inequality & the Equality Unfolding Tree¶
Absolute vs. Relative Quantity, Multiple, Superparticular, Superpartient & Compound Ratios, and the Three Prostagma Commands Generating All Ratios from Equality (Book I, Chs. 17–23)
«ἵνα δέ σοι ἀποδειχθῇ σαφῶς καὶ ἀδιστάκτως, ὅτι ἄρα ἐξ ἰσότητος ἡ τῶν ἐν ταῖς σχέσεσιν εἰδῶν σύμπασα σύστασις πρόεισι πρῶτόν τε καὶ στοιχειῶδες ὑφισταμένης, ἔφοδός τις ἔσται ἡμῖν γλαφυρὰ καὶ ἀναγκαιοτάτη πρὸς πᾶσαν φυσιολογίαν…»
“And that you may have demonstrated to you clearly and indisputably that the whole system of species in relations proceeds from equality as its primordial and elementary root, there will be for us a graceful method, most indispensable for every study of nature…” — Book I, Ch. 23.4 (Hoche p. 65)
Book I, Ch. 17 — Equality & the Five Species of Relative Inequality¶
English Translation by Martin Luther D’Ooge (1926) · Hoche pp. 44–46 · D’Ooge pp. 212–214
1–2. Now that we have given a preliminary systematic account of absolute quantity (to kath’ heauto poson)—even, odd, prime, composite, superabundant, deficient, and perfect—it is next in turn to examine relative quantity (to pros ti poson). Of relative quantity there are two highest generic divisions: Equality (isotēs) and Inequality (anisotēs). For everything viewed in comparison with another thing is either equal to it or unequal, and there is no third besides these.
3–5. Equality (1 : 1) occurs when of the things compared neither exceeds nor falls short of the other, as a hundred to a hundred, ten to ten, two to two, a mina to a mina, a talent to a talent, a cubit to a cubit; and as a peculiar property this relation is natural and indivisible, incapable of being split into subspecies—for there is not one kind of equality and another kind, since equality remains one and identical.
6–8. The unequal, on the other hand, is split by division, and one part of it is the greater (meizon), the other the less (elatton), which have opposite names and relations to one another. Again, the greater is subdivided into five species, and the less correspondingly into five opposite (hypenantia) species:
Multiple (pollaplasios) — opposite to submultiple (hypopollaplasios)
Superparticular (epimorios) — opposite to subsuperparticular (hypepimorios)
Superpartient (epimerēs) — opposite to subsuperpartient (hypepimerēs)
Multiple superparticular (pollaplasiepimorios) — opposite to submultiple-superparticular
Multiple superpartient (pollaplasiepimerēs) — opposite to submultiple-superpartient
Book I, Ch. 18 — First Species: The Multiple (Pollaplasios)¶
Hoche pp. 46–49 · D’Ooge pp. 214–216
1–4. Again, then, the multiple (pollaplasios) is the first and most original by nature of all the species of the greater, as straightway we shall see, and it is a number which, when observed in comparison with another, contains the whole of that number more than once (\(m \cdot b : b\) for \(m \ge 2\)). For instance, compared with unity, all the successive numbers beginning with \(2\) generate the regular forms of multiplicity in turn: \(2\) in the first place is and is called double (diplasiōn), \(3\) triple (triplasiōn), \(4\) quadruple (tetraplasiōn), \(5\) quintuple, and so on indefinitely; and opposite to these stand half (hēmisys), third (tritos), fourth (tetartos), and so on.
Book I, Ch. 19 — Second Species: The Superparticular (Epimorios) & Musical Consonances¶
Hoche pp. 49–55 · D’Ooge pp. 216–219
1–3. The second species of the greater, both naturally and in order, is the superparticular (epimorios), which contains within itself the whole of the number compared plus some one aliquot factor (morion) of that lesser number (\((n + 1) : n\) for \(n \ge 2\)):
If this factor is half, the greater of the terms compared is called specifically sesquialter (hēmiolios, \(3:2, 6:4, 9:6, 12:8\)), governing the musical fifth (diapente).
If it is a third, it is called sesquitertian (epitritos, \(4:3, 8:6, 12:9\)), governing the musical fourth (diatessaron).
If it is a fourth, epitetartos (\(5:4\)); if an eighth, sesquioctave (epogdoos, \(9:8\)), governing the whole tone (tonos) by which the fifth exceeds the fourth (\(\frac{3}{2} \div \frac{4}{3} = \frac{9}{8}\)).
4–9. Observe the wonderful order of nature: when the natural series from \(1\) is compared with the even series \(2, 4, 6, 8, \dots\) vs. the triple series \(3, 6, 9, 12, \dots\), the root terms (\(3:2, 4:3, 5:4, \dots\)) admit no intervening integer and reveal the pure root ratios, while their multiples (\(6:4, 9:6, 12:8, \dots\)) preserve the exact same relation across an infinite multiplication table.
Book I, Chs. 20–22 — Superpartient & the Two Compound Ratio Species¶
Hoche pp. 55–64 · D’Ooge pp. 219–225
I.20 (Superpartient / Epimerēs). We have the superpartient relation (epimeres) when a number contains within itself the whole of the number compared and in addition more than one aliquot part of it (\((n + r) : n\) where \(1 < r < n\) and \(\gcd(n, r) = 1\))—beginning with two parts (epidimerēs / superbipartient, \(5:3\) or disepitetartos in the looser speech for \(5:3\) as \(3 + 2\)), three parts (epitrimerēs / supertripartient, \(7:4\)), four parts (\(9:5\)), and in general comparing successive odd numbers from \(5\) (\(5, 7, 9, 11, \dots\)) with consecutive integers from \(3\) (\(3, 4, 5, 6, \dots\)).
I.21–22 (Multiple Superparticular & Multiple Superpartient). From the blending of the simple species arise the two compound classes:
The multiple superparticular (pollaplasiepimorios) occurs when the greater term contains the lesser more than once plus one aliquot part (\((m \cdot n + 1) : n\) with \(m \ge 2, n \ge 2\)), such as duplex sesquialter (diplasiepēmiolios, \(5:2 = 2 \cdot 2 + 1 : 2\)), duplex sesquitertian (\(7:3\)), or triplex sesquialter (\(7:2\)).
The multiple superpartient (pollaplasiepimerēs) occurs when the greater term contains the lesser more than once plus several aliquot parts (\((m \cdot n + r) : n\) with \(m \ge 2, 1 < r < n, \gcd(n, r) = 1\)), such as duplex superbipartient (\(8:3 = 2 \cdot 3 + 2 : 3\)) or duplex supertripartient (\(11:4\)).
Book I, Ch. 23 — The Three Prostagma Commands Unfolding All Ratios from Equality¶
Hoche pp. 64–72 · D’Ooge pp. 225–229
I.23.4–8 (The Three Commands). That the whole system of species in relations proceeds from equality as its first and elementary root will become indisputable by the following graceful method. Let there be given to us three terms in equality—either three units (1, 1, 1), or three twos \((2, 2, 2)\), or three threes, or any three equal numbers whatever—and from them let three new terms be molded according to three unvarying commands (tria prostagmata) as if by a law of nature:
Make the first equal to the first: \(a' = a\)
Make the second equal to the first plus the second: \(b' = a + b\)
Make the third equal to the first, plus twice the second, plus the third: \(c' = a + 2b + c\)
I.23.8–15 (Generation of Every Species in Exact Order). Watch how every species of inequality springs forth in natural sequence without our contrivance:
Step 1 — All Multiples from Equality: Apply the three commands to (1, 1, 1): first \(= 1\), second \(= 1 + 1 = 2\), third \(= 1 + 2(1) + 1 = 4\), producing the double progression (1, 2, 4). Apply the same commands directly to \((1, 2, 4)\) without rearrangement, and out comes the triple (1, 3, 9); from \((1, 3, 9)\) springs the quadruple (1, 4, 16), and so on forever through all multiples!
Step 2 — Superparticulars from Reversed Multiples (*Anastrophe*): Now take those very multiples and reverse their order (anastrophe, placing the greatest term first). From reversed double \((4, 2, 1)\), the three commands yield \(4,\; 4+2=6,\; 4+4+1=9\), namely (4, 6, 9)—the first superparticular (hemiolios, \(3:2\))! From reversed triple \((9, 3, 1)\) springs (9, 12, 16) (epitritos, \(4:3\)); from reversed quadruple \((16, 4, 1)\) springs \((16, 20, 25)\) (epitetartos, \(5:4\)), and so on!
Step 3 — Compound & Superpartient Species from Superparticulars: From the superparticulars thus born, applying the three commands directly (starting from the least term) generates the multiple superparticulars—for \((4, 6, 9)\) directly yields (4, 10, 25) (duplex sesquialter, \(5:2\)), and \((9, 12, 16)\) yields \((9, 21, 49)\) (duplex sesquitertian, \(7:3\))—whereas applying the three commands to the reversed superparticulars (\((9, 6, 4)\) and \((16, 12, 9)\)) generates the superpartients: (9, 15, 25) (superbipartient, \(5:3\)) and \((16, 28, 49)\) (supertripartient, \(7:4\)), while direct children of superpartients such as \((9, 15, 25)\) yield multiple superpartients like (9, 24, 64) (\(8:3\))! Notice too the universal marvel: in every generated triple, the extreme terms are invariably squares (\(1, 4, 9, 16, 25, 49, 64, \dots\)) surrounding their geometric mean!