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Fractions are normally given in normalized form (since easier to lookup and link to). The information in the table doesn't say anything about the direction (i.e. if the exponent is positive or negative).
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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]].  
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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]].  


The Functional Just System can be seen as an extension of the pythagorean system: the base name of a note (G, D, Ab, 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.”''<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 ==
* [https://misotanni.github.io/fjs/en/index.html The FJS website]
* [https://misotanni.github.io/fjs/en/index.html The FJS website]
* [https://misotanni.github.io/fjs/en/calc.html Calculator]
* [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 ==
== Quick reference ==
=== Formal commas ===
=== Formal commas ===
{| class="wikitable center-all"
{| class="wikitable center-all"
|+Formal commas below 32-limit
|+ style="font-size: 105%;" | Formal commas and intervals up to the 89-limit
! Prime
! rowspan="2" |Prime
! Formal Comma
! colspan="3" |Formal comma
! rowspan="2" |Interval
! colspan="3" |Reduced prime harmonic
|-
! Standard
!FloraC
!Neutral
!Standard
!FloraC
!Neutral
|-
|-
| [[5-limit|5]]
| [[5-limit|5]]
| [[81/80]]
| colspan="3" | [[81/80|80/81]]
|[[5/4]]
| colspan="3" |M3^5
|-
|-
| [[7-limit|7]]
| [[7-limit|7]]
| [[64/63]]
| 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"
|+Overtones 1–32 with root C
|+ style="font-size: 105%;" | Overtones 1–32 with root C [Default]
|-
! 1–8
! 1–8
| C
| C
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| E<sup>5</sup>
| E<sup>5</sup>
| G
| G
| Bb<sup>7</sup>
| B♭<sup>7</sup>
| C
| C
|-
|-
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| F<sup>11</sup>
| F<sup>11</sup>
| G
| G
| Ab<sup>13</sup>
| A♭<sup>13</sup>
| Bb<sup>7</sup>
| B♭<sup>7</sup>
| B<sup>5</sup>
| B<sup>5</sup>
| C
| C
|-
|-
! 17–24
! 17–24
| Db<sup>17</sup>
| D♭<sup>17</sup>
| D
| D
| Eb<sup>19</sup>
| E♭<sup>19</sup>
| E<sup>5</sup>
| E<sup>5</sup>
| F<sup>7</sup>
| F<sup>7</sup>
| F<sup>11</sup>
| F<sup>11</sup>
| F#<sup>23</sup>
| F♯<sup>23</sup>
| G
| G
|-
|-
! 25–32
! 25–32
| G#<sup>25</sup>
| G♯<sup>25</sup>
| Ab<sup>13</sup>
| A♭<sup>13</sup>
| A
| A
| Bb<sup>7</sup>
| B♭<sup>7</sup>
| Bb<sup>29</sup>
| B♭<sup>29</sup>
| B<sup>5</sup>
| B<sup>5</sup>
| B<sup>31</sup>
| B<sup>31</sup>
| 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 ==
* [[Neutral FJS]]
* [[User:FloraC/Critique on Functional Just System|Flora Canou's proposal]]
{{Navbox notation}}


[[Category:Notation]]
[[Category:Notation]]
[[Category:Just intonation]]
[[Category:Just intonation]]
Retrieved from "https://en.xen.wiki/w/Functional_Just_System"