Just major triad on C.mid Play just . Pythagoreantuning lang el is a system of musical ... generated above gives seven notes, a diatonic major scale on C in Pythagoreantuning, shown in notation on the top right. Method Pythagoreantuning is based on a stack of intervals called perfect fifth ..., in Pythagoreantuning they have different ratios with respect to D, which means they are at a different ... as a Pythagorean comma . To get around this problem, Pythagoreantuning ignores A music b , and uses ... of Pythagoreantuning, all the fifths are 701.96 cents wide, in the exact ratio 3 2, except the wolf ..., there will always be one wolf fifth in Pythagoreantuning, making it impossible to play in all key ... Pythagoreantuning powers for large numbers .PNG frame right Frequency ratio of the 144 intervals in D based Pythagoreantuning. Interval music Quality Interval names are given in their shortened form ... symmetric Pythagoreantuning powers of 2 26 3 .PNG here and http commons.wikimedia.org wiki File Interval ... Pythagoreantuning. Interval music Quality Interval names are given in their shortened form. 5 limit ..., except for the unisons and the octaves, has two different sizes in Pythagoreantuning. This is the price ... systems this table . By definition, in Pythagoreantuning 11 perfect fifths P5 in the table have a size ... of the above mentioned intervals take a specific name in Pythagoreantuning. In the following table ... and semiditone are specific for Pythagoreantuning, while tone and tritone are used generically for all ... rowspan 2 Pythagoreantuning rowspan 2 5 limit tuning rowspan 2 Quarter comma meantone 1 4 comma ... will be heard in such pieces. Because most fifths in Pythagoreantuning are in the simple ratio of 3 ..., Pythagoreantuning is particularly well suited to music which treats fifths as consonances ... challenges, and its wolf intervals proved to be even worse than those of the Pythagoreantuning so much so that it often required 19 keys to the octave as opposed to the 12 in Pythagoreantuning , so ... more details
File Pythagore chartres.jpg thumb Sculpture of Pythagoras at Chartres , France Pythagorean means of or pertaining to the ancient Ionia n mathematician, philosopher, and music theorist Pythagoras . See Philosophy Pythagoreanism is a term used for the esoteric and metaphysical beliefs purported to have been held by Pythagoras. Neo Pythagoreanism was a school of philosophy reviving Pythagorean doctrines that became prominent in the 1st and 2nd centuries AD. Pythagoreanism Vegetarianism Pythagorean diet was the name for vegetarianism prior to the nineteenth century Mathematics Pythagorean theorem Pythagorean triple Pythagorean prime Pythagorean trigonometric identity Table of Pythagoras, another name for the Multiplication table Music Pythagorean comma Pythagorean hammers Pythagoreantuning Drinkware Pythagorean cup Graphics Upsilon Symbolism Pythagorean letter Sports statistics Pythagorean expectation disambig ... more details
Wiktionary tuningTuning can refer to Musical tuning , musical systems of tuning, and the act of tuning an instrument or voice Guitar tunings Piano tuning , adjusting the pitch of pianos using a tuning fork or a frequency counter Radio tuning The process of tuning a tuned filter Performance tuning the optimization of systems, especially computer systems, may include Category Computer hardware tuning Computer hardware tuning Database tuning Car tuning , an industry and hobby involving modifying automobile engines to improve their performance Engine tuning , the adjustment, modification, or design of internal combustion engines to yield more performance Neuronal tuning , the property of brain cells to selectively represent a particular kind of sensory, motor or cognitive information Self tuningTuning , a song by Avail from their 1994 album Dixie album Dixie Tuning, a psychokinetic ability in the 1998 film Dark City 1998 film Dark City See also lookfrom intitle Tune disambiguation Tuner disambiguation Disambig da Tuning de Tuning es Tuning fr Tuning homonymie ja pl Tuning sk Tuning ... more details
Lead rewrite date November 2010 Image Perfect fifth on C.png thumb right Pythagorean perfect fifth on C audio Just perfect fifth on C.mid Play C G 3 2 1 1 3 2 . In musical tuning theory, a Pythagorean ... to 2 sup 2 sup 3 sup 1 sup are Pythagorean intervals. All the intervals between the notes of a scale are Pythagorean if they are tuned using the Pythagoreantuning system. However, some Pythagorean intervals are also used in other tuning systems. For instance, the above mentioned Pythagorean perfect ..., or about 400 cents . Image Interval ratios in D based symmetric Pythagoreantuning powers for large numbers .PNG frame right Frequency ratio of the 144 intervals in D based Pythagoreantuning ... commons.wikimedia.org wiki File Interval ratios in D based symmetric Pythagoreantuning powers of 2 ... tuned with D based symmetric Pythagoreantuning. Further details about this table can be found in Pythagoreantuning Size of intervals Size of Pythagorean intervals . Image Just perfect fifth on D.png ... ratio of Pythagoreantuning, as shown by St rmer s theorem . Two tones make a ditone , a dissonantly ... tuning, and their difference, 531441 524288, is known as the Pythagorean comma . Contrast ... Schulter Intervals state expanded DEFAULTSORT Pythagorean Interval Category 3 limit tuning Category ... Explorations , p.56. ISBN 9780195144369. The frequency ratio of every Pythagorean interval is a ratio between a power of two and a power of three...confirming the Pythagorean requirements that all ... center 23.460 align center 0 Audio Pythagorean comma on C.mid play 12 perfect unison P1 align center 1 1 align center align center 1 1 align center 0.000 align center 0 Audio Unison on C.mid play 0 Pythagorean ... 0 Audio Pythagorean comma on C.mid play 12 minor second m2 Pythagorean limma limma , br diatonic semitone ... center 90.225 align center 100 Audio Pythagorean minor semitone on C.mid play 5 augmented unison A1 Pythagorean apotome apotome , br chromatic semitone, br major semitone align center 2187 2048 align ... more details
tuning , the Pythagorean comma or ditonic comma ref not to be confused with the syntonic comma ... s often refer to tempering is the Pythagorean comma. ref Coyne, Richard 2010 . The Tuning of Place Sociable Spaces and Pervasive Digital Media , p.45. ISBN 9780262013918. ref The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma ref ... a chromatic and a diatonic semitone , as determined in Pythagoreantuning , or the difference between twelve Just intonation just perfect fifth s and seven octave s, or the difference between three Pythagorean ditone s and one octave this is the reason why the Pythagorean comma is also called ditonic comma . The diminished second , in Pythagoreantuning, is defined as the difference between limma ... ratio frequency ratio of 3 2. It is used in Pythagoreantuning , together with the octave, as a yardstick ... and limma are the two kinds of semitone s defined in Pythagoreantuning . Namely, the apotome about ... or major third is an interval formed by two major tone s. In Pythagoreantuning, a major tone ... tone of Pythagoreantuning with the ratio of 9 8, the octave with the ratio of 2 1, and a number ...Image Pythagorean comma difference A1 m2 .PNG thumb right 400px Pythagorean comma PC defined in Pythagoreantuning as difference between semitones A1&ndash m2 , or interval between enharmonic enharmonically ... direction from to C Music to D Music b . Image Pythagorean comma on C.png thumb right Pythagorean comma on C. audio Pythagorean comma on C.mid Play . The note depicted as lower on the staff B ... and philosopher Pythagoras , is the small Interval music interval or Comma music comma existing in Pythagoreantuning between two enharmonic enharmonically equivalent notes such as C and B Music audio Pythagorean comma on C.mid Play , or D Music b and C Music . ref Apel, Willi 1969 . Harvard Dictionary of Music , p.188. ISBN 9780674375017. ...the difference between the two semitones of the Pythagorean ... more details
Pythagorean tree may refer to Tree of primitive Pythagorean triples Pythagoras tree fractal disambig Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... more details
In algebra, a Pythagorean field is a field mathematics field in which every sum of two squares is a square. A Pythagorean extension of a field F is an extension obtained by adjoining an element mrad 1 &lambda sup 2 sup for some in F . So a Pythagorean field is one closure mathematics closed under taking Pythagorean extensions. For any field there is a minimal Pythagorean field containing it, unique up to isomorphism , called its Pythagorean closure . Pythagorean fields can be used to construct models for some of Hilbert s axioms for geometry harv Ito 1980 loc 163 C . The analytic geometry given by F sup n sup for F a Pythagorean field satisfies many of Hilbert s axioms, such as the incidence axioms, the congruence axioms and the axioms of parallels. However, in general this geometry need not satisfy all Hilbert s axioms unless the field F has extra properties for example, if the field is also ordered then the geometry will satisfy Hilbert s ordering axioms, and if the field is also complete the geometry will satisfy Hilbert s completeness axiom. The Pythagorean closure of a non archimedean ordered field , such as the Pythagorean closure of the field of rational function s Q t in one variable over the rational numbers Q , can be used to construct non archimedean geometries that satisfy many of Hilbert s axioms but not his axiom of completeness harv Ito 1980 loc 163 D . Dehn used such a field to construct a non Legendrian geometry and a semi Euclidean geometry in which there are many lines though a point not intersecting a given line. The Witt group Witt ring of a Pythagorean field is of order 2 if the field is not Formally real field formally real , and torsion free otherwise. See also Euclidean field References Citation last1 Elman first1 Richard last2 Lam first2 T. Y. title Quadratic forms over formally real fields and pythagorean fields jstor 2373568 mr ... year 2005 volume 67 chapter Chapter VIII section 4 Pythagorean fields pages 255 264 citation first1 ... more details
Refimprove date November 2007 A Pythagorean prime is prime number of the form 4 n 1. These are exactly the primes that can be the hypotenuse of a Pythagorean triangle . The first few Pythagorean primes are 5 number 5 , 13 number 13 , 17 number 17 , 29 number 29 , 37 number 37 , 41 number 41 , 53 number 53 , 61 number 61 , 73 number 73 , 89 number 89 , 97 number 97 , 101 number 101 , 109 number 109 , 113 number 113 , OEIS id A002144 . Fermat s theorem on sums of two squares states that these primes can be represented as sums of two squares uniquely up to order , and that no other primes can be represented this way, aside from 2 1 sup 2 sup 1 sup 2 sup . Thus these primes and 2 occur as norms of Gaussian integers , while other primes do not. The law of quadratic reciprocity says that if p and q are distinct odd primes, at least one of which is Pythagorean, then p is a quadratic residue mod q if and only if q is a quadratic residue mod p by contrast, if neither p nor q is Pythagorean, then p is a quadratic residue mod q if and only if q is not a quadratic residue mod p . &minus 1 is a quadratic residue mod p if and only if p is a Pythagorean prime or 2 . In the field Z p with p a Pythagorean prime, the polynomial x 2 1 has two solutions. Category Classes of prime numbers Prime number classes state collapsed numtheory stub da Pythagor iske primtal es N mero primo pitag rico sl Pitagorejsko pra tevilo zh ... more details
thumb upright A Pythagorean cup sold on Crete File Pythagoras cup 2.jpg thumb upright A Pythagorean cup sold on Samos The Pythagorean cup is credited as an invention to Pythagoras. For this reason ... to the city. Hero of Alexandria c. 10 70 AD used Pythagorean cups as hydraulic components in his ... more details
Multiple issues cleanup January 2009 original research January 2009 Image Pythagorean.svg thumb The Pythagorean ... the simplest case of the Pythagorean Triple 3 sup 2 sup 4 sup 2 sup 5 sup 2 sup . A Pythagorean triple consists of three positive integer s a , b , and c , such that nowrap ... known example is nowrap 3, 4, 5 . If nowrap a , b , c is a Pythagorean triple, then so is ka , kb , kc for any positive integer k . A primitive Pythagorean triple PPT is one in which a , b and c are pairwise coprime . A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle . The name is derived from the Pythagorean theorem , stating that every right triangle has side lengths satisfying the formula nowrap a sup 2 sup b sup 2 sup c sup 2 sup thus, Pythagorean ... integer sides do not form Pythagorean triples. For instance, the triangle with sides nowrap a b 1 and c 2 is right, but 1, 1, 2 is not a Pythagorean triple because 2 is not an integer ... and k 1. Please note the definition of a primitive Pythagorean triple, one in which the terms are relatively prime. Image Pythagorean triple scatterplot2.png thumb A scatter plot of the legs a , b of the Pythagorean triples with c less than 6000. Negative values are included to illustrate the parabolic patterns in the plot more clearly. There are 16 primitive Pythagorean triples with nowrap c ... radiating lines in the scatter plot. Additionally these are all the primitive Pythagorean triples ... main Formulas for generating Pythagorean triples Image Pythagorean Triples from Grapher.png 300px ... Pythagorean triples given an arbitrary pair of positive integers m and n with nowrap m n . The formula states that the integers math a m 2 n 2 , , b 2mn , , c m 2 n 2 math form a Pythagorean .... ref Mitchell, Douglas W., An alternative characterisation of all primitive Pythagorean triples, Mathematical ... many primitive Pythagorean triples. This relationship of a and b to m and n from Euclid s formula ... more details
A Pythagoras Pythagorean quadruple is a tuple of integer s a , b , c and d , such that d 0 and math a 2 b 2 c 2 d 2 math , and is often denoted math a,b,c,d math . Geometrically, a Pythagorean quadruple math a,b,c,d math defines a cuboid with side Norm mathematics Euclidean norm lengths a , b , and c , whose space diagonal has integer length d . Pythagorean quadruples are thus also called Pythagorean Boxes . ref R.A. Beauregard and E. R. Suryanarayan, Pythagorean boxes , Math. Magazine 74 2001 , 222 227. ref Parametrization of primitive quadruples The Set mathematics set of all primitive Pythagorean quadruples, i.e., those for which gcd a , b , c 1, where gcd denotes the greatest common divisor of a , b , and c , is parametrized by, ref R.D. Carmichael, Diophantine Analysis , New York John Wiley & Sons, 1915. ref ref L.E. Dickson, Some relations between the theory of numbers and other branches of mathematics , in Villat Henri , ed., Conf rence g n rale, Comptes rendus du Congr s international des math maticiens, Strasbourg, Toulouse, 1921, pp. 41 56 reprint Nendeln Liechtenstein Kraus Reprint Limited, 1967 Collected Works 2, pp. 579 594. ref ref R. Spira, The diophantine equation math x 2 y 2 z 2 m 2 math , Amer. Math. Monthly 69 1962 , 360 365. ref math a m 2 n 2 p 2 q 2, , math math b 2 mq np , , math math c 2 nq mp , , math math d m 2 n 2 p 2 q 2, , math where m , n , p , q are non negative integer s and gcd m , n , p , q 1 and m n p q 1 mod 2 . Thus, all primitive Pythagorean ... Pythagorean quadruple math a,b,c,d math Parametrization parametrized by math m,n,p,q math corresponds ..., Letter to the Editor , Amer. Math. Monthly 94 1987 , 757 758. ref Pythagorean quadruples with small ... Alternate parametrization All Pythagorean quadruples including non primitives, and with repetition ... a and b are odd. See also Pythagorean triple Quaternions and spatial rotation SO 4 The Euler Rodrigues ... links MathWorld urlname PythagoreanQuadruple title Pythagorean Quadruple MathWorld urlname LebesgueIdentity ... more details
Ecphantus or Ecphantos lang grc is a shadowy Greek pre Socratic philosopher. He may not have actually existed. ref Some scholars have argued that Hicetas and Ecphantus, both of Syracuse, were not historical figures at all but rather characters in dialogues written by Heraclides of Pontus . From the Stanford Encyclopedia of Philosophy , article http plato.stanford.edu entries pythagoreanism Pythagoreanism . ref He is identified as a Pythagoreanism Pythagorean of the 4th century BCE, and as a supporter of the heliocentric theory . Described as from Syracuse, Sicily Syracuse , this may or may not be the same figure as the attested Ecphantus of Croton . Notes references External links fr icon http www.cosmovisions.com Ecphante.htm DEFAULTSORT Ecphantus The Pythagorean Category 4th century BC Greek people Category 4th century BC philosophers Category Ancient Syracusians Category Ancient Greek philosophers Category Pythagoreans of Magna Graecia Category Ancient Greek astronomers Philosopher stub de Ekphantos el it Ecfanto di Siracusa pt Ecfanto, o Pitag rico ru sk Ekfantos zo Syrak z sl Ekfant ... more details
Pythagorean expectation is a formula invented by Bill James to estimate how many games a baseball team ... s actual and Pythagorean winning percentage can be used to evaluate how lucky that team was by examining ... to the Pythagorean theorem . The basic formula is math mathrm Win frac text runs scored 2 text ...?articleid 342 Baseball Prospectus Articles Revisiting the Pythagorean Theorem Bot generated title ref ... into the pythagorean formula, one can generate second order wins, the number of wins a team ..., Hein Hundal provided an inexact derivation of the formula and showed that the Pythagorean exponent ... of James Pythagorean Formula Long url http groups.google.com group rec.puzzles browse thread ... Chance Magazine 20 2007 , no. 1, 40 48 pages 9698 title A Derivation of the Pythagorean Won Loss Formula ... cite journal first Steven J last Miller title A Derivation of the Pythagorean Won Loss Formula in Baseball ... via James Pythagorean expectation. The formula says that a team s win percentage equals the following ... James Pythagorean expectation to professional basketball while a researcher at STATS, Inc. . He ... Daryl s Modified Pythagorean Theorem was first published in http morey.org pythbook.gif STATS Basketball ... James Pythagorean theory to professional basketball. The result was similar. Another noted basketball statistician, John Hollinger , uses a similar Pythagorean formula except with 16.5 as the exponent ... website and publisher Football Outsiders , where it is known as Pythagorean projection . The 2011 ... Football League NFL in Pythagorean wins, while only seven were one by the team with the most actual victories. Super Bowl champions that led the league in Pythagorean wins but not actual wins include ... from 2005 2008, it reasserted itself in 2009 and 2010. Furthermore, t he Pythagorean projection ... full game more than their Pythagorean projection tend to regress the following year teams that win a minimum ... Orleans Saints season 2008 New Orleans Saints went 8 8 despite 9.5 Pythagorean wins, hinting at the improvement ... more details
File Distorted truncated square tiling.png thumb A Pythagorean tiling File Jacob Ochtervelt Street Musicians ... Ochtervelt , 1665. As observed by Nelsen ref name rbn the floor tiles in this painting are set in the Pythagorean tiling In geometry , the Pythagorean tiling or two squares tessellation is a tessellation ... Tile Inc. date August 2008 . ref Topology and symmetry The Pythagorean tiling is the unique tiling ... b39h2mar.ps.gz volume 39 year 1998 . ref Topologically, the Pythagorean tiling has the same structure ... page 171 publisher W. H. Freeman title Tilings and Patterns year 1987 . ref The smaller squares in the Pythagorean ... tiling, while the larger squares in the Pythagorean tiling are adjacent to eight neighbors that alternate ... symmetry around the center of each tile, while the Pythagorean tiling has a smaller cyclic group cyclic ... triangles and regular hexagons, and one more, the Pythagorean tiling. ref harvtxt Gr nbaum Shephard 1987 , pp. 73 74. ref Pythagorean theorem and dissections File Pythagorean dissections.svg ... and by Henry Perigal right This tiling is called the Pythagorean tiling because it has been used as the basis of proofs of the Pythagorean theorem by the ninth century Arabic mathematicians Al Nayrizi ..., the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, and the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple 5,12,13 . By overlaying a square grid of side length c onto the Pythagorean tiling, it may be used to generate a five ... two Pythagorean tilings may be used to generate a six piece dissection of two unequal squares ... tilings File Pythagorean tiling section.svg thumb 240px An aperiodic sequence generated from tilings by two squares whose side lengths form the golden ratio Although the Pythagorean tiling is itself ... construction for aperiodic sequences Klotz is a German word for a block , one forms a Pythagorean ... 211 title What is known about unit cubes volume 42 year 2005 . ref None of the squares in the Pythagorean ... more details
ALLTYPES PPA98,M1 online Struck regards the magico religious performative power of the Pythagorean symbol as exceeding the methodology methodological confines of conventional semiotics . ref The Pythagorean symbols Walter Burkert featured Androcydes in his Textual criticism Stemmatics stemma of Pythagorean ... beans see Kenneth Sylvan Guthrie, The Pythagorean Sourcebook and Library Red Wheel Weiser, 1987 , p ... 3&as pt ALLTYPES online. ref or the quasi monasticism monastic rule governing the Pythagorean way of life. His view that wine and meat harm the mind conforms to Pythagoreanism Vegetarianism Pythagorean ... modes of discourse, in that he cited passages from Hesiod in interpreting the Pythagorean symbols ... was physician to Alexander the Great . ref Gillian Clark, Iamblichus On the Pythagorean Life Liverpool ... ANdrokydes&lr &num 100&as brr 3&as pt ALLTYPES PPA64,M1 online. ref Pythagorean dietary discipline ... Category Pythagoreans Category Pythagorean philosophy Category Ancient philosophers ... more details
pp move small yes pp semi small yes See also Pythagorean trigonometric identity File Pythagorean.svg thumb The Pythagorean theorem The sum of the areas of the two squares on the legs a and b equals the area of the square on the hypotenuse c . Trigonometry In mathematics , the Pythagorean theorem or Pythagoras ... the Pythagorean equation ref name Sally0 cite book title Roots to research a vertical development of mathematical problems author Judith D. Sally, Paul Sally page 63 chapter Chapter 3 Pythagorean triples ..., and a and b represent the lengths of the other two sides. The Pythagorean theorem is named ... Tablet and the Babylonian Method of Generating Pythagorean Triples author Abdulrahman A. Abdulaziz ... that are not triangles at all, but n dimensional solids. The Pythagorean theorem has attracted ... the lengths of the other two sides, the Pythagorean theorem can be expressed as the Pythagorean equation ... or math b sqrt c 2 a 2 . , math The Pythagorean equation relates the sides of a right triangle ... angle, the law of cosines reduces to the Pythagorean equation. Proofs This theorem may have more ... the book The Pythagorean Proposition contains 370 proofs. ref Harv Loomis 1968 ref Proof using ... math a 2 b 2 c times e c times d c times d e c 2 , , math which, tidying up, is the Pythagorean theorem ... s proof File Illustration to Euclid s proof of the Pythagorean theorem.svg thumb Proof in Euclid ... Pythagorean theorem by shear mapping , Saint Louis University website Java applet ref File Illustration to Euclid s proof of the Pythagorean theorem2.svg thumb Illustration including the new lines File Illustration to Euclid s proof of the Pythagorean theorem3.PNG thumb Showing the two congruent ... url http www.cut the knot.org pythagoras index.shtml 10 title Pythagorean Theorem, proof number 10 ... proof by rearrangement File Pythagorean graphic 2 .PNG thumb Proof using an elaborate rearrangement ... http www.cut the knot.org pythagoras index.shtml 3 title Cut the knot.org Pythagorean theorem and its ... more details
In mathematics , Pythagorean addition is the following binary operation on the real number s math a oplus b sqrt a 2 b 2 . math The name recalls the Pythagorean theorem , which states that the length of the hypotenuse of a right triangle is nowrap a b , where a and b are the lengths of the other sides. This operation provides a simple notation and terminology when the summands are complicated for example, the energy momentum relation in physics becomes math E mc 2 oplus pc. math Properties The operation is associative and commutative, and math sqrt x 1 2 x 2 2 cdots x n 2 x 1 oplus x 2 oplus cdots oplus x n math . This is enough to form the real numbers into a commutative semigroup . However, is not a Group mathematics group operation for the following reasons. The only element which could potentially act as an identity element is 0, since an identity e must satisfy e e e . This yields the equation math sqrt 2 e e math , but if e is nonzero then math sqrt 2 1 math is a contradiction, so e could only be zero. Unfortunately 0 does not work as an identity element after all, since 0 1 1. This does indicate, however, that if the operation is restricted to nonnegative real numbers, then 0 does act as an identity. Consequently the operation acting on the nonnegative real numbers forms a commutative monoid . See also Euclidean distance Hypot function Further reading expand further date February 2012 cite journal author Moler, Cleve and Donald Morrison title Replacing Square Roots by Pythagorean Sums journal IBM Journal of Research and Development volume 27 issue 6 pages 577 581 year 1983 url http www.research.ibm.com journal rd 276 ibmrd2706P.pdf doi 10.1147 rd.276.0577 Try also http citeseerx.ist.psu.edu viewdoc download?doi 10.1.1.90.5651&rep rep1&type pdf here . cite journal first Augustin A. last Dubrulle title A Class of Numerical Methods for the Computation of Pythagorean Sums journal IBM Journal of Research and Development volume 27 issue 6 pages ... more details
fact. As such, it is symbolic of, and perhaps leads to, the Pythagorean conception of mathematics ... books.google.co.uk books?id 8vw9z mnzH8C&pg PA517&lpg PA517&dq pythagorean hammers chisels&source ... v onepage&q pythagorean 20hammers 20chisels&f false ref As Aristotle was later to write, the Pythagoreans ... reflist DEFAULTSORT Pythagorean Hammers Category Acoustics Category Music theory Category Ancient Greek science Category Pythagorean philosophy Category Mathematics of music Link GA de de Pythagoras ... more details
Image MathematicalMeans.svg thumb right A geometric construction of the Quadratic mean and the Pythagorean means of two numbers a and b . Harmonic mean denoted by H , Geometric by G , Arithmetic by A and Quadratic mean also known as Root mean square denoted by Q . In mathematics, the three classical Pythagorean means are the arithmetic mean A , the geometric mean G , and the harmonic mean H . They are defined by math A x 1, ldots, x n frac 1 n x 1 cdots x n math math G x 1, ldots, x n sqrt n x 1 cdots x n math math H x 1, ldots, x n frac n frac 1 x 1 cdots frac 1 x n math Each mean has the following properties Value preservation math M x,x, ldots,x x math First order homogeneous function homogeneity math M bx 1, ldots, bx n b M x 1, ldots, x n math Invariance under exchange math M ldots, x i, ldots, x j, ldots M ldots, x j, ldots, x i, ldots math for any math i math and math j math . Averaging math min x 1, ldots,x n leq M x 1, ldots,x n leq max x 1, ldots,x n math These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians Thomas Heath, History of Ancient Greek Mathematics because of their importance in geometry and music. There is an ordering to these means if all of the math x i math are positive , along with the quadratic mean math Q sqrt frac x 1 2 x 2 2 cdots x n 2 n math math min leq H leq G leq A leq Q leq max math with equality holding if and only if the math x i math are all equal. This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized mean s. This inequality sequence can be proved for the math n 2 math case for the numbers a and b using a sequence of right triangle s x , y , z with hypotenuse z and the Pythagorean theorem , which states that math x 2 y 2 z 2 math and implies that math z x math and math z y math . The right triangles are ref Kung ... links http mathworld.wolfram.com PythagoreanMeans.html Pythagorean means on MathWorld Category Means ... more details
Eurytus lang el , an eminent Pythagoreanism Pythagorean philosopher, lived c. 400 BC, who Iamblichus in one passage ref Iamblichus, de Vit. Pyth. 28 ref describes as a native of Croton , while in another, ref Iamblichus, de Vit. Pyth. 36 ref he enumerates him among the Taranto Tarentine Pythagoreans. He was a disciple of Philolaus , and Diogenes La rtius ref Diogenes La rtius iii. 6, viii. 46 ref mentions him among the teachers of Plato , though this statement is very doubtful. It is uncertain whether Eurytus was the author of any work, unless we suppose that the fragment in Stobaeus , ref Stobaeus, Phys. Ecl. i. ref which is there ascribed to one Eurytus, belongs to this Eurytus. Aristotle , Metaphysics Aristotle Metaphysics 1092b mentions Eurytus, speaking about points as limits of spatial magnitude It was in this sense that Eurytus determined the number of anything for he computed the number of a man or that of a horse or of any living thing by outlining its shape with pebbles, as one would number the sides of a triangle or a square, ref Aristotle, Metaphysics , 1092b. translated by Richard Hope, p. 314, Columbia University Press, 2008. ref Notes reflist SmithDGRBM Persondata Metadata see Wikipedia Persondata . NAME Eurytus ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH PLACE OF BIRTH DATE OF DEATH PLACE OF DEATH DEFAULTSORT Eurytus Category Ancient Greek philosophers Category Pythagoreans of Magna Graecia Category Ancient Crotonians Category 4th century BC philosophers Category 4th century BC deaths ca Eurit de Eurytos Philosoph it Eurito pitagorico fi Eurytos filosofi roa tara Eurito, pitagoriche uk ... more details
Cross tuning or cross tuning is an alternative Musical tuningtuning used for the open strings music strings of a string instrument . The term refers to the practice of retuning the strings it also refers to the various tunings commonly used, or in some contexts it may refer to the AEAE fiddle tuning. In folk music traditions cross tunings are used to give the instrument a different sound by altering ... position as if played in regular tuning, while the sounded pitch is altered. In American folk fiddle music Cross tuning is commonly used on the fiddle in folk music of Appalachia , the southern United ... common re tunings. While the standard tuning for open strings of the violin is GDAE&mdash with the G being the tuning of the lowest pitched string and the E being the tuning for the highest pitched string&mdash fiddlers playing tunes in the key of D major sometimes employ a tuning of ADAE. In this tuning the open G string is raised to the A directly above it. Even more frequently used is a cross tuning of AEAE for music played in the key of A major. ref cite book title Old Time ... after tunes usually played in that tuning. ref Among fiddlers this is referred to as cross tuning ... tuning include Breaking Up Christmas, Cluck Old Hen, Hangman s Reel, Horse and Buggy, and Ways ... Talian Italian tuning, the implication being that it is only one of many possibilities. Other tunings include FCGD Cajun Tuning one whole step down from GDAE GDGB Open G Tuning GDGD Sawmill Tuning or Cross G GDAD Gee Dad DDAD Dead Man s Tuning, or Open D Tuning, or Bonaparte s Retreat Tuning, or Dee Dad ADAE Old Timey D Tuning AEAE Cross Tuning, Cross A , High Bass, High Counter or High Bass, High Tenor similar to Sawmill Tuning AEAC music sharp Black Mountain Rag Tuning, Calico Tuning, Open A Tuning, or Drunken Hiccups Tuning AEAD for Old Sledge, Silver Lake EDAE for Glory in the Meeting House EEAE for Get up in the Cool See also Scordatura Notes references Category Musical tuning ... more details
context date July 2010 unreferenced date July 2010 Augmented tuning is a system of choosing the reeds for a diatonic wind instrument such as a harmonica or accordion in which the blow notes repeat a sequence of C musical note C E musical note E G musical note G perhaps shifted to begin with E or with G and draw notes follow a repeating sequence of E musical note E G musical note G B musical note B though perhaps with a different initial sequence. For example class wikitable style text align center hole 1 2 3 4 5 6 7 8 9 10 11 12 blow note C E G music sharp C E G music sharp C E G music sharp C E G music sharp draw note E music flat G B E music flat G B E music flat G B E music flat G B See also Country tuning Diminished tuning Dorian Cross tuning Harmonic minor tuning Major seventh tuning Melody Maker tuning Melody Maker is a trademarked name Natural minor tuning Paddy Richter tuning Richter tuning Scale tuning Solo tuning Category Musical tuning music theory stub ... more details
Diminished tuning is a system of choosing the reeds for a diatonic wind instrument such as a harmonica or accordion in which the blow notes repeat a sequence of C musical note C E musical note E F musical note F A musical note A and draw notes follow a repeating sequence of D musical note D F musical note F G musical note G B musical note B perhaps shifted to begin with E and F, with F and G , or with A and B . For example class wikitable style text align center hole 1 2 3 4 5 6 7 8 9 10 11 12 blow note C E music flat F music sharp A C E music flat F music sharp A C E music flat F music sharp A draw note D F G B D F G B D F G B See also Augmented tuning Country tuning Dorian Cross tuning Harmonic minor tuning Major seventh tuning Melody Maker tuning Melody Maker is a trademarked name Natural minor tuning Paddy Richter tuning Richter tuning Scale tuning Solo tuning Category Musical tuning music stub ... more details
Multiple issues orphan December 2008 unreferenced December 2008 notability December 2008 Seton Tuning Ltd. is a Colchester based motorcycle and race preparation company. Seton Tuning also manage the Linxcel Seton Tuning Motorcycle Race Team currently running in The British Supersport 600 Championship. Phil Seton, director of Seton Tuning won the MRO Supersport 600 Championship in 2006 and has is responsible for the machines that won the same championship in 2007 and 2008. Seton is pronounced Seat on. Category Motorcycle racing motorcycle stub ... more details
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