Functional Just System

The Functional Just System (FJS) is a logical notation system for ∞-limit 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.”) Depending on the radius of tolerance used, some primes will differ in formal commas. Below is a table of formal commas calculated with the standard lambda, Flora Canou's proposal (λ = sqrt(2187/2048)), and neutral FJS (λ = sqrt(134217728/129140163)).

Weblinks

• The FJS website
• Calculator
• Custom FJS Explorer
• misotanni [old] - YouTube – (abandoned channel)

Quick reference

Formal commas

Formal commas and intervals up to the 89-limit

Prime	Formal comma			Interval	Reduced prime harmonic
	Standard	FloraC	Neutral		Standard	FloraC	Neutral
5	80/81			5/4	M3^5
7	63/64			7/4	m7^7
11	33/32		sqrt(242/243)	11/8	P4^11		sA4^11
13	1053/1024		sqrt(507/512)	13/8	m6^13		n6^13
17	4131/4096			17/16	m2^17
19	513/512			19/16	m3^19
23	736/729			23/16	A4^23
29	261/256		sqrt(841/864)	29/16	m7^29		n7^29
31	248/243	31/32	sqrt(2101707/2097152)	31/16	M7^31	P8^31	sd8^31
37	37/36		sqrt(175232/177147)	37/32	M2^37		sA2^37
41	82/81			41/32	M3^41
43	129/128			43/32	P4^43
47	47/48		sqrt(536787/524288)	47/32	P5^47		sd5^47
53	53/54			53/32	M6^53
59	236/243		sqrt(3481/3456)	59/32	M7^59		n7^59
61	244/243			61/32	M7^61
67	16281/16384			67/64	m2^67
71	71/72			71/64	M2^71
73	73/72			73/64	M2^73
79	79/81		sqrt(6241/6144)	79/64	M3^79		n3^79
83	249/256		sqrt(135596187/134217728)	83/64	P4^83		sd4^83
89	712/729		64881/65536	89/64	A4^89		d5^89

Flora's version differs from standard only for the primes 31, 157, 353...

Harmonic series

Overtones 1–32 with root C [Default]

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

Overtones 1–32 with root C [FloraC]

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	C31	C

Overtones 1–32 with root C [Neutral]

1–8	C	C	G	C	E5	G	B♭7	C
9–16	D	E5	F‡11	G	Ad13	B♭7	B5	C
17–24	D♭17	D	E♭19	E5	F7	F‡11	F♯23	G
25–32	G♯25	Ad13	A	B♭7	Bd29	B5	Cd31	C

Boldened notes denote deviations from default.

See also

• Neutral FJS
• Flora Canou's proposal