It has been seen that certain Compositional Values or Patterns persist even through the scaling such as the reductions or enlargements. These compositional values, whatever scales or measures they may relate, are of pure numbers.
Compositions of pure numbers have a degree of order, called the System of Proportions. When the order occurs as a pervasive system, whole to the parts or parts to the whole, a System of Modulation occurs.
A System of Modulation inherently will have some kind of System of Proportion, but a System of Proportion may not have any apparent System of Modulation. A System of Modulation is an intellectual contribution to the composition, whereas most of the Proportion Systems defy such definition, and so seem intuitive.
According to the arithmetical definition, a Proportion is the equality of Ratios. A Proportionate Ratio manifests with numbers that have some contextual relationship, such as adjacent numbers in sequence, in a matrix or in multi lateral composition (between length, width and height). A proportion is an ideal relationship between two numbers, defined as the division of one number by the other.
Golden Section in Architectural Theory|
Golden Section and other Systems of Proportions: Historically many Systems of Proportions and Systems of Modulation have been attempted.
Golden Section, is an order of a Geometric Proportion based on a specific ratio in which the Whole relates to the Larger Part, just as the Larger Part relates to the Smaller Part. For example a line AC (whole) is divided into two unequal parts, AB (larger part) and BC (smaller part). The ratio of AC / AB (whole / larger part) is same as the ratio of AB / BC (lager part / smaller part). Mathematical this reads as AC/AB = AB/BC or inversely as AB/AC = BC/AB.
This ratio is known as the Divine Proportion. The Golden Rectangle, whose length and width are the segments of a line divided according to the Golden Section, occupies an important position in paintings, sculpture, and architecture, because its proportions have long been considered the most attractive to the eye.
Another Proportioning System is the Ratio of √2 : 1 = 1.4142 : 1 The simplicity of this derivation (a square root of 2 is the diagonal, in a square of side length 1) is paralleled by the ease of maintaining the proportion through division or multiplication of the proportioned rectangles.
Many complex Numeric Orders have been devised and tried, but none has yet proved to be a universal system. The most common are the various Arithmetic Orders, in which through a specific formula (equation) the numbers are sequenced to form a Logical Series. The Fibonacci Series is an arithmetic order (1,2,3,5,8,13,21,34,55...) That has been shown to have an Order of Proportion between adjoining two numbers (3/5, 5/8, 8/13, 13/21, 21/34...).
There is an on going search for an Order or Modulation System that coordinates various limb sizes (anthropometric measures), of not only an average or a perfect human being, but people of different races (different stature).
|Corbusier's Modulor System (https://www.flickr.com/photos/eager/5031911411)|
The 'Modulor System' was essentially a linear system. Human perception of solid - 3D forms are conditioned by the perspective or converging view. The perspective view depends on the distance and angle of vision of the object. From every point in space one gets a different perspective, and so our perceptions of objects' measures are ever changing.
No definite system that truly works for such a dynamic situation has yet been devised. A Modular Measure System based on the Ergonomics (usage through human limbs), may not work, for the visual and other sensorial (aesthetic) needs.