In music theory

Music theory

Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...

 and tuning

Musical tuning

In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...

, an Euler–Fokker genus (plural: genera), named after Leonhard Euler

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 and Adriaan Fokker

Adriaan Fokker

Adriaan Daniël Fokker , was a Dutch physicist and musician.Fokker was born in Buitenzorg, Dutch East Indies ; he was a cousin of the aeronautical engineer Anthony Fokker...

, is a musical scale in just intonation

Just intonation

In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

 whose pitches can be expressed as products

Multiplication

Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....

 of some of the members of some multiset

Multiset

In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...

 of generating prime

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

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

Octave

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

s.

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." 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

Unison

In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...

, 9/8, 21/16, 3/2

Perfect fifth

In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

, 7/4

Harmonic seventh

The harmonic seventh interval , also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio . This is somewhat narrower than and is "sweeter in quality" than an "ordinary" minor seventh, which has a just-intonation ratio of 9:5 , or an equal-temperament ratio of...

, 63/32

Septimal comma

A septimal comma is a small musical interval in just intonation that contains the number seven in its prime factorization. There is more than one such interval, so the term septimal comma is ambiguous, but it most commonly refers to the interval 64/63....

.

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 [X^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).

Adriaan Fokker

Adriaan Fokker

Adriaan Daniël Fokker , was a Dutch physicist and musician.Fokker was born in Buitenzorg, Dutch East Indies ; he was a cousin of the aeronautical engineer Anthony Fokker...

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

31 equal temperament

In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...

. Alan Ridout

Alan Ridout

-Life:Born at West Wickham, Greater London, England, Alan Ridout studied briefly at the Guildhall School of Music before commencing four years of study at the Royal College of Music, London with Herbert Howells and Gordon Jacob...

 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 are either multiples of the fundamental or divisors of the guide tone (otonality and utonality

Otonality and Utonality

Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones or "undertones" of a given fixed tone. For example: 1/1, 2/1, 3/1,.....

). For example, given 1 as fundamental and choosing 15 as guide tone, one obtains: 1:3:5:15 (genus [35]).

Further reading

• Franck Jedrzejewski (2006). Mathematical Theory of Music, p.157. ISBN 9782752100238.