More on Runsums Integer Sums or Partitions of an integer The number 2016 The number 2021 Fractions Fractions and Decimals - their periods and patterns and in non-decimal bases.įarey Fractions and Stern-Brocot Tree Calculators Two ways of arranging all fractionsĮgyptian fractions The Egyptians only had unit fractions of the form 1/n. Runsums Numbers which are the sum of a run of consecutive whole numbers More on Polygonal Numbers Central polygonal numbers, matchstick figures, 3D solid shapes and higher dimensions Integer Palindromes Polygonal and Figurate Numbers Triangular, Square, Pentagonal. Primes & Factors Calculator Integer Bases many kinds of bases to represent integers Introduction to trilinear and barycentric coordinates, links to Clark Kimberling'sĮncyclopedia of Triangle Centers (ETC) with automatic lookup 3, 4, 5.Įxact Trig Values for Simple Angles Which angles have a simple exact value for their sine,cosine or tangent?Ī Triangle Convertor for Cartesian, Trilinear and Barycentric Coordinates Pythagorean triangles Right-angled triangles with integer sides, e.g. Find out how here.Other Maths Pages at this site: Triangles and Geometry Some people think this is one of the reasons it sounds so good.Īs well as being used to craft violins, the Golden Ratio that comes from the Fibonacci Sequence is also used for saxophone mouthpieces, in speaker wires, and even in the acoustic design of some cathedrals.Įven Lady Gaga has used it in her music. The Golden Ratio can be found throughout the violin by dividing lengths of specific parts of the violin. Stradivari used the Fibonacci Sequence and the Golden Ratio to make his violins. There's a reason a Stradivarius violin would cost you a few million pounds to buy – and its value is partly down to the Fibonacci Sequence and its Golden Ratio. Read more: To save the sound of a Stradivarius, this entire Italian city is keeping quiet Hailed as the master of violin making, Antonio Stradivari has made some of the most beautiful and sonorous violins in existence. The first movement as a whole consists of 100 bars.Ħ2 divided by 38 equals 1.63 (approximately the Golden Ratio)Įxperts claim that Beethoven, Bartók, Debussy, Schubert, Bach and Satie (to name a few) also used this technique to write their sonatas, but no one is exactly sure why it works so well. The exposition consists of 38 bars and the development and recapitulation consists of 62. ![]() ![]() In the above diagram, C is the sonata's first movement as a whole, B is the development and recapitulation, and A is the exposition. The Golden Ratio in Mozart's Piano Sonata No. Let's take the first movement of Mozart's Piano Sonata No. Mozart arranged his piano sonatas so that the number of bars in the development and recapitulation divided by the number of bars in the exposition would equal approximately 1.618, the Golden Ratio. Development and recapitulation – where the theme is developed and repeated.Mozart, for instance, based many of his works on the Golden Ratio – especially his piano sonatas.Įxposition – where the musical theme is introduced ![]() The Fibonacci Sequence can be seen on a piano keyboard.Ĭomposers and instrument makers have been using the Fibonacci Sequence and the Golden Ratio for hundreds of years to compose and create music. Starting to see a pattern? These are all numbers in the Fibonacci Sequence: 3, 5, 8, 13.
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