Unit IV — The Ten Neo-Pythagorean Means & the Cube’s Most Perfect Proportion¶
Arithmetic, Geometric & Harmonic Proportions, the Complete System of Ten Means, Philolaus’s Cube, and the Teleiotatē Analogia (6 : 8 :: 9 : 12) (Book II, Chs. 21–29)
«λοιπὸν ἐπὶ τέλει τῆς εἰσαγωγῆς καιρὸς ἂν εἴη τὴν τελειοτάτην καὶ τριχῇ διαστατὴν καὶ πασῶν περιεκτικωτάτην ἀναλογίαν ἐν τέσσαρσιν ὅροις… ἐκθέσθαι, χρησιμεύουσαν εἴς τε πάσας τὰς ἐν μουσικῇ προκοπὰς καὶ εἰς πάσας φυσιολογίας.»
“Finally, at the close of our Introduction, it would be seasonable to set forth the most perfect proportion—three-dimensional, four-termed, and most comprehensive of all—serviceable for all advances in music and all investigations of nature.” — Book II, Ch. 29.1 (Hoche p. 144)
Book II, Chs. 21–24 — Proportion (Analogia) & the Arithmetic and Geometric Means¶
English Translation by Martin Luther D’Ooge (1926) · Hoche pp. 119–131 · D’Ooge pp. 264–273
II.21–22 (Proportion Strictly vs. Broadly; Continuous vs. Disjunct). After these matters, it would be the proper time to introduce the nature of proportions (analogiai), a subject most essential for speculation on the nature of the universe and for the doctrines of music, astronomy, geometry, and perusal of the works of the ancients. A proportion is, strictly speaking, the bringing together to the same point of two or more ratios; more broadly, of two or more relations even if they are subjected not to the same ratio, but to a regulated difference or some similar law. The first three proportions, acknowledged by all the ancients—Pythagoras, Plato, and Aristotle—are the Arithmetic, Geometric, and Harmonic (originally called subcontrary, hypenantia); three more exist directly opposite to them (Means 4, 5, 6), and four others discovered by later investigators (Myonides and Euphranor) bring the total to ten—the Pythagorean Decad, most perfect of numbers! A proportion in three terms (\(a : b :: b : c\), middle term taken twice) is continuous (synēgmenē), whereas one in four distinct terms (\(a : b :: c : d\), e.g. \(6 : 8 :: 9 : 12\) in arithmetic difference or geometric/harmonic ratio) is disjunct (diezeugmenē).
II.23 (The Arithmetic Proportion & Its Subtle Square Property). We have an arithmetic proportion whenever three or more terms exceed and are exceeded by equal differences (\(c - b = b - a = d\)), as \(1, 2, 3\) or \(6, 9, 12\) (where \(b = \frac{a + c}{2}\), geometrically the radius OR = A equal to PO and OS on semicircle PR). Notice here two remarkable properties:
While the differences are equal, the ratios are larger among the smaller terms and smaller among the larger terms (in \(1, 2, 3\), \(2:1 > 3:2\); in \(6, 9, 12\), \(9:6 = 3:2 > 12:9 = 4:3\)).
A subtle property missed by most: the product of the extremes falls short of the square of the middle term by the exact square of the common difference (\(b^2 - ac = d^2\))! Thus in \(6, 9, 12\) (difference \(d = 3\)), \(6 \times 12 = 72\), which plus \(3^2 = 9\) equals \(9^2 = 81\)!
II.24 (The Geometric Proportion — The Only Proportion Strictly So Called). The geometric proportion, the only one strictly entitled to the name analogia, occurs when the terms stand to one another in the same ratio (\(c : b = b : a\)), as \(1, 2, 4\) or \(4, 6, 9\), so that the product of the extremes equals the exact square of the mean (\(ac = b^2\), realized geometrically by the perpendicular ordinate QS = G to semicircle PR). Moreover, in continuous geometric progressions, consecutive differences retain the exact same ratio as the terms themselves, plane numbers are bonded by one geometric mean (e.g. \(4, 6, 9\)), and solid cubes are bonded by two geometric means in continuous proportion (e.g. \(8, 12, 18, 27\) between \(2^3\) and \(3^3\), cf. plato:timaeus:27d-36d, 31c–32b; euclid:elements:vii-ix, VIII.11–12).
Book II, Chs. 25–27 — The Harmonic Mean, Philolaus’s Cube & Musical Consonances¶
Hoche pp. 131–139 · D’Ooge pp. 273–281
II.25 (The Harmonic Proportion). The third proportion, the harmonic (harmonikē), occurs whenever among three terms the greatest stands to the smallest in the same ratio as the difference between greatest and middle stands to the difference between middle and smallest:
as in \(3, 4, 6\) (where \(6 : 3 = (6 - 4) : (4 - 3) = 2 : 1\)) or \(6, 8, 12\) (where \(12 : 6 = (12 - 8) : (8 - 6) = 2 : 1\), geometrically realized on semicircle PR by radial projection segment QF = H satisfying \(G^2 = A \cdot H\)). Contrary to the arithmetic mean, in the harmonic proportion the ratios among the larger terms are greater than those among the smaller terms (in \(3, 4, 6\), \(6:4 = 3:2 > 4:3\); in \(6, 8, 12\), \(12:8 = 3:2 > 8:6 = 4:3\)—which is why it was originally called subcontrary!), and adding the extremes and multiplying by the mean gives twice the product of the extremes (\((a + c)b = 2ac\)).
II.26 (Musical Consonances & Philolaus’s Cube as “Geometric Harmony”). Why was this proportion named harmonic? First, because in \(6, 8, 12\) its extreme ratio \(12 : 6 = 2 : 1\) is the fundamental consonance diapason (octave), \(12 : 8 = 3 : 2\) is the diapente (fifth), and \(8 : 6 = 4 : 3\) is the diatessaron (fourth)! Second, and most strikingly, according to Philolaus, because of its presence in all geometric harmony—which Philolaus identifies with the Cube, harmonious across all three dimensions (platē, mēkē, bathē): for every cube has 6 plane faces, 8 solid angles (vertices), and 12 bounding edges, forming the exact harmonic triple 6, 8, 12!
II.27 (Rules for Inserting the Three Classical Means). Between any two extremes (both even or both odd, like \(10\) and \(40\)), the arithmetic mean is \(\frac{a + c}{2} = 25\), the geometric mean is \(\sqrt{ac} = \sqrt{400} = 20\), and the harmonic mean is \(\frac{2ac}{a + c} = \frac{800}{50} = 16\) (\(16 < 20 < 25\) with \(20^2 = 16 \times 25 = 400\)).
Book II, Ch. 28 — The Seven Additional Means Completing the Ten Proportions¶
Hoche pp. 139–144 · D’Ooge pp. 281–284
II.28.1–11 (The Complete Decad of Proportions). Following the three classical means (Means 1–3), three subcontrary means arise by reversing their defining ratios, and four more discovered by later Pythagoreans (Myonides and Euphranor) compare the difference of extremes \((c - a)\) against one of the inner differences:
4th Mean (Subcontrary to Harmonic): \((c - b) : (b - a) = a : c\), exemplified in \(3, 5, 6\) (\((6-5):(5-3) = 1:2 = 3:6\)).
5th Mean (Subcontrary to Geometric I): \((c - b) : (b - a) = a : b\), exemplified in \(2, 4, 5\) (\(1:2 = 2:4\)).
6th Mean (Subcontrary to Geometric II): \((c - b) : (b - a) = b : c\), exemplified in \(1, 4, 6\) (\(2:3 = 4:6\)).
7th Mean: \((c - a) : (b - a) = c : a\), exemplified in \(6, 8, 9\) (\((9-6):(8-6) = 3:2 = 9:6\)).
8th Mean: \((c - a) : (c - b) = c : a\), exemplified in \(6, 7, 9\) (\((9-6):(9-7) = 3:2 = 9:6\)).
9th Mean: \((c - a) : (b - a) = b : a\), exemplified correctly in \(4, 6, 7\) (\((7-4):(6-4) = 3:2 = 6:4\); see Critical Notes on the manuscript variant \(4, 7, 8\)).
10th Mean (Additive / Fibonacci Identity): \((c - a) : (c - b) = b : a\), which simplifies identically for \(b \neq c\) to \(c - a = b \iff a + b = c\), exemplified in 3, 5, 8 (\((8-3):(8-5) = 5:3\) and \(3 + 5 = 8\))!
Book II, Ch. 29 — The Most Perfect Proportion (Teleiotatē Analogia, 6 : 8 :: 9 : 12)¶
Hoche pp. 144–147 · D’Ooge pp. 284–286
II.29.1–5 (Crown of the Treatise). Let us crown our Introduction with the Most Perfect Proportion (teleiotatē analogia)—three-dimensional, four-termed, containing within a single tetrad all three classical proportions together with every musical consonance and the three-dimensional solid architecture of the Cube:
Let there be taken two solid cubes—for example the even cube \(8 = 2 \times 2 \times 2\) and the odd cube \(27 = 3 \times 3 \times 3\) (or most simply in root integers, the extreme terms 6 and 12, which themselves arise from the Cube’s 6 plane faces, 8 solid vertices, and 12 linear edges). Between 6 and 12 insert both interior means on the single four-term monochord 6 : 8 :: 9 : 12:
Geometric Proportion across all four terms: \(12 : 9 = 8 : 6\) (\(4 : 3\)) and alternately \(12 : 8 = 9 : 6\) (\(3 : 2\)), while the product of the extremes equals the product of the means: \(6 \times 12 = 8 \times 9 = \mathbf{72}\)!
Arithmetic Proportion: 9 exceeds 6 by the exact same difference (\(3\)) by which 12 exceeds 9 (\(6, 9, 12\)), and equals half the sum of the extremes (\(\frac{6 + 12}{2} = 9\)).
Harmonic Proportion: 8 exceeds 6 and is exceeded by 12 by the exact same fraction (one-third) of each extreme (\(8 - 6 = 2 = \frac{1}{3} \cdot 6\) and \(12 - 8 = 4 = \frac{1}{3} \cdot 12\)), matching the Cube’s 6 faces, 8 vertices, and 12 edges!
Complete Pythagorean Musical Scale: - Extreme to extreme \(12 : 6 = 2 : 1\) sounds the Diapason (Octave). - Greater extreme to harmonic mean (\(12 : 8\)) and arithmetic mean to lesser extreme (\(9 : 6\)) both equal \(3 : 2\), sounding the Diapente (Fifth). - Greater extreme to arithmetic mean (\(12 : 9\)) and harmonic mean to lesser extreme (\(8 : 6\)) both equal \(4 : 3\), sounding the Diatessaron (Fourth). - Arithmetic mean to harmonic mean (\(9 : 8\)) sounds the Epogdoos Whole Tone, the common measure of musical intervals!