Functional Just System: Difference between revisions

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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''' ('''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&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>)
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 ==
== Weblinks ==
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=== Formal commas ===
=== Formal commas ===
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Formal commas up to the 31-limit
|+ style="font-size: 105%;" | Formal commas up to the 89-limit
! rowspan="2" |Prime
! colspan="3" |Formal comma
|-
|-
! Prime
! Standard
! Formal comma
!FloraC
!Neutral
|-
|-
| [[5-limit|5]]
| [[5-limit|5]]
| [[81/80|80/81]]
| colspan="3" | [[81/80|80/81]]
|-
|-
| [[7-limit|7]]
| [[7-limit|7]]
| [[64/63|63/64]]
| colspan="3" | [[64/63|63/64]]
|-
|-
| [[11-limit|11]]
| [[11-limit|11]]
| [[33/32]]
| colspan="2" | [[33/32]]
|[[180224/177147]]
|-
|-
| [[13-limit|13]]
| [[13-limit|13]]
| [[1053/1024]]
| colspan="2" | [[1053/1024]]
|[[6656/6561]]
|-
|-
| [[17-limit|17]]
| [[17-limit|17]]
| [[4131/4096]]
| colspan="3" | [[4131/4096]]
|-
|-
| [[19-limit|19]]
| [[19-limit|19]]
| [[513/512]]
| colspan="3" | [[513/512]]
|-
|-
| [[23-limit|23]]
| [[23-limit|23]]
| [[736/729]]
| colspan="3" | [[736/729]]
|-
|-
| [[29-limit|29]]
| [[29-limit|29]]
| [[261/256]]
| colspan="2" | [[261/256]]
|59392/59049
|-
|-
| [[31-limit|31]]
| [[31-limit|31]]
| [[248/243]]
| [[248/243]]
|[[31/32]]
|16474671/16777216
|-
|[[37-limit|37]]
| colspan="2" |[[37/36]]
|4849664/4782969
|-
|[[41-limit|41]]
| colspan="3" |[[82/81]]
|-
|[[43-limit|43]]
| colspan="3" |[[129/128]]
|-
|[[47-limit|47]]
| colspan="2" |47/48
|8325909/8388608
|-
|[[53-limit|53]]
| colspan="3" |53/54
|-
|[[59-limit|59]]
| colspan="2" |236/243
|129033/131072
|-
|[[61-limit|61]]
| colspan="3" |244/243
|-
|[[67-limit|67]]
| colspan="3" |16281/16384
|-
|[[71-limit|71]]
| colspan="3" |71/72
|-
|[[73-limit|73]]
| colspan="3" |73/72
|-
|[[79-limit|79]]
| colspan="2" |79/81
|518319/524288
|-
|[[83-limit|83]]
| colspan="2" |249/256
|132328809/134217728
|-
|[[89-limit|89]]
| colspan="2" |712/729
|64881/65536
|}
|}


=== Harmonic series ===
=== Harmonic series ===
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Overtones 1–32 with root C
|+ style="font-size: 105%;" | Overtones 1–32 with root C [Default]
|-
|-
! 1–8
! 1–8
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| B<sup>5</sup>
| B<sup>5</sup>
| B<sup>31</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>
| '''E♯<sup>11</sup>'''
| G
| '''G♯<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>
| '''E♯<sup>11</sup>'''
| '''F♯<sup>23</sup>'''
| G
|-
! 25–32
| G♯<sup>25</sup>
| '''G♯<sup>13</sup>'''
| A
| B♭<sup>7</sup>
| '''A♯<sup>29</sup>'''
| B<sup>5</sup>
| '''D𝄫<sup>31</sup>'''
| C
| C
|}
|}
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== See also ==
== See also ==
* [[Neutral FJS]]
* [[Neutral FJS]]
* [[User:FloraC/Critique on Functional Just System|Flora Canou's proposal]]


{{Navbox notation}}
{{Navbox notation}}

Revision as of 13:36, 9 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 up to the 89-limit
Prime Formal comma
Standard FloraC Neutral
5 80/81
7 63/64
11 33/32 180224/177147
13 1053/1024 6656/6561
17 4131/4096
19 513/512
23 736/729
29 261/256 59392/59049
31 248/243 31/32 16474671/16777216
37 37/36 4849664/4782969
41 82/81
43 129/128
47 47/48 8325909/8388608
53 53/54
59 236/243 129033/131072
61 244/243
67 16281/16384
71 71/72
73 73/72
79 79/81 518319/524288
83 249/256 132328809/134217728
89 712/729 64881/65536

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 E♯11 G G♯13 B♭7 B5 C
17–24 D♭17 D E♭19 E5 F7 E♯11 F♯23 G
25–32 G♯25 G♯13 A B♭7 A♯29 B5 D𝄫31 C

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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