Unit IV — The Ten Neo-Pythagorean Means & the Cube’s Most Perfect Proportion — Structural Map

Semicircle Geometry, the Decad of Means & the Cube’s Teleiotatē Analogia · Book II, Chs. 21–29

Nicomachus II.29 — The Most Perfect Proportion (Τελειοτάτη Ἀναλογία): 6 : 8 :: 9 : 12 ϛ, η, ϑ, ιβ · Cube Topology (6 Faces, 8 Vertices, 12 Edges) & Pythagorean Musical Consonances Philolaus's "Geometric Harmony" 6 Faces · 8 Vertices · 12 Edges Harmonic Cube: 8 = 2·6·12 / (6 + 12) 6 = 1×2×3 (scalene) · 8 = 2×2×2 (cube) 9 = 1×3×3 (brick) · 12 = 2×2×3 (beam) Euler: F − E + V = 6 − 12 + 8 = 2 Projective Harmonic Conjugate & Monochord Consonances K (Steiner apex) Z=0 6 (ϛ) 8 (η) 9 (ϑ) 12 (ιβ) 9:8 Tone 4:3 Fourth 12:9 = 4:3 Diapente (Fifth) 9:6 = 12:8 = 3:2 Diapason 12:6 = 2:1 · 6×12 = 8×9 = 72 · CR(6,12;8,0) = −1
Book II, Ch. 29: Unification of polyhedral topology (6 faces, 8 vertices, 12 edges), three proportions (A=9, H=8, G^2=72), and musical consonances (2:1, 3:2, 4:3, 9:8).

Architectural Thesis

Unit IV crowns the Introductio Arithmetica by uniting arithmetic, geometry, stereometry, and harmonics in the theory of means:

  1. The Three Classical Means & Semicircle Identity (II.21–27): Arithmetic (\(A = \frac{a+c}{2}\)), Geometric (\(G = \sqrt{ac}\)), and Harmonic (\(H = \frac{2ac}{a+c}\)) means satisfy \(H \le G \le A\) and the exact geometric mean law \(G^2 = A \cdot H\), constructed on semicircle PR via radius OR = A, ordinate QS = G, and projection QF = H.

  2. The Complete Decad of Ten Proportions (II.28): Adding the three subcontrary means (4–6) and the four outer-difference means of Myonides and Euphranor (7–10) completes the Pythagorean Decad, culminating in Mean 10 (3, 5, 8) where \(a + b = c\).

  3. The Most Perfect Proportion 6 : 8 :: 9 : 12 (II.29): Built upon Philolaus's Cube (6 faces, 8 vertices, 12 edges), the four-term teleiotatē analogia embeds Arithmetic (\(9\)), Harmonic (\(8\)), and Geometric (\(6 \cdot 12 = 8 \cdot 9 = 72\)) symmetry alongside the octave 2:1, fifth 3:2, fourth 4:3, and whole tone 9:8.

Synoptic Table of the Ten Means

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Defining Proportion (\(a < b < c\))

Closed Formula for \(b\)

Canonical Triple

1

\((c - b) : (b - a) = 1 : 1\) (Arithmetic)

\(b = \frac{a + c}{2}\)

\((1, 2, 3)\) / \((6, 9, 12)\)

2

\((c - b) : (b - a) = c : b = b : a\) (Geometric)

\(b = \sqrt{ac}\)

\((1, 2, 4)\) / \((4, 6, 9)\)

3

\((c - b) : (b - a) = c : a\) (Harmonic)

\(b = \frac{2ac}{a + c}\)

\((3, 4, 6)\) / \((6, 8, 12)\)

4

\((c - b) : (b - a) = a : c\) (Subcontrary)

\(b = \frac{a^2 + c^2}{a + c}\)

\((3, 5, 6)\)

5

\((c - b) : (b - a) = a : b\)

\(b = \frac{c - a + \sqrt{(c-a)^2 + 4a^2}}{2}\)

\((2, 4, 5)\)

6

\((c - b) : (b - a) = b : c\)

\(b = \frac{a - c + \sqrt{(c-a)^2 + 4c^2}}{2}\)

\((1, 4, 6)\)

7

\((c - a) : (b - a) = c : a\)

\(b = \frac{a^2 + ac - a^2 \text{ [}2ac - a^2\text{]}}{c} = \frac{a(2c - a)}{c}\)

\((6, 8, 9)\)

8

\((c - a) : (c - b) = c : a\)

\(b = \frac{a^2 + c^2 - ac}{c}\)

\((6, 7, 9)\)

9

\((c - a) : (b - a) = b : a\)

\(b = \frac{a + \sqrt{a(4c - 3a)}}{2}\)

\((4, 6, 7)\) (corr.)

10

\((c - a) : (c - b) = b : a\) (Fibonacci)

\(b = c - a \iff a + b = c\)

\((3, 5, 8)\)