geometric

(-rĭ-kəl)
adj.

1. 1. Of or relating to geometry and its methods and principles.
   2. Increasing or decreasing in a geometric progression.
2. Using simple geometric forms such as circles and squares in design and decoration.
3. Of or relating to properties in algebraic geometry involving algebraically closed fields.

geometrically ge'o·met'ri·cal·ly adv.

[Etymology: Gk: ‘Earth’ + ‘measure’] mathematics The geometric mean of n numbers is the nth root of their collective product; compare arithmetic.

Applied to a series of numbers, ‘geometric’ indicates that adjacent members differ by a constant multiplier, the ‘common ratio’ (any finite number). The geometric series with common ratio b has the form

a, a·b¹, a·b², a·b³, …

for some value a. Compare arithmetic.

For measurement scales, geometric means that a step of any one size in the scale value represents a given amount of multiplicative change in the measured item, regardless of place on the scale. The scale for musical pitch is geometric relative to frequency, essentially so for just intonation but precisely so for the scale of equal temperament, in which a rise of 1 semitone anywhere involves a multiplication of frequency (pitch) by


= 1.0595~, a rise of 1 full tone multiplication by the square of this factor, i.e. 1.1225~, a rise of three semitones multiplication by the cube and so on.

Since

log(a·bⁿ) = log a + n·log b

logarithms transform multiplication into addition. Taking logarithms of the above series converts it into the arithmetic series

log a + 0 log b, log a + 1·log b, log a + 2·log b, log a + 3·log b, …

with common difference log b. Thus the modified savart scale for musical interval, expressed as

number of savarts = 996.578~ × log₁₀ (f₂/f₁)

for the interval between frequencies f₁ and f₂, gives 25 savarts for the semitone, everywhere. A rise of 1 full tone equals 25 + 25 = 50, a rise of 3 semitones equals 75 savarts and so on. Thus, relative to any initial note, the value in savarts for a sequence of notes separated by 1 semitone form an arithmetic series. However, because of what its values represent, the savart scale is referred to as being a geometric scale.

As illustrated by the familiar piano, the notes of music are discrete, i.e. there is some relevant space between them (minor physically, but distinct in frequency terms). However, the range of frequencies that can be generated, e.g. by a violin, the human voice, or a machine, forms a continuum. The logarithmic transformation of the savart could be applied to any interval, very small to very large, were it required, with consistent results. The Ancient Grecian scale for stellar magnitude had the discrete values 1, 2, 3, 4, 5, and 6. Though an arithmetic series of itself, each step forward in the scale involved a diminishing of brightness by about 60%, hence it was, roughly, a geometric scale. Adoption of a logarithmic basis has allowed consistent representation of any magnitude. The geometric bel scale, in contrast, was initiated on a logarithmic basis; each increase of 1 represents a multiplication of the measured power level by 1.26~ =


.

Most scales for measurement using named units are essentially geometric, though the multiplier might not be constant; many volume scales have successive units doubling, while the traditional division of the various foot units of Europe was by 12, down to inch, then to line, then to point. The original metric system went seven decimal steps, from milli- up to kilo-, then myria-. Many scales are essentially geometric but not precisely so, convenience of manufacture often demanding compromise (see preferred numbers).

The various proposed logarithmic scales of pressure (logarithmic in this sense being synonymous with geometric) are particular examples, addressing the fact that a change in pressure of a given absolute magnitude depends for its significance on the absolute pressure to which it is applied. These are all akin to the decibel scheme, varying in the units for expressing absolute pressure and the comparative reference pressure. The boyle scheme uses the torr or mm of mercury as its unit, but the characteristic atmospheric pressure of 1 bar (100 kPa, 750.062~ torr) as its reference. The number of deciboyles (coincidentally labelled dB) for pressure p is:

10 log (p/750.062~) = 10 log p - 10 log 750.062~ = (10 log p) - 28.751~.

For a generic scheme, see preferred numbers.

Note: click on a word meaning below to see its connections and related words.

The adjective has 2 meanings:

Meaning #1: (fine arts) characterized by simple geometric forms in design and decoration
  Synonym: geometrical

Meaning #2: of or relating to or determined by geometry
  Synonym: geometrical
  Pertains to noun: geometry (meaning #1)

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Dansk (Danish)
adj. - geometrisk

Nederlands (Dutch)
meetkundig

Français (French)
adj. - géométrique

Deutsch (German)
adj. - geometrisch

Ελληνική (Greek)
adj. - γεωμετρικός

Italiano (Italian)
geometrico

Português (Portuguese)
adj. - geométrico

Русский (Russian)
геометрический

Español (Spanish)
adj. - geométrico

Svenska (Swedish)
adj. - geometrisk

中文（简体）(Chinese (Simplified))
几何学的, 几何学图形的, 几何学上的

中文（繁體）(Chinese (Traditional))
adj. - 幾何學的, 幾何學圖形的, 幾何學上的

한국어 (Korean)
adj. - 기하학[상]의

日本語 (Japanese)
adj. - 幾何学の, 幾何学的図形の

العربيه (Arabic)
‏(صفه) هندسي‏

עברית (Hebrew)
adj. - ‮גיאומטרי‬

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Geometric Duvet	Deco Geometric Ring
Area Rugs Modern Geometric	Geometric School Stencils