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


See [https://misotanni.github.io/fjs/en/index.html the FJS website] for details.
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&nbsp;=&nbsp;[[65/63]] by default (in ''“The radius of tolerance is a constant, by definition equal to 65/63.”''<ref>[https://misotanni.github.io/fjs/en/rules.html The Complete Formal FJS Description]</ref>) 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&nbsp;=&nbsp;sqrt(2187/2048)), and neutral FJS (λ&nbsp;=&nbsp;sqrt(134217728/129140163)).
 
== Weblinks ==
* [https://misotanni.github.io/fjs/en/index.html The FJS website]
* [https://misotanni.github.io/fjs/en/calc.html Calculator]
* [https://www.yacavone.net/fjs-explorer/ Custom FJS Explorer]
* [https://www.youtube.com/channel/UCrKUfsh5r1uMEx8EBevPflw misotanni &#91;old&#93; &#45; YouTube] – (abandoned channel)
 
== Quick reference ==
=== Formal commas ===
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Formal commas and intervals up to the 89-limit
! rowspan="2" |Prime
! colspan="3" |Formal comma
! rowspan="2" |Interval
! colspan="3" |Reduced prime harmonic
|-
! Standard
!FloraC
!Neutral
!Standard
!FloraC
!Neutral
|-
| [[5-limit|5]]
| colspan="3" | [[81/80|80/81]]
|[[5/4]]
| colspan="3" |M3^5
|-
| [[7-limit|7]]
| colspan="3" | [[64/63|63/64]]
|[[7/4]]
| colspan="3" |m7^7
|-
| [[11-limit|11]]
| colspan="2" | [[33/32]]
|sqrt([[243/242|242/243]])
|[[11/8]]
| colspan="2" |P4^11
|sA4^11
|-
| [[13-limit|13]]
| colspan="2" | [[1053/1024]]
|sqrt([[512/507|507/512]])
|[[13/8]]
| colspan="2" |m6^13
|n6^13
|-
| [[17-limit|17]]
| colspan="3" | [[4131/4096]]
|[[17/16]]
| colspan="3" |m2^17
|-
| [[19-limit|19]]
| colspan="3" | [[513/512]]
|[[19/16]]
| colspan="3" |m3^19
|-
| [[23-limit|23]]
| colspan="3" | [[736/729]]
|[[23/16]]
| colspan="3" |A4^23
|-
| [[29-limit|29]]
| colspan="2" | [[261/256]]
|sqrt(841/864)
|[[29/16]]
| colspan="2" |m7^29
|n7^29
|-
| [[31-limit|31]]
| [[248/243]]
|[[32/31|31/32]]
|sqrt(2101707/2097152)
|[[31/16]]
|M7^31
|P8^31
|sd8^31
|-
|[[37-limit|37]]
| colspan="2" |[[37/36]]
|sqrt(175232/177147)
|[[37/32]]
| colspan="2" |M2^37
|sA2^37
|-
|[[41-limit|41]]
| colspan="3" |[[82/81]]
|[[41/32]]
| colspan="3" |M3^41
|-
|[[43-limit|43]]
| colspan="3" |[[129/128]]
|[[43/32]]
| colspan="3" |P4^43
|-
|[[47-limit|47]]
| colspan="2" |47/48
|sqrt(536787/524288)
|[[47/32]]
| colspan="2" |P5^47
|sd5^47
|-
|[[53-limit|53]]
| colspan="3" |53/54
|[[53/32]]
| colspan="3" |M6^53
|-
|[[59-limit|59]]
| colspan="2" |236/243
|sqrt(3481/3456)
|[[59/32]]
| colspan="2" |M7^59
|n7^59
|-
|[[61-limit|61]]
| colspan="3" |244/243
|[[61/32]]
| colspan="3" |M7^61
|-
|[[67-limit|67]]
| colspan="3" |16281/16384
|[[67/64]]
| colspan="3" |m2^67
|-
|[[71-limit|71]]
| colspan="3" |71/72
|[[71/64]]
| colspan="3" |M2^71
|-
|[[73-limit|73]]
| colspan="3" |73/72
|[[73/64]]
| colspan="3" |M2^73
|-
|[[79-limit|79]]
| colspan="2" |79/81
|sqrt(6241/6144)
|[[79/64]]
| colspan="2" |M3^79
|n3^79
|-
|[[83-limit|83]]
| colspan="2" |249/256
|sqrt(135596187/134217728)
|[[83/64]]
| colspan="2" |P4^83
|sd4^83
|-
|[[89-limit|89]]
| colspan="2" |712/729
|64881/65536
|[[89/64]]
| colspan="2" |A4^89
|d5^89
|}
Flora's version differs from standard only for the primes 31, 157, 353...
 
=== Harmonic series ===
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Overtones 1–32 with root C [Default]
|-
! 1–8
| C
| C
| G
| C
| E<sup>5</sup>
| G
| B♭<sup>7</sup>
| C
|-
! 9–16
| D
| E<sup>5</sup>
| F<sup>11</sup>
| G
| A♭<sup>13</sup>
| B♭<sup>7</sup>
| B<sup>5</sup>
| C
|-
! 17–24
| D♭<sup>17</sup>
| D
| E♭<sup>19</sup>
| E<sup>5</sup>
| F<sup>7</sup>
| F<sup>11</sup>
| F♯<sup>23</sup>
| G
|-
! 25–32
| G♯<sup>25</sup>
| A♭<sup>13</sup>
| A
| B♭<sup>7</sup>
| B♭<sup>29</sup>
| B<sup>5</sup>
| B<sup>31</sup>
| C
|}
{| class="wikitable center-all"
|+Overtones 1–32 with root C [FloraC]
|-
! 1–8
| C
| C
| G
| C
| E<sup>5</sup>
| G
| B♭<sup>7</sup>
| C
|-
! 9–16
| D
| E<sup>5</sup>
| F<sup>11</sup>
| G
| A♭<sup>13</sup>
| B♭<sup>7</sup>
| B<sup>5</sup>
| C
|-
! 17–24
| D♭<sup>17</sup>
| D
| E♭<sup>19</sup>
| E<sup>5</sup>
| F<sup>7</sup>
| F<sup>11</sup>
| F♯<sup>23</sup>
| G
|-
! 25–32
| G♯<sup>25</sup>
| A♭<sup>13</sup>
| A
| B♭<sup>7</sup>
| B♭<sup>29</sup>
| B<sup>5</sup>
| '''C<sup>31</sup>'''
| C
|}
{| class="wikitable center-all"
|+Overtones 1–32 with root C [Neutral]
|-
! 1–8
| C
| C
| G
| C
| E<sup>5</sup>
| G
| B♭<sup>7</sup>
| C
|-
! 9–16
| D
| E<sup>5</sup>
| '''F‡<sup>11</sup>'''
| G
| '''Ad<sup>13</sup>'''
| B♭<sup>7</sup>
| B<sup>5</sup>
| C
|-
! 17–24
| D♭<sup>17</sup>
| D
| E♭<sup>19</sup>
| E<sup>5</sup>
| F<sup>7</sup>
| '''F‡<sup>11</sup>'''
| F♯<sup>23</sup>
| G
|-
! 25–32
| G♯<sup>25</sup>
| '''Ad<sup>13</sup>'''
| A
| B♭<sup>7</sup>
| '''Bd<sup>29</sup>'''
| B<sup>5</sup>
| '''Cd<sup>31</sup>'''
| C
|}
Boldened notes denote deviations from default.
 
== See also ==
* [[Neutral FJS]]
* [[User:FloraC/Critique on Functional Just System|Flora Canou's proposal]]
 
{{Navbox notation}}
 
[[Category:Notation]]
[[Category:Just intonation]]

Latest revision as of 23:52, 27 November 2025

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.”[1]) 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

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


View • Talk • EditMusical notation
Universal Sagittal notation
Just intonation Functional Just System • Ben Johnston's notation (Johnston–Copper notation) • Helmholtz–Ellis notation • Color notation
MOS scales Diamond-mos notation • KISS notation (Quasi-diatonic MOS notation)
Temperaments Chain-of-fifths notation • Stein–Zimmermann–Gould notation • Ups and downs notation • Syntonic–rastmic subchroma notation • Extended meantone notation • Fractional sharp notation
See musical notation for a longer list of systems by category. See Category:Notation for the most complete, comprehensive list, but not sorted by category.
  1. ↑ The Complete Formal FJS Description
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