Gate 116: זל — RATIO

זל

Pillar 5: NUMBER AND PATTERN

[116:1] "Of all bonds the best is that which makes itself

[116:2] and the terms it connects a unity in the fullest sense;

and it is of the nature of a continued geometrical proportion to effect

this most perfectly."

[116:3] --- Plato, Timaeus 31c

[116:4] "The musical proportion 6:8::9:12 exemplifies the

science*

[116:5] Plato labels stereometry---the gauging of solids."

[116:6] --- Ernest McClain, The Pythagorean Plato

[116:7] "Ratio is the relation of two magnitudes of the same

kind."

[116:8] --- Euclid, Elements V

[116:9] [116:1] A ratio is a relationship between two

quantities---and relationship is more fundamental than quantity.

[116:10] [116:2] The number 6 alone tells you little. The number

12 alone tells you little. But the ratio 6:12 (which is 1:2) tells you

something precise and eternal: the octave relationship, the doubling,

the return to the same at a different level. Ratios are the language of

form because form is structure, and structure is relationship.

[116:11] [116:3] The musical proportion 6:8::9:12 was the

cornerstone of Pythagorean philosophy. Between 6 and 12 (the octave),

there are two means: the arithmetic mean (9) and the harmonic mean (8).

These divide the octave into a fourth (6:8, or 3:4) and a fifth (8:12,

or 2:3). All of Greek music theory---and by extension, all of Platonic

cosmology---rests on this proportion.

[116:12] [116:4] Why these numbers? Because they are the

smallest integers that exhibit the structure. You could use 12:16::18:24

(doubling everything) and preserve the same ratios. The Pythagorean

aesthetic demanded the smallest integers possible---what mathematicians

call "lowest terms." Simplicity reveals essence.

[116:13] [FIGURE 116.1: The musical proportion mapped onto a

circle: 6 at bottom, 12 at top (octave), 8 and 9 as the two means that

divide the space. The octave as the "mother," the means as the

children.] [116:5] Ratios connect number to geometry. The ratio 1:2

is not only an octave; it is the relationship between a square and a

square twice its area. The ratio 1:√2 is the diagonal of a unit

square---irrational, incommensurable, but geometrically precise.

Geometry is ratio made visible; ratio is geometry abstracted.

[116:14] [116:6] The three means---arithmetic, geometric, and

harmonic---are the fundamental tools of proportion. The arithmetic mean

(A+B)/2 divides the distance equally. The harmonic mean 2AB/(A+B)

divides inversely. The geometric mean √(AB) is the "mean

proportional"---the term that makes a continuous proportion (A is to

√(AB) as √(AB) is to B). Plato said this geometric mean is "the best

bond."

[116:15] [116:7] In music, the arithmetic and harmonic means

produce different intervals depending on whether you measure up or down.

This reciprocity---what McClain calls "dialectical tonal

meanings"---is essential to Platonic mathematics. Every ratio has two

aspects, like two sides of a coin, and wisdom requires holding both

simultaneously.

[116:16] [116:8] The golden ratio φ (approximately 1.618) is the

ratio where the whole is to the larger part as the larger part is to the

smaller: (A+B)/A = A/B = φ. This self-similar ratio appears throughout

nature---in spirals, in branching patterns, in the proportions of the

human body---because it is the ratio of balanced growth.

[116:17] [116:9] When the traditions speak of "proportion" and

"harmony," they mean ratio. A beautiful face is not absolutely large

or small; it is well-proportioned---the parts in harmonic ratio with

each other. A just society is not rich or poor absolutely; it is

balanced---the classes in right relationship. Ratio is the key to

beauty, justice, and truth.

[116:18] [116:10] To think in ratios is to think relationally.

Instead of asking "how much?" ask "in what proportion?" Instead of

"how big?" ask "relative to what?" This shift transforms perception.

The universe reveals itself not as a collection of things but as a web

of relationships, a dance of ratios, a symphony of proportion.

[116:19] See Also: • Gate 27: בח (Bach, "Test, Examine") --- The

Gate of Polarity (ratio as relationship of poles) • Gate 86: הח --- The

Gate of Vibration (ratio as frequency relationship) • Gate 115: וש

(Vash) --- The Gate of Number (the terms that ratios relate) • Gate 118:

זט --- The Gate of Fibonacci (ratios approaching φ)