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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 below 32-limit
|+ 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
|-
|-
! Prime
! Standard
! Formal Comma
!FloraC
!Neutral
!Standard
!FloraC
!Neutral
|-
|-
| [[5-limit|5]]
| [[5-limit|5]]
| [[81/80|80/81]]
| colspan="3" | [[81/80|80/81]]
|[[5/4]]
| colspan="3" |M3^5
|-
|-
| [[7-limit|7]]
| [[7-limit|7]]
| [[64/63|63/64]]
| colspan="3" | [[64/63|63/64]]
|[[7/4]]
| colspan="3" |m7^7
|-
|-
| [[11-limit|11]]
| [[11-limit|11]]
| [[33/32]]
| colspan="2" | [[33/32]]
|sqrt([[243/242|242/243]])
|[[11/8]]
| colspan="2" |P4^11
|sA4^11
|-
|-
| [[13-limit|13]]
| [[13-limit|13]]
| [[1053/1024]]
| colspan="2" | [[1053/1024]]
|sqrt([[512/507|507/512]])
|[[13/8]]
| colspan="2" |m6^13
|n6^13
|-
|-
| [[17-limit|17]]
| [[17-limit|17]]
| [[4131/4096]]
| colspan="3" | [[4131/4096]]
|[[17/16]]
| colspan="3" |m2^17
|-
|-
| [[19-limit|19]]
| [[19-limit|19]]
| [[513/512]]
| colspan="3" | [[513/512]]
|[[19/16]]
| colspan="3" |m3^19
|-
|-
| [[23-limit|23]]
| [[23-limit|23]]
| [[736/729]]
| colspan="3" | [[736/729]]
|[[23/16]]
| colspan="3" |A4^23
|-
|-
| [[29-limit|29]]
| [[29-limit|29]]
| [[261/256]]
| colspan="2" | [[261/256]]
|sqrt(841/864)
|[[29/16]]
| colspan="2" |m7^29
|n7^29
|-
|-
| [[31-limit|31]]
| [[31-limit|31]]
| [[248/243]]
| [[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 ===
=== 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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| C
| 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 ==
== See also ==
* [[Neutral FJS]]
* [[Neutral FJS]]
* [[User:FloraC/Critique on Functional Just System|Flora Canou's proposal]]


{{Navbox notation}}
{{Navbox notation}}
Retrieved from "https://en.xen.wiki/w/Functional_Just_System"