User:Eufalesio/Moture's Extended Functional Just System

This is a machine translation of the following article created by Moture, with some extra proposals down below.

Limitations of Existing Systems and Motivation

A typical FJS notation consists of three parts:

1. The accidental (d, m, P, M, A, etc.) describing the alteration;
2. The nominal degree number;
3. Superscripts/subscripts that describe otonalities/utonality involving primes 5 and higher.

The limitation of standard FJS is that it cannot intuitively describe intervals such as “half a perfect fifth.” For example, 11/9 is very close to half a perfect fifth, yet FJS notates it as m3¹¹ (a minor third raised by the 11-limit comma), which can feel counter-intuitive. To address this, NFJS was proposed, introducing “n” for neutral intervals and additional symbols like sA, sd for half-sharps/flats, neutral thirds, etc.

Even NFJS is still insufficient for cases such as “half a perfect fourth,” “one-third of a perfect fifth,” or treating 7/4 as functionally lying between a sixth and a seventh. The core purpose of this notation is to elegantly describe these subtle fractional and intermediate relationships.

Re-analysis of FJS

The Degree (Nominal) Component

Octave equivalence holds, and the additive property remains unchanged: second + second = third, etc. The most distinctive feature of the degree system is that there are seven degrees per octave — this strongly suggests 7-EDO. We therefore define the degree span d, where the displayed nominal degree is d+1, and the generating ratio for the pure degree is [math]\displaystyle{ 2^{d/7} }[/math].

• d = 0 → nominal degree 1 (unison)
• d = 1 → nominal degree 2 (second)
• d = 2 → nominal degree 3 (third)
• … and so on.

The Chroma Component

The unit of the accidentals is intimately tied to 2187/2048. Every minor-to-major step, augmented-vs-perfect/major step, and diminished-vs-perfect/minor step differs by exactly one apotome. We therefore define chroma c, measured in apotomes:

[math]\displaystyle{ \left(\frac{2187}{2048}\right)^c }[/math]

• c = 0 → P1 (1/1)
• c = +1 → A1 (2187/2048)
• c = −1 → d1 (2048/2187)
• etc.

Connection and Extension to FJS

Any interval ratio is expressed as

[math]\displaystyle{ R = 2^{d/7} \cdot \left(\frac{2187}{2048}\right)^c }[/math]

Given a ratio in 3-limit form [math]\displaystyle{ R = 2^x \cdot 3^y }[/math], we solve:

[math]\displaystyle{ \begin{cases} x = \dfrac{d}{7} - 11c \\ y = 7c \end{cases} \implies c = \frac{y}{7},\quad d = 7x + 11y }[/math]

Example: M2 = 9/8 = [math]\displaystyle{ 2^{-3} \cdot 3^{2} }[/math]

[math]\displaystyle{ c = \frac{2}{7},\quad d = 7(-3) + 11(2) = 1 }[/math]

→ second (d+1 = 2) with chroma [math]\displaystyle{ c = +\frac{2}{7} }[/math]

Natural Interval Chroma Table

Nominal	Interval	Ratio	x	y	Degree	Chroma
1	P1	1/1	0	0	0	0
2	m2	256/243	8	−5	1	−5/7
2	M2	9/8	−3	2	1	+2/7
3	m3	32/27	5	−3	2	−3/7
3	M3	81/64	−6	4	2	+4/7
4	P4	4/3	2	−1	3	−1/7
4	A4	729/512	−9	6	3	+6/7
5	d5	1024/729	10	−6	4	−6/7
5	P5	3/2	−1	1	4	+1/7
6	m6	128/81	7	−4	5	−4/7
6	M6	27/16	−4	3	5	+3/7
7	m7	16/9	4	−2	6	−2/7
7	M7	243/128	−5	5	6	+5/7

Baseline Chroma (Brightness) and Naming Rules

Baseline Chroma for Integer Degrees

Two standards:

• P-standard (degrees 1, 4, 5 and octave equivalents): baseline chroma = chroma of the perfect interval.
• n-standard (degrees 2, 3, 6, 7 and octave equivalents): baseline chroma = arithmetic mean of minor and major chroma.

Degree	Standard	Baseline chroma (calculation)	Value
1	P1	[math]\displaystyle{ c_{P1} = 0 }[/math]	0
2	n2	[math]\displaystyle{ \frac{-5/7 + 2/7}{2} = -3/14 }[/math]	−3/14
3	n3	[math]\displaystyle{ \frac{-3/7 + 4/7}{2} = +1/14 }[/math]	+1/14
4	P4	[math]\displaystyle{ c_{P4} = -1/7 }[/math]	−1/7
5	P5	[math]\displaystyle{ c_{P5} = +1/7 }[/math]	+1/7
6	n6	[math]\displaystyle{ \frac{-4/7 + 3/7}{2} = -1/14 }[/math]	−1/14
7	n7	[math]\displaystyle{ \frac{-2/7 + 5/7}{2} = +3/14 }[/math]	+3/14

Octave rule: adding 7n to the degree keeps the same standard (P1, P8, P15 … all use P-standard; n2, n9, n16 … all use n-standard).

Naming Rules Based on Chroma Deviation

P-standard degrees (1, 4, 5 etc.)

• c = baseline → P
• c > baseline → xA (x = c − baseline)
• c < baseline → xd (x = baseline − c)
• x = 1 may omit the number; x = 3 may be written AAA or ddd.

n-standard degrees (2, 3, 6, 7 etc. and all fractional degrees)

• c = baseline → n
• 0 < Δc ≤ 0.5 (brighter) → (2Δc)M
• 0 < |Δc| ≤ 0.5 (darker) → (2|Δc|)m
• Δc > 0.5 → (Δc − 0.5)A
• |Δc| > 0.5 (darker) → (|Δc| − 0.5)d

Coefficient 1 may omit the number; larger integers may be stacked (AAA, mm, etc.).

Baseline Chroma for Non-Integer (Fractional) Degrees

For a fractional degree p/q, the baseline is the distance-weighted arithmetic mean of the two nearest integer-degree baselines:

[math]\displaystyle{ c_{\text{baseline}} = \frac{(R - p/q)c_L + (p/q - L)c_R}{R - L} }[/math]

All fractional degrees use the n-standard naming rules.

Example: degree 4/3 ≈ 1.333

Adjacent: L = 1 (c = 0), R = 2 (c = −3/14) Weights: w_L = 2 − 4/3 = 2/3, w_R = 4/3 − 1 = 1/3 → baseline chroma = −1/14 → baseline name n4/3

Negative Degrees

When d+1 < 1, three conventions (“flavours”):

1. Invert the interval and prefix the degree with “−” (e.g. downward d-2).
2. Prefix the entire name with “−” (e.g. −d2 or −(d2)).
3. Octave-shift upward until the nominal is 1–7 and annotate octave offset (lo, lolo, hi, hihi or “−1 oct”, “+2 oct”, etc.).

Examples

1. Half perfect fifth [math]\displaystyle{ \sqrt{3/2} }[/math]

[math]\displaystyle{ x = -0.5,\ y = 0.5 \to c = 1/14,\ d = 2 }[/math] → nominal degree 3, baseline +1/14 → exactly neutral → n3

2. Half perfect fourth [math]\displaystyle{ \sqrt{4/3} }[/math]

[math]\displaystyle{ x = 1,\ y = -0.5 \to c = -1/14,\ d = 1.5 }[/math] → nominal degree 2.5  
Baseline (interpolated) = −1/14 → exactly neutral → n2.5

3. Porcupine trichord third [math]\displaystyle{ (4/3)^{1/3} }[/math]

[math]\displaystyle{ x = 2/3,\ y = -1/3 \to c = -1/21,\ d = 1 }[/math] → nominal degree 2  
Baseline n2 = −3/14  
Δc = +1/6 (brighter, ≤0.5) → coefficient 2×(1/6) = 1/3 → ⅓M2 or (1/3)M2

This system thus allows intuitive, continuous naming of any 3-limit interval (including fractional degrees and sub-major/minor shadings) while remaining fully compatible with standard FJS for ordinary just intervals.

Formal Commas

Below is the nomenclature and mathematical identity of all primes' formal commas up to 31, with radius of tolerance = 1/2d2 (Neutral FJS). Everything is the same as

Prime	Formal Comma	EFJS	Fifth shift
5	80/81	M3^5	+4
7	63/64	m7^7	-2
11	(242/243)^1/2	1/2A4^11	+2.5
13	(507/512)^1/2	n6^13	-0.5
17	4131/4096	m2^17	-5
19	513/512	m3^19	-3
23	736/729	A4^23	+6
29	sqrt(841/864)	n7^29	+1.5
31	sqrt(2101707/2097152)	1/2d8^31	-3.5

Irrational intervals

For irrational intervals, such as edosteps, i degrees are used, where i(x) = x-1 steps of 7edo, x being any real number.

Comma stacking is written with parentheses for large numbers or nonintegers; 3125/3072 = -dd2^5(5), sqrt(5/4) = M2^5(1/2).

Below is a table of some examples to which this notation can be taken. Some of these examples are absurd beyond use, but the aim is merely illustrative; to show how far this notation can be taken.

Edosteps		Edonoisteps and others
1/1	P1	(81/80)^3/2	P1^5(3/2)
1\7	i2	(64/63)^2	P1^7(2)
5\14	i3.5	Vavoom comma	-12d7^5(17)
9\56	i1.125	5\13{3}	i(log2(3)35/13) ≈ i5.2672
31\53	i(4+5/53) ≈ i5.0943	1\5{5}	i(log2(5)7/5) ≈ i4.2507
18\31	i(4+2/31) ≈ i5.0645	[1/2 -1/5 1/7]	i(log2(3)7/5+3.5)^5(1/7) ≈ i4.603
1157\3613	i(2+873/3613) ≈ i3.2416	|-1 4/11 -14/37 23/41 -1/44⟩	i(log2(3)28/11-7)^7(23/41)_5(14/37)_11(1/44) ≈ i2.3581