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	Pythagorean tuning - Wikipedia, the free encyclopedia
		Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2.
		Pythagorean tuning is based on a stack of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1, which is the ratio of an octave.
		In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a semitone flatter.
	en.wikipedia.org /wiki/Pythagorean_tuning (937 words)

	Pythagorean triple - Wikipedia, the free encyclopedia
		A primitive Pythagorean triple is one in which a, b and c are coprime.
		The name is derived from the Pythagorean theorem, of which every Pythagorean triple is a solution.
		In every Pythagorean triple, the radius of the in-circle and the radii of the three ex-circles are natural numbers.
	en.wikipedia.org /wiki/Pythagorean_triple (1837 words)

	The Pythagorean Theorem
		The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations.
		According to the Pythagorean Theorem, the sum of the areas of the two red squares, squares A and B, is equal to the area of the blue square, square C.
		Therefore, the square on c is equal to the sum of the squares on a and b.
	jwilson.coe.uga.edu /emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay.1/Pythagorean.html (2131 words)

	Pythagorean theorem Info - Encyclopedia WikiWhat.com (Site not responding. Last check: 2007-11-06)
		The Pythagorean theorem or Pythagoras' theorem is named after and commonly attributed to the 6th century BC Greek philosopher and mathematician Pythagoras, though the facts of the theorem were known before he lived.
		The Pythagorean theorem stated in Cartesian coordinates is the formula for the distance between points in the plane -- if (a, b) and (c, d) are points in the plane, then the distance between them is given by
		Since the Pythagorean theorem is derived from the axioms of Euclidean geometry, and physical space may not always be Euclidean, it need not be true of triangles in physical space.
	www.wikiwhat.com /encyclopedia/p/py/pythagorean_theorem.html (892 words)

	Pythagorean Theorems - Some 'Not So Familiar' Implications
		Still, the Pythagorean relationship holds, the sum of the areas of the rectangles drawn on the two legs is equal to the area of the rhombus drawn on the hypotenuse of the right triangle.
		In these figures the Pythagorean relationship still holds, the sum of the areas of the parallelograms drawn on the two sides is equal to the area of the parallelogram drawn on the third side of the triangle.
		Similar to Pythagorean Triples, quadruples are sets of 4 integers, such that the sum of the squares of the smaller three equals the square of the fourth larger integer.
	contracosta.edu /math/pythagoras.htm (1637 words)

	Pythagorean theorem -- Encyclopædia Britannica
		For an arbitrary triangle, the Pythagorean theorem is generalized to the law of cosines: a2 + b2 = c2 2ab cos (ACB).
		He founded the Pythagorean brotherhood, a group of his followers whose beliefs and ideas were rediscovered during the Renaissance and contributed to the development of mathematics and Western rational...
		The Pythagorean Theorem is used to calculate the relationship between the legs and angles of a triangle.
	www.britannica.com /eb/article-9343833 (793 words)

	Secret Teachings of All Ages: Pythagorean Mathematics
		From the study of the mysterious Pythagorean monad, Leibnitz evolved his magnificent theory of the world atoms--a theory in perfect accord with the ancient teachings of the Mysteries, for Leibnitz himself was an initiate of a secret school.
		By the Pythagoreans monad was called chaos, obscurity, chasm, Tartarus, Styx, abyss, Lethe, Atlas, Axis, Morpho (a name for Venus), and Tower or Throne of Jupiter, because of the great power which abides in the center of the universe and controls the circular motion of the planers about itself.
		The Pythagoreans taught that the elements of earth, fire, air, and water were permeated by a substance called ether--the basis of vitality and life.
	www.sacred-texts.com /eso/sta/sta16.htm (6241 words)

	Pythagorean Physics - Writings by Todd Matthews Kelso
		Pythagorean Physics postulates the existence of a basic unit of matter, the Pythagorean atom.
		Unlike classical mechanics, Pythagorean Physics considers mass to be a variable and has a different concept of what a particle is. Pythagorean Physics employs an axiomatic system that incorporates both philosophy and science in order to achieve meaning.
		Pythagorean Physics follows an axiomatic system that starts with definitions and proceeds step by step from there in a logical fashion that provides meaning in a way that other approaches can not.
	home.att.net /~zei/TMKelso (1465 words)

	BAIN: A Pythagorean tuning of the diatonic scale
		The Pythagorean cult's preference for proportions involving whole numbers is evident in this scale's construction, as all of its tones may be derived from interval frequency ratios based on the first three counting numbers: 1, 2, and 3.
		For example, a Pythagorean tuning of the 12-note chromatic scale on C is shown in Fig.
		The Pythagorean semitone was also called the limma (left over), as it was calculated by the Greeks to be the difference (or amount left over) between a fourth and two whole tones.
	www.music.sc.edu /fs/bain/atmi02/pst (1303 words)

	Pythagorean Theorem
		To begin, the Pythagorean theorem states that the square on the hypotenuse of a right triangle has an area equal to the combined areas of the squares on the other two sides.
		The Pythagorean theorem was a mathematical fact that the Babylonians knew and used.
		With the Pythagorean theorem being such a popular topic, it is no wonder high school students study the theorem.
	www.ms.uky.edu /~lee/ma502/pythag/pythag.htm (488 words)

	Search Results for Pythagorean - Encyclopædia Britannica
		The Orphic creeds were the basis of the Pythagorean brotherhood, which flourished in southern Italy beginning in the 6th century BC.
		A Pythagorean triple is formed by the measures of the sides of an...
		The Pythagoreans used geometrical figures to illustrate their slogan that all is number—thus their “triangular numbers” (), “square numbers”; (n2), and “altar numbers”; (n3), some of which are shown in...
	www.britannica.com /search?query=Pythagorean&submit=Find&source=MWTAB (479 words)

	Pythagorean Tuning - Basic concepts (Site not responding. Last check: 2007-11-06)
		As mentioned above, Pythagorean tuning defines all notes and intervals of a scale from a series of pure fifths with a ratio of 3:2.
		In fact, Pythagorean tuning is described in the medieval sources as being based on four numbers: 12:9:8:6.
		The following table shows how the standard intervals of Pythagorean tuning except the pure unison (1:1) and octave (2:1) are derived primarily from superimposed fifths (3:2), thus having ratios which are powers of 3:2, or secondarily from the differences between these primary intervals and the octave.
	www.medieval.org /emfaq/harmony/pyth2.html (422 words)

	Pythagorean Tuning and Medieval Polyphony - Table of Contents
		Providing a simple and elegant way of generating a musical scale, this tuning system may have a special appeal for styles of harmony where fifths and fourths are the most favored intervals, as is true in the ensemble music of Chinese and related traditions, for example, as well as in medieval European polyphony.
		Section 2 presents some basic concepts of Pythagorean tuning as applied to Gothic music, while Section 3 explores how this system nicely fits in with the subtle spectrum of harmonic tension in the 13th century.
		Section 3, on stylistic considerations, is linked in many ways to a companion article on 13th-century polyphony, and owes a special debt of gratitude to studies by Vincent Corrigan on the Notre Dame conductus repertory, and by Mark Lindley on the later 13th and 14th centuries, although any flaws or infelicities are of course mine.
	www.medieval.org /emfaq/harmony/pyth.html (766 words)

	Pythagorean Triples
		A Pythagorean triple which is not a multiple of a smaller one is called a primitive Pythagorean triple.
		This is a Pythagorean triple since, as a triangle, is it just 3 times the 3-4-5 triangle (by which we mean that we just triple the lengths of each side of a 3-4-5 triangle, which we already know is right-angled).
		Pythagorean Triples Projects is Eric Rowland's useful page of further ideas for your own investigations together with some hints and solutions.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Pythag/pythag.html (6318 words)

	Read This: Pythagorean Triangles
		Chapter 5: We find that at least one of the arms of a Pythagorean triangle is a multiple of 3, while at least one of the three sides (arms or hypotenuse) must be a multiple of 5.
		We are also given the exact number of primitive Pythagorean triangles with a given integer as the radius of the inscribed circle.
		It is dedicated to proving Fermat's assertion that the smallest Pythagorean triangle in which the hypotenuse and the sum of the arms are squares is the triangle (456548602761, 1061652293520, 4687298610289).
	www.maa.org /reviews/pythtriangles.html (1089 words)

	Pythagorean Theorem and its many proofs
		It is known that the Pythagorean Theorem is Equivalent to Parallel Postulate.
		It generalizes the Pythagorean Theorem in two ways: the triangle ABC is not required to be right-angled and the shapes built on its sides are arbitrary parallelograms instead of squares.
		This is a rather convoluted way to prove the Pythagorean Theorem that, nonetheless reflects on the centrality of the Theorem in the geometry of the plane.
	www.cut-the-knot.org /pythagoras/index.shtml (7548 words)

	Pythagorean Triples
		Although the Pythagorean theorem arose in geometry, we will be concerned strictly with the number theoretic properties of the Pythagorean equation, using the connection to geometry only as a jumping off point.
		Remark 1 If (x, y, z) is a primitive Pythagorean triple, then y ≡ 0 mod 4, z ≡ 1 mod 4, exactly one of {x, y} satisfies k ≡ 0 mod 3, and exactly one of {x, y, z} satisfies k ≡ 0 mod 5.
		In general, the number of primitive Pythagorean triples of hypotenuse n is dependent on the number of prime factors of n that are congruent to 1 modulo 4.
	www.math.rutgers.edu /~erowland/pythagoreantriples.html (3628 words)

	Pythagoras and the Pythagoreans (Site not responding. Last check: 2007-11-06)
		Pythagorean philosophy was the prime source of inspiration for Plato and Aristotle; the influence of these philosophers is without question and is immeasurable.
		In right-angled triangles, the square upon the hypotenuse is equal to the sum of the squares upon the legs.
		One account gives that the Pythagorean s were at sea at the time and when Hippasus produced an element which denied virtually all of Pythagorean doctrine, he was thrown overboard.
	www.math.tamu.edu /~don.allen/history/pythag/pythag.html (2531 words)

	CATHOLIC ENCYCLOPEDIA: Neo-Pythagorean Philosophy
		Their original aim — to save the pagan world from moral and social ruin by the introduction of the religious element into philosophy and into conduct — was, of course, conceived without any reference to the claims of Christianity.
		Next, they interpreted the Pythagorean doctrine in a Platonic sense, when they taught that numbers are the thoughts of God.
		Finally, it should be remembered that the Pythagorean biographers openly acknowledged "the principle of permitting exaggeration and deceit in the cause of philosophy" (Newman).
	www.newadvent.org /cathen/10745a.htm (1247 words)

	Introduction to the Pythagorean Tarot
		Although the Pythagorean Tarot began as a personal project, I have been encouraged to make it generally available, since its interpretive framework is not the same as many other tarots, and so serious tarotists may find it useful both in itself, and as a starting point for their own designs.
		All this is relevant to the proper interpretation of the Pythagorean Y. According to a legend, Y was called "the Pythagorean letter" (littera Pythagorica) because he himself had been responsible for adding it to the Greek alphabet (Persius/Conington 61n56; Persius/Gildersleeve 130-1nn56-7; Persius/Koenig 81nn56, 240n56).
		This interpretation of the Pythagorean Y is clearly judgmental and may seem moralistic: the right-hand path is the better; this was part of the message the ancient Pythagoreans offered to their time.
	www.cs.utk.edu /~mclennan/BA/PT/Intro.html (6259 words)

	Math Forum: Ask Dr. Math FAQ: Pythagorean Triples
		Pythagorean Triples (Sums of 3 cubes equal a cube; sums of 4th powers equal a 4th power)
		Pythagorean Triples (One of (a,b,c) is divisible by 3, one by 4, and one by 5)
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /dr.math/faq/faq.pythag.triples.html (780 words)

	PlanetMath: Pythagorean triplet
		It follows that there are countably many Pythagorean triplets.
		All the primitive Pythagorean triplets are given by
		This is version 6 of Pythagorean triplet, born on 2001-10-06, modified 2005-12-15.
	planetmath.org /encyclopedia/PythagoreanTriple2.html (116 words)

	The Pythagorean Theorem
		Thus, implicit in this particular constructivist approach to the Pythagorean theorem is the notion that the student should build his or her own knowledge by "eyeballing" right angles.
		Another problem arising in this constructivist approach to the Pythagorean theorem is that of calculating the areas of the squares built on the sides and hypotenuse of a geoboard triangle.
		Experiments with the geoboard correspond to a cumbersome verification of the Pythagorean theorem in rather special circumstances (the geoboard's discrete structure is well suited to experimentation, but it fails to represent the more general structure of the Euclidean plane).
	www.mathematicallycorrect.com /pythag.htm (980 words)

	MathSteps: Grade 7: Pythagorean Theorem: What Is It?
		The Pythagorean (puh thag or ee un) Theorem, also called the Pythagorean Property, says that the sum of the squares of the lengths of the legs of any right triangle is equal to the square of the length of the hypotenuse,
		Another way of looking at the Pythagorean Theorem is to think about actual squares of the lengths of the sides of a right triangle.
		Pythagorean Triples are groups of three whole numbers that make the Pythagorean Theorem true (and therefore define a true right triangle).
	www.eduplace.com /math/mathsteps/7/c (545 words)

	Pythagorean History
		Pythagoras gained his famous status by founding a group, the Brotherhood of Pythagoreans, which was devoted to the study of mathematics.
		The Pythagoreans wrote many geometric proofs, but it is difficult to ascertain who proved what, as the group wanted to keep their findings secret.
		It is thought that the Babylonians saw this pattern of tiles to be a proof of the Pythagorean Theorem.
	www.geom.uiuc.edu /~demo5337/Group3/hist.html (688 words)

	Pythagorean Tarot Review
		The Pythagorean Tarot is a complex and deep set of tarot cards, based on ancient Greek paganism, alchemy and Pythagorean numerology.
		The Pythagorean Tarot, therefore, brings together Pythagorean numerology and the Tarot to create a 'unique system of divination and transformation'.It draws on a pre-Christian framework in the form of ancient Greek Paganism, making a tarot deck that appeals to contemporary Pagans by removing medieval Christian religious symbolism common to many tarot decks.
		Despite this, however, no previous knowledge of Pythagorean philosophy is needed to use the deck and gain unique insights from it.
	www.aeclectic.net /tarot/cards/pythagorean/review.shtml (1054 words)

	Pythagoras, the Father of Numerology.
		The Pythagorean system, is among the most enduring and popular of all self-help methods ever created.
		Pythagorean numerology was organized by Greek philosopher and mathematician Pythagoras, who combined the mathematical disciplines of the Arabic, Druid, Phoenician, Egyptian, and Essene sciences.
		The Pythagorean system is today the most commonly used system of numerology in the West.
	www.decoz.com /pythagoras.htm (447 words)

	The Pythagorean Sourcebook and Library
		The Pythagorean ethical and political tractates are especially interesting for they are based on the premise that the universal principles of Harmony, Proportion, and Justice govern the physical cosmos, and these writings show how individuals and societies alike attain their peak of excellence when informed by these same principles.
		The comprehensive introduction by David Fideler outlines the history of the Pythagorean school in the Classical and Hellenistic periods and carefully examines its philosophy and teachings, helping to situate the writings of this anthology in their proper historical and philosophical contexts.
		Contains over 20 engravings and illustrations, in addition to four appendices dealing with the formation of the musical scale in Pythagorean tuning, Pythagorean mathematical discoveries, etc. Many of the writings are prefaced by brief introductory notes; annotations explicate obscure passages and unfamiliar terms.
	phanes.com /pytsou.html (428 words)

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