View source for User:Eufalesio/Moture's Extended Functional Just System

This is a machine translation of the [[User:FilterNashi/Moture's Extended Functional Just System (Simplified Chinese)|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:

# The accidental (d, m, P, M, A, etc.) describing the alteration;
# The nominal degree number;
# 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 m3<sup>11</sup> (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>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>\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>R = 2^{d/7} \cdot \left(\frac{2187}{2048}\right)^c</math>

Given a ratio in 3-limit form <math>R = 2^x \cdot 3^y</math>, we solve:

:<math>\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>2^{-3} \cdot 3^{2}</math>

:<math>c = \frac{2}{7},\quad d = 7(-3) + 11(2) = 1</math>

→ second (''d''+1 = 2) with chroma <math>c = +\frac{2}{7}</math>

=== Natural Interval Chroma Table ===

{| class="wikitable" style="text-align:center;"
! 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.

{| class="wikitable" style="text-align:center;"
! Degree !! Standard !! Baseline chroma (calculation) !! Value
|-
| 1 || P1 || <math>c_{P1} = 0</math> || 0
|-
| 2 || n2 || <math>\frac{-5/7 + 2/7}{2} = -3/14</math> || −3/14
|-
| 3 || n3 || <math>\frac{-3/7 + 4/7}{2} = +1/14</math> || +1/14
|-
| 4 || P4 || <math>c_{P4} = -1/7</math> || −1/7
|-
| 5 || P5 || <math>c_{P5} = +1/7</math> || +1/7
|-
| 6 || n6 || <math>\frac{-4/7 + 3/7}{2} = -1/14</math> || −1/14
|-
| 7 || n7 || <math>\frac{-2/7 + 5/7}{2} = +3/14</math> || +3/14
|}

'''Octave rule:''' adding 7''n'' 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 → ''x''A  (''x'' = ''c'' − baseline)
* ''c'' < baseline → ''x''d  (''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 < Δ''c'' ≤ 0.5 (brighter) → (2Δ''c'')M
* 0 < |Δ''c''| ≤ 0.5 (darker) → (2|Δ''c''|)m
* Δ''c'' > 0.5 → (Δ''c'' − 0.5)A
* |Δ''c''| > 0.5 (darker) → (|Δ''c''| − 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>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<sub>L</sub> = 2 − 4/3 = 2/3, w<sub>R</sub> = 4/3 − 1 = 1/3  
→ baseline chroma = −1/14 → baseline name '''n4/3'''

=== Negative Degrees ===

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

# Invert the interval and prefix the degree with “−” (e.g. downward d-2).
# Prefix the entire name with “−” (e.g. −d2 or −(d2)).
# 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>\sqrt{3/2}</math>  
 <math>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>\sqrt{4/3}</math>  
 <math>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>(4/3)^{1/3}</math>  
 <math>x = 2/3,\ y = -1/3 \to c = -1/21,\ d = 1</math> → nominal degree 2  
 Baseline n2 = −3/14  
 Δ''c'' = +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.

=== ––– Everything below this point is not in the original document; Eufalesio's addition. –––– ===

== 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 
{| class="wikitable"
|+
!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.
{| class="wikitable"
! colspan="2" |Edosteps
! colspan="2" |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
|<nowiki>|-1 4/11 -14/37 23/41 -1/44⟩</nowiki>
|i(log2(3)28/11-7)^7(23/41)_5(14/37)_11(1/44) ≈ i2.3581
|}

*