In music theory and

tuning Tuning can refer to: Common uses * Tuning, the process of tuning a tuned amplifier or other electronic component * Musical tuning, musical systems of tuning, and the act of tuning an instrument or voice ** Guitar tunings ** Piano tuning, adjusti ...

, 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 Adriaan Daniël Fokker (; 17 August 1887 – 24 September 1972) was a Dutch physicist. He worked in the fields of special relativity and statistical mechanics. He was the inventor of the Fokker organ, a 31-tone equal-tempered (31-TET) organ. ...

,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 the ...

in just intonation whose pitches can be expressed as

products Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...

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 e ...

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 ways ...

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

wrote much of his music in Euler–Fokker genera expressed in 31-tone equal temperament.

Alan Ridout Alan Ridout (9 December 1934 – 19 March 1996) was a British composer and teacher. Life Born in West Wickham, Kent, England, Alan Ridout studied briefly at the Guildhall School of Music before commencing four years of study at the Royal C ...

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 Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...

or as divisors of the guide tone (

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 , , ,.... Definitio ...

). 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