Euler–Fokker genus
An Euler–Fokker genus is a simple way to build a musical scale in just intonation using products of certain prime factors. Powers of two are ignored because octaves are heard as the same pitch.
- How it works: A genus uses generating prime factors, and each extra factor adds another dimension. A genus with x dimensions uses x factors. It can be written as a list like [3 3 7] or in the compact form [XxYy], meaning x copies of X and y copies of Y. The degree of a genus is the total number of factors, and the number of pitches is (x+1)(y+1) for two-factor examples.
- Example: The multiset {3, 3, 7} gives the genus [3, 3, 7], which yields six pitches within an octave: 1/1, 9/8, 21/16, 3/2, 7/4, 63/32.
- Generating primes: Euler genera come from the primes 3 and 5. Euler–Fokker genera can include higher primes like 7 or more.
- Tuning and naming: The degree is the total number of generator factors. The notation [XxYy] is another way to show the same idea.
- History and use: Adriaan Fokker used these ideas in music written in 31-tone equal temperament. Alan Ridout also used Euler–Fokker genera.
- Complete chord idea: A complete chord is seen as two pitches— the fundamental and a guide tone— with other pitches relating as multiples of the fundamental or divisors of the guide tone. For example, with fundamental 1 and guide tone 15, you get 1:3:5:15, a genus written as [3 5]. The ratio of the guide to the fundamental is called the tension number.
- Related concepts: Erv Wilson expanded these ideas into Combination Product Sets (CPS), formed by multiplying sets of factors (including 1). This leads to structures like the Hexany, Eikosany, and Hebdomekontany.
This page was last edited on 2 February 2026, at 08:07 (CET).