Functional Just System

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The Functional Just System (FJS) is a logical notation system for the entirety of just intonation (JI) which claims to be both more coherent and more succinct than both Helmholtz–Ellis notation and Ben Johnston's notation.

The Functional Just System can be seen as an extension of the Pythagorean system: the base name of a note (G, D, A♭, etc.) or interval (P5, M2, m6) is calculated by a fifth distance superscript or subscript numbers are added to mark the deviation from the pythagorean base. The chain of fifths used is controlled by a threshold value (or "radius of tolerance") that is λ = 65/63 by default (in “The radius of tolerance is a constant, by definition equal to 65/63.”[1])


Quick reference

Formal commas

Formal commas below 32-limit
Prime Formal Comma
5 80/81
7 63/64
11 33/32
13 1053/1024
17 4131/4096
19 513/512
23 736/729
29 261/256
31 248/243

Harmonic series

Overtones 1–32 with root C
1–8 C C G C E5 G B♭7 C
9–16 D E5 F11 G A♭13 B♭7 B5 C
17–24 D♭17 D E♭19 E5 F7 F11 F♯23 G
25–32 G♯25 A♭13 A B♭7 B♭29 B5 B31 C