In music theory and tuning, an Euler–Fokker genus (plural: genera), named after

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

and Adriaan Fokker,Rasch, Rudolph (2000). ''Harry Partch'', p.31-2. Dunn, David, ed. . is a

musical scale In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Often, especially in t ...

just intonation In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and ...

whose pitches can be expressed as products of some of the members of some

multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...

of generating

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

factors. Powers of two are usually ignored, because of the way the human ear perceives

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

s as equivalent. 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 5 5 5 7 3 5 5 7 7 11 11 etc." For example, the multiset yields the Euler–Fokker genus , 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. 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 XXYYYmay also be notated ^xY^y "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).What is an Euler–Fokker genus?
, ''Huygens-Fokker.org''. Adriaan Fokker wrote much of his music in Euler–Fokker genera expressed in 31-tone equal temperament. Alan Ridout also used Euler-Fokker genera.

Complete contracted chord

The Euler–Fokker genus may also be called a complete contracted chord. Euler coined the term ''complete chord'', while Fokker coined the entire term. 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 Jazz improvisation is the spontaneous invention of melodic solo lines or accompaniment parts in a performance of jazz music. It is one of the defining elements of jazz. Improvisation is composing on the spot, when a singer or instrumentalist inv ...

(

otonality and utonality ''Otonality'' and ''utonality'' are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone ( identity), respectively. For example: , , ,... or , , ,.... Definition ...

). For example, given 1 as fundamental and choosing 15 as guide tone, one obtains: 1:3:5:15 (genus 5. The quotient of the guide tone divided by the fundamental is the "tension number" or "exponens" (Euler: ''Exponens consonantiae'').

List of Euler genera

...

Complete contracted chord

List of Euler genera

See also

References

Further reading