An x-dimensional tone-dimension contains x factors. "An Euler-Fokker genus with two dimensions may be represented in a two-dimensional (rectangular) tone-grid, one with three dimensions in a three-dimensional (block-shaped) tone-lattice. Euler-Fokker genera are characterized by a listing of the number of steps in each dimension. The number of steps is represented by a repeated mention of the dimension, so that there arise descriptions such as [3 3 5 5], [3 5 7], [3 3 5 5 7 7 11 11], etc."[1] For example, the multiset {3, 3, 7} yields the Euler–Fokker genus [3, 3, 7], which contains these pitches:
Euler genera are generated from the prime factors 3 and 5, whereas an Euler–Fokker genus can have factors of 7 or any higher prime number. The degree is the number of intervals which generate a genus. However, not all genera of the same degree have the same number of tones since [XXXYYY] may also be notated [XxYy], "the degree is thus the sum of the exponents," and the number of pitches is obtained adding one to each exponent and then multiplying those ((X+1)×(Y+1)=Z).[2]
The Euler–Fokker genus may also be called a complete contracted chord. Euler coined the term complete chord, while Fokker coined the entire term.[2]
A complete chord has two pitches, the fundamental and a guide tone, the guide tone being a multiple of the fundamental. In between are other pitches which can be seen either as multiples of the fundamental or as divisors of the guide tone (otonality and utonality). For example, given 1 as fundamental and choosing 15 as guide tone, one obtains: 1:3:5:15 (genus [35]). The quotient of the guide tone divided by the fundamental is the "tension number" or "exponens" (Euler: Exponens consonantiae).[2]
^ abcd"What is an Euler–Fokker genus?", Huygens-Fokker.org.
Further reading
Franck Jedrzejewski (2006). Mathematical Theory of Music, p. 157. ISBN978-2-7521-0023-8.
January 29, 2023
euler, fokker, genus, music, theory, tuning, plural, genera, named, after, leonhard, euler, adriaan, fokker, musical, scale, just, intonation, whose, pitches, expressed, products, some, members, some, multiset, generating, prime, factors, powers, usually, igno. In music theory and tuning an Euler Fokker genus plural genera named after Leonhard Euler and Adriaan Fokker 1 is a musical scale in just intonation whose pitches can be expressed as products of some of the members of some multiset of generating prime factors Powers of two are usually ignored because of the way the human ear perceives octaves as equivalent Euler Fokker genus 3 3 7 as rectangular parallelepiped Euler genus 3 5 as rectangleEuler genus 3 3 5 as parallelepipedEuler genus 3 5 5 as parallelepiped An x dimensional tone dimension contains x factors An Euler Fokker genus with two dimensions may be represented in a two dimensional rectangular tone grid one with three dimensions in a three dimensional block shaped tone lattice Euler Fokker genera are characterized by a listing of the number of steps in each dimension The number of steps is represented by a repeated mention of the dimension so that there arise descriptions such as 3 3 5 5 3 5 7 3 3 5 5 7 7 11 11 etc 1 For example the multiset 3 3 7 yields the Euler Fokker genus 3 3 7 which contains these pitches 1 3 3 7 7 3 3 9 3 7 21 3 3 7 63 Normalized to fall within an octave these become 1 1 9 8 21 16 3 2 7 4 63 32 Play help info Euler genera are generated from the prime factors 3 and 5 whereas an Euler Fokker genus can have factors of 7 or any higher prime number The degree is the number of intervals which generate a genus However not all genera of the same degree have the same number of tones since XXXYYY may also be notated XxYy the degree is thus the sum of the exponents and the number of pitches is obtained adding one to each exponent and then multiplying those X 1 Y 1 Z 2 Adriaan Fokker wrote much of his music in Euler Fokker genera expressed in 31 tone equal temperament Alan Ridout also used Euler Fokker genera 2 Contents 1 Complete contracted chord 2 List of Euler genera 3 See also 4 References 5 Further readingComplete contracted chord EditThe Euler Fokker genus may also be called a complete contracted chord Euler coined the term complete chord while Fokker coined the entire term 2 A complete chord has two pitches the fundamental and a guide tone the guide tone being a multiple of the fundamental In between are other pitches which can be seen either as multiples of the fundamental or as divisors of the guide tone otonality and utonality For example given 1 as fundamental and choosing 15 as guide tone one obtains 1 3 5 15 genus 35 The quotient of the guide tone divided by the fundamental is the tension number or exponens Euler Exponens consonantiae 2 List of Euler genera EditSecond degreeGenera Notes33 1 1 9 8 3 235 1 1 5 4 3 2 15 855 1 1 5 4 25 16Third degreeGenera Notes333 1 1 9 8 3 2 27 16335 1 1 9 8 5 4 45 32 3 2 15 8355 1 1 75 64 5 4 3 2 25 16 15 8555 1 1 5 4 25 16 125 64Fourth degreeGenera Notes3333 1 1 9 8 81 64 3 2 27 163335 1 1 135 128 9 8 5 4 45 32 3 2 27 16 15 83355 1 1 9 8 75 64 5 4 45 32 3 2 25 16 27 16 225 128 15 83555 1 1 75 64 5 4 375 256 3 2 25 16 15 8 125 645555 1 1 625 512 5 4 25 16 125 64 See also EditHexanyReferences Edit a b Rasch Rudolph 2000 Harry Partch p 31 2 Dunn David ed ISBN 978 90 5755 065 2 a b c d What is an Euler Fokker genus Huygens Fokker org Further reading EditFranck Jedrzejewski 2006 Mathematical Theory of Music p 157 ISBN 978 2 7521 0023 8 Retrieved from https en wikipedia org w index php title Euler Fokker genus amp oldid 1070819374, wikipedia, wiki, book, books, library,
