Helmholtz–Ellis notation: Difference between revisions

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The '''~~Helmholtz-Ellis~~ JI pitch notation''' ('''HEJI''') is a [[musical notation]] system for [[just intonation]] intervals up to the [[47-limit]]. ~~It consists~~ of a set of [[~~Nominal~~-accidental chain|accidentals]] defined by formal ~~commas for~~ each [[prime harmonic]].

The '''Helmholtz–Ellis JI pitch notation''' ('''HEJI''') is a [[musical notation]] system for [[just intonation]] intervals up to the [[47-limit]]. Based on [[chain-of-fifths notation]] with default staff notation representing [[Pythagorean tuning]], it adds a set of [[nominal-accidental chain|accidentals]] defined by [[formal comma]]s to reach each [[prime harmonic]].

Further accidentals were designed by [[richie]] for primes up to the [[89-limit]]; see [[richie's HEJI extensions]].

== Introductory materials ==

* [https://~~marsbat~~.~~space~~/~~pdfs/HEJI2legend+series.pdf~~ The Helmholtz-Ellis JI Pitch Notation (HEJI)] by Marc Sabat and Thomas Nicholson from Plainsound Music Edition – 2020 version with revised symbols for primes up to 47 entirely based on alterations of Pythagorean notes

* [https://heji.plainsound.org/ The Helmholtz-Ellis JI Pitch Notation (HEJI)] by Marc Sabat and Thomas Nicholson from Plainsound Music Edition – 2020 version with revised symbols for primes up to 47 entirely based on alterations of Pythagorean notes

* [http://www.plainsound.org/pdfs/ji_notation.pdf Extended ~~Helmholtz-Ellis~~ JI Pitch Notation] by Marc Sabat and Wolfgang von Schweinitz from Plainsound Music Edition – deprecated<ref>~~See~~ [https://~~marsbat~~.~~space~~/#writing Marc Sabat : Music & Writings]. </ref> 2004 version

* [http://www.plainsound.org/pdfs/ji_notation.pdf Extended Helmholtz–Ellis JI Pitch Notation] by Marc Sabat and Wolfgang von Schweinitz from Plainsound Music Edition – deprecated<ref>[https://masa.plainsound.org/work.html#writing Marc Sabat : Music & Writings].</ref> 2004 version

== Quick reference ==

~~=== Formal~~ commas ~~===~~

Below is a quick guide to the accidentals and commas used in Helmholtz–Ellis notation.

{| class="wikitable center-all"

~~! Prime~~

{| class="wikitable center-all" style="margin: auto auto auto auto;"

! Formal ~~Comma~~

|+ style="font-size: 105%;" | Formal commas

|-

~~| [[5-limit|5]]~~

! Prime !! Formal comma

~~| [[81/80]]~~

|-

| [[7-limit|7]]

| [[5-limit|5]] || [[81/80]]

| [[64/63]]

|-

| [[11-limit|11]]

| [[7-limit|7]] || [[64/63]]

| [[33/32]]

|-

| [[13-limit|13]]

| [[11-limit|11]] || [[33/32]]

| [[27/26]]

|-

| [[17-limit|17]]

| [[13-limit|13]] || [[27/26]]

| [[~~2187~~/~~2176~~]]

|-

| [[19-limit|19]]

| [[17-limit|17]] || [[2187/2176]]

| [[~~513~~/~~512~~]]

|-

| [[23-limit|23]]

| [[19-limit|19]] || [[513/512]]

| [[~~736~~/~~729~~]]

|-

| [[29-limit|29]]

| [[23-limit|23]] || [[736/729]]

| [[~~261~~/~~256~~]]

|-

| [[31-limit|31]]

| [[29-limit|29]] || [[261/256]]

| [[32/31]]

|-

| [[37-limit|37]]

| [[31-limit|31]] || [[32/31]]

| [[37/36]]

|-

| [[41-limit|41]]

| [[37-limit|37]] || [[37/36]]

| [[82/81]]

|-

| [[43-limit|43]]

| [[41-limit|41]] || [[82/81]]

| [[~~129~~/~~128~~]]

|-

| [[47-limit|47]]

| [[43-limit|43]] || [[129/128]]

| [[752/729]]

|-

| [[47-limit|47]] || [[48/47]]<ref group="note" name="v2025">In the recent March 2025 revision of HEJI, the 47 comma was changed from [[752/729]] to [[48/47]], which has yet to gain popular support. See [https://heji.plainsound.org The Helmholtz–Ellis JI Pitch Notation (HEJI)].</ref>

|}

~~=== Prime harmonics ===~~

{| class="wikitable sortable center-all left-6" style="margin: auto auto auto auto;"

{| class="wikitable sortable center-all left-5"

|+ style="font-size: 105%;" | Odd harmonics in the harmonic series

~~! rowspan~~="2" ~~|Prime~~

~~! rowspan="2"~~ |~~Just Ratio~~

~~! colspan~~="2" |~~Notation (assuming 1/1 is C)~~

~~! rowspan="2" |Comments~~

|-

! 2020 version

! rowspan="2" | Harmonic

! rowspan="2" | [[Octave reduced]] ratio

! colspan="3" | Notation (assuming 1/1 is C)

! rowspan="2" | Comments

|-

! New 2020 version<ref group="note" name="v2025"/>

! Old 2020 version

! 2004 version

|-

| 1

| 1/1

| ~~[[File:HejiC.svg|48x48px]]~~

| colspan="3" | [[File:HejiC.svg|48x48px]]

| [[File:HejiC.svg|48x48px]]

|

| ~~Default staff notation represents Pythagorean tuning~~

|-

| 3

| 3/2

| ~~[[File:HejiG.svg|48x48px]]~~

| colspan="3" | [[File:HejiG.svg|48x48px]]

| [[File:HejiG.svg|48x48px]]

|

| ~~Default staff notation represents Pythagorean tuning~~

|-

| 5

| 5/4

| ~~[[File:Heji17.svg|48x48px]][[File:HejiE.svg|48x48px]]~~

| colspan="3" | [[File:Heji17.svg|54x54px]][[File:HejiE.svg|48x48px]]

| [[File:Heji17.svg|~~48x48px~~]][[File:HejiE.svg|48x48px]]

|

|-

| 7

| 7/4

| [[File:Heji37.svg|~~48x48px~~]][[File:Heji11.svg|~~48x48px~~]][[File:HejiB.svg|48x48px]]

| ~~[[File:Heji37.svg~~|~~48x48px]][[File:Heji11.svg~~|~~48x48px]]~~[[File:~~HejiB~~.svg|48x48px]]

|

|-

| 9

| 9/8

| colspan="3" | [[File:HejiD.svg|48x48px]]

| style="text-align: left"; |

|-

| 11

| 11/8

| [[File:Heji41.svg|~~48x48px~~]][[File:HejiF.svg|48x48px]]

| colspan="3" | [[File:Heji41.svg|54x54px]][[File:HejiF.svg|48x48px]]

| [[File:Heji41.svg|~~48x48px~~]][[File:~~HejiF~~.svg|~~48x48px~~]]

|

|-

| 13

| 13/8

| [[File:Heji42.svg|~~48x48px~~]][[File:HejiA.svg|48x48px]]

| colspan="3" | [[File:Heji42.svg|54x54px]][[File:HejiA.svg|48x48px]]

| [[File:~~Heji42~~.svg|~~48x48px~~]][[File:~~HejiA~~.svg|48x48px]]

|

|-

| 15

| 15/8

| colspan="3" | [[File:Heji17.svg|54x54px]][[File:HejiB.svg|48x48px]]

|

|-

| 17

| 17/16

| [[File:Heji44.svg|~~48x48px~~]][[File:Heji25.svg|~~48x48px~~]][[File:HejiC.svg|48x48px]]

| [[File:Heji44.svg|~~48x48px~~]][[File:Heji12.svg|~~48x48px~~]][[File:HejiD.svg|48x48px]]

| [[File:Heji44.svg|54x54px]][[File:Heji12.svg|54x54px]][[File:HejiD.svg|48x48px]]

| Definition of the accidental is revised from 256/255 to 2187/2176~~<br>~~in the 2020 version by Plainsound Music Edition

| style="text-align: left"; | Definition of the accidental is revised from 256/255 to 2187/2176 in the 2020 version by Plainsound Music Edition

|-

| 19

| 19/16

| [[File:Heji47.svg|~~48x48px~~]][[File:Heji11.svg|~~48x48px~~]][[File:HejiE.svg|48x48px]]

| ~~[[File:Heji47.svg~~|~~48x48px]]~~[[File:~~Heji11~~.svg|~~48x48px~~]][[File:~~HejiE~~.svg|48x48px]]

|

|-

| 21

| 21/16

| colspan="3" | [[File:Heji37.svg|54x54px]][[File:HejiF.svg|48x48px]]

|

|-

| 23

| 23/16

| [[File:Heji49.svg|~~48x48px~~]][[File:Heji25.svg|48x48px]][[File:HejiF.svg|48x48px]]

| [[File:~~Heji49~~.svg|~~48x48px~~]][[File:~~Heji25~~.svg|48x48px]][[File:~~HejiF~~.svg|48x48px]]

|

|-

| 25

| 25/16

| colspan="3" | [[File:Heji23.svg|54x54px]][[File:HejiG.svg|48x48px]]

|

|-

| 27

| 27/16

| colspan="3" | [[File:HejiA.svg|48x48px]]

|

|-

| 29

| 29/16

| ~~(todo)~~

| [[File:Heji51.svg|~~48x48px~~]][[File:Heji12.svg|~~48x48px~~]][[File:HejiB.svg|48x48px]]

| [[File:Heji51.svg|54x54px]][[File:Heji12.svg|54x54px]][[File:HejiB.svg|48x48px]]

| Definition of the accidental is revised from 145/144 to 261/256~~<br>~~in the 2020 version by Plainsound Music Edition

| style="text-align: left"; | Definition of the accidental is revised from 145/144 to 261/256 in the 2020 version by Plainsound Music Edition

|-

| 31

| 31/16

| [[File:Heji52.svg|48x48px]][[File:HejiC.svg|48x48px]]

| colspan="2" | [[File:Heji52.svg|54x54px]][[File:HejiC.svg|48x48px]]

| ~~(todo)~~

| [[File:HejiaccidentalCombiningLower31SchismaOld.svg|48x48px]][[File:Heji40.svg|54x54px]][[File:HejiC.svg|48x48px]]

| Definition of the accidental is revised from 1024/1023 to 32/31~~<br>~~in the 2020 version by Plainsound Music Edition

| style="text-align: left"; | Definition of the accidental is revised from 1024/1023 to 32/31 in the 2020 version by Plainsound Music Edition

|-

| 33

| 33/32

| colspan="3" | [[File:Heji41.svg|54x54px]][[File:HejiC.svg|48x48px]]

|

|-

| 35

| 35/32

|

|-

| 37

| 37/32

| ~~(todo)~~

| colspan="2" | [[File:HejiaccidentalCombiningRaise37Quartertone.svg|48x48px]][[File:HejiD.svg|48x48px]]

| ~~(todo)~~

| -

|

|-

| 39

| 39/32

| colspan="3" | [[File:Heji42.svg|54x54px]][[File:HejiE.svg|48x48px]]

|

|-

| 41

| 41/32

| ~~(todo)~~

| colspan="2" | [[File:HejiaccidentalCombiningRaise41Comma.svg|48x48px]][[File:HejiE.svg|48x48px]]

| ~~(todo)~~

| -

|

|-

| 43

| 43/32

| ~~(todo)~~

| colspan="2" | [[File:HejiaccidentalCombiningRaise43Comma.svg|48x48px]][[File:HejiF.svg|48x48px]]

| ~~(todo)~~

| -

|

|-

| 45

| 45/32

| colspan="3" | [[File:Heji24.svg|54x54px]][[File:HejiF.svg|48x48px]]

|

|-

| 47

| 47/32

| ~~(todo)~~

| [[File:HejiLowerByOne47limitSixthtone.png|24x36px]][[File:HejiG.svg|48x48px]]

| ~~(todo)~~

| [[File:HejiaccidentalCombiningRaise47Quartertone.svg|48x48px]][[File:Heji25.svg|54x54px]][[File:HejiF.svg|48x48px]]

|

| -

| In pre–March-2025 notation, [[File:HejiaccidentalCombiningRaise47Quartertone.svg|36x36px]] represents [[752/729]]; in the update March 2025 of version 2020, [[File:HejiRaiseByOne47limitSixthtone.png|24x36px]] represents [[48/47]].

|}

== ~~Helmholtz-Ellis~~ glyphs ==

== Helmholtz–Ellis glyphs ==

~~{{Todo~~|~~update~~|~~inline=1~~|~~comment=Update the~~ 29-limit comma. }}

File:Heji1.svg | Double flat lowered by three syntonic commas

File:Heji2.svg | Double flat lowered by two syntonic commas

File:Heji3.svg | Double flat lowered by one syntonic comma

File:Heji4.svg | Double flat

File:Heji5.svg | Double flat raised by one syntonic comma

File:Heji6.svg | Double flat raised by two syntonic commas

File:Heji7.svg | Double flat raised by three syntonic commas

File:Heji8.svg | Flat lowered by three syntonic commas

File:Heji9.svg | Flat lowered by two syntonic commas

File:Heji10.svg | Flat lowered by one syntonic comma

File:Heji11.svg | Flat

File:Heji12.svg | Flat raised by one syntonic comma

File:Heji13.svg | Flat raised by two syntonic commas

File:Heji14.svg | Flat raised by three syntonic commas

File:Heji15.svg | Natural lowered by three syntonic commas

File:Heji16.svg | Natural lowered by two syntonic commas

File:Heji17.svg | Natural lowered by one syntonic comma

File:Heji18.svg | Natural

File:Heji19.svg | Natural raised by one syntonic comma

File:Heji20.svg | Natural raised by two syntonic commas

File:Heji21.svg | Natural raised by three syntonic commas

File:Heji22.svg | Sharp lowered by three syntonic commas

File:Heji23.svg | Sharp lowered by two syntonic commas

File:Heji24.svg | Sharp lowered by one syntonic comma

File:Heji25.svg | Sharp

File:Heji26.svg | Sharp raised by one syntonic comma

File:Heji27.svg | Sharp raised by two syntonic commas

File:Heji28.svg | Sharp raised by three syntonic commas

File:Heji29.svg | Double sharp lowered by three syntonic commas

File:Heji30.svg | Double sharp lowered by two syntonic commas

File:Heji31.svg | Double sharp lowered by one syntonic comma

File:Heji32.svg | Double sharp

File:Heji33.svg | Double sharp raised by one syntonic comma

File:Heji34.svg | Double sharp raised by two syntonic commas

File:Heji35.svg | Double sharp raised by three syntonic commas

File:Heji36.svg | Lower by two septimal commas

File:Heji37.svg | Lower by one septimal comma

File:Heji38.svg | Raise by one septimal comma

File:Heji39.svg | Raise by two septimal commas

File:Heji40.svg | Lower by one undecimal quartertone

File:Heji41.svg | Raise by one undecimal quartertone

File:Heji42.svg | Lower by one tridecimal third tone

File:Heji43.svg | Raise by one tridecimal third tone

File:Heji44.svg | Combining lower by one 17-limit schisma

File:Heji45.svg | Combining raise by one 17-limit schisma

File:Heji46.svg | Combining lower by one 19-limit schisma

File:Heji47.svg | Combining raise by one 19-limit schisma

File:Heji48.svg | Combining lower by one 23-limit comma

File:Heji49.svg | Combining raise by one 23-limit comma

File:Heji accidentalCombiningLower29LimitComma.svg | Combining lower by one 29-limit comma

File:HejiaccidentalCombiningRaise29LimitComma.svg | Combining raise by one 29-limit comma

File:Heji52.svg | Combining lower by one 31-limit quartertone

File:Heji53.svg | Combining raise by one 31-limit quartertone

File:HejiaccidentalCombiningLower37Quartertone.svg | Combining lower by one 37-limit quartertone

File:HejiaccidentalCombiningRaise37Quartertone.svg | Combining raise by one 37-limit quartertone

File:HejiaccidentalCombiningLower41Comma.svg | Combining lower by one 41-limit comma

File:HejiaccidentalCombiningRaise41Comma.svg | Combining raise by one 41-limit comma

File:HejiaccidentalCombiningLower43Comma.svg | Combining lower by one 43-limit comma

File:HejiaccidentalCombiningRaise43Comma.svg | Combining raise by one 43-limit comma

File:HejiRaiseByOne47limitSixthtone.png | Raise by one 47-limit sixthtone (update 03.2025)

File:HejiLowerByOne47limitSixthtone.png | Lower by one 47-limit sixthtone (update 03.2025)

</gallery>

</div>

=== Old/deprecated symbols ===

File:~~Heji1.svg|Double flat lowered by three syntonic commas~~

File:Heji50.svg | Combining lower by one 29-limit schisma (old)

~~File:Heji2.svg|Double flat lowered by two syntonic commas~~

File:Heji51.svg | Combining raise by one 29-limit schisma (old)

~~File:Heji3.svg|Double flat lowered by one syntonic comma~~

File:HejiaccidentalCombiningLower31SchismaOld.svg | Combining lower by one 31-limit schisma (old)

~~File:Heji4.svg|Double flat~~

File:HejiaccidentalCombiningRaise31SchismaOld.svg | Combining raise by one 31-limit schisma (old)

~~File:Heji5.svg|Double flat raised by one syntonic comma~~

File:HejiaccidentalCombiningLower47Quartertone.svg | Combining lower by one 47-limit quartertone (old)

~~File:Heji6.svg|Double flat raised by two syntonic commas~~

File:HejiaccidentalCombiningRaise47Quartertone.svg | Combining raise by one 47-limit quartertone (old)

~~File:Heji7.svg|Double flat raised by three syntonic commas~~

~~File:Heji8.svg|Flat lowered by three syntonic commas~~

~~File:Heji9.svg|Flat lowered by two syntonic commas~~

~~File:Heji10.svg|Flat lowered by one syntonic comma~~

~~File:Heji11.svg|Flat~~

~~File:Heji12.svg|Flat raised by one syntonic comma~~

~~File:Heji13.svg|Flat raised by two syntonic commas~~

~~File:Heji14.svg|Flat raised by three syntonic commas~~

~~File:Heji15.svg|Natural lowered by three syntonic commas~~

~~File:Heji16.svg|Natural lowered by two syntonic commas~~

~~File:Heji17.svg|Natural lowered by one syntonic comma~~

~~File:Heji18.svg|Natural~~

~~File:Heji19.svg|Natural raised by one syntonic comma~~

~~File:Heji20.svg|Natural raised by two syntonic commas~~

~~File:Heji21.svg|Natural raised by three syntonic commas~~

~~File:Heji22.svg|Sharp lowered by three syntonic commas~~

~~File:Heji23.svg|Sharp lowered by two syntonic commas~~

~~File:Heji24.svg|Sharp lowered by one syntonic comma~~

~~File:Heji25.svg|Sharp~~

~~File:Heji26.svg|Sharp raised by one syntonic comma~~

~~File:Heji27.svg|Sharp raised by two syntonic commas~~

~~File:Heji28.svg|Sharp raised by three syntonic commas~~

~~File:Heji29.svg|Double sharp lowered by three syntonic commas~~

~~File:Heji30.svg|Double sharp lowered by two syntonic commas~~

~~File:Heji31.svg|Double sharp lowered by one syntonic comma~~

~~File:Heji32.svg|Double sharp~~

~~File:Heji33.svg|Double sharp raised by one syntonic comma~~

~~File:Heji34.svg|Double sharp raised by two syntonic commas~~

~~File:Heji35.svg|Double sharp raised by three syntonic commas~~

~~File:Heji36.svg|Lower by two septimal commas~~

~~File:Heji37.svg|Lower by one septimal comma~~

~~File:Heji38.svg|Raise by one septimal comma~~

~~File:Heji39.svg|Raise by two septimal commas~~

~~File:Heji40.svg|Lower by one undecimal quartertone~~

~~File:Heji41.svg|Raise by one undecimal quartertone~~

~~File:Heji42.svg|Lower by one tridecimal third tone~~

~~File:Heji43.svg|Raise by one tridecimal third tone~~

~~File:Heji44~~.svg|Combining lower by one 17-limit schisma

File:~~Heji45~~.svg|Combining raise by one 17-limit schisma

File:~~Heji46~~.svg|Combining lower by one ~~19-limit schisma~~

~~File:Heji47.svg|Combining raise by one 19-limit schisma~~

~~File:Heji48.svg|Combining lower by one 23-limit comma~~

~~File:Heji49.svg|Combining raise by one 23-limit comma~~

~~File:Heji50.svg|Combining lower by one 29~~-limit schisma (old)

File:~~Heji51~~.svg|Combining raise by one 29-limit schisma (old)

File:~~Heji52~~.svg|Combining lower by one 31-limit quartertone

File:~~Heji53~~.svg|Combining raise by one 31-limit quartertone

</gallery>

</div>

== External links ==

* [http://tonalsoft.com/enc/h/hewm.aspx HEWM Notation (Helmholtz-Ellis-Wolf-Monzo)] – Tonalsoft enyclopedia of microtonal music theory

* [https://~~www~~.plainsound.org~~/HEJI~~/ Plainsound Harmonic Space Calculator]

* [https://hejicalc.plainsound.org/ Plainsound Harmonic Space Calculator]

* [https://w3c.github.io/smufl/latest/tables/extended-helmholtz-ellis-accidentals-just-intonation-supplement.html Glyphs in SMuFL]

== See also ==

* [[Functional Just System]] (FJS) ~~– a logical notation system for the entirety of just intonation~~

* [[Color notation]]

* [[Functional Just System]] (FJS)

* [[Ben Johnston's notation]]

* [[:Category: SVG]]

== Notes ==

== References ==

[[Category:~~Helmholtz-Ellis~~ notation| ]]

[[Category:Helmholtz–Ellis notation| ]]

[[Category:Notation]]

[[Category:Just intonation]]

@@ Line 1: / Line 1: @@
-The '''Helmholtz-Ellis JI pitch notation''' ('''HEJI''') is a [[musical notation]] system for [[just intonation]] intervals up to the [[47-limit]]. It consists of a set of [[Nominal-accidental chain|accidentals]] defined by formal commas for each [[prime harmonic]].
+The '''Helmholtz–Ellis JI pitch notation''' ('''HEJI''') is a [[musical notation]] system for [[just intonation]] intervals up to the [[47-limit]]. Based on [[chain-of-fifths notation]] with default staff notation representing [[Pythagorean tuning]], it adds a set of [[nominal-accidental chain|accidentals]] defined by [[formal comma]]s to reach each [[prime harmonic]].
 Further accidentals were designed by [[richie]] for primes up to the [[89-limit]]; see [[richie's HEJI extensions]].
 == Introductory materials ==
-* [https://marsbat.space/pdfs/HEJI2legend+series.pdf The Helmholtz-Ellis JI Pitch Notation (HEJI)] by Marc Sabat and Thomas Nicholson from Plainsound Music Edition – 2020 version with revised symbols for primes up to 47 entirely based on alterations of Pythagorean notes
+* [https://heji.plainsound.org/ The Helmholtz-Ellis JI Pitch Notation (HEJI)] by Marc Sabat and Thomas Nicholson from Plainsound Music Edition – 2020 version with revised symbols for primes up to 47 entirely based on alterations of Pythagorean notes
-* [http://www.plainsound.org/pdfs/ji_notation.pdf Extended Helmholtz-Ellis JI Pitch Notation] by Marc Sabat and Wolfgang von Schweinitz from Plainsound Music Edition – deprecated<ref>See [https://marsbat.space/#writing Marc Sabat : Music & Writings]. </ref> 2004 version
+* [http://www.plainsound.org/pdfs/ji_notation.pdf Extended Helmholtz–Ellis JI Pitch Notation] by Marc Sabat and Wolfgang von Schweinitz from Plainsound Music Edition – deprecated<ref>[https://masa.plainsound.org/work.html#writing Marc Sabat : Music & Writings].</ref> 2004 version
 == Quick reference ==
-=== Formal commas ===
+Below is a quick guide to the accidentals and commas used in Helmholtz–Ellis notation.
-{| class="wikitable center-all"
-! Prime
+{| class="wikitable center-all" style="margin: auto auto auto auto;"
-! Formal Comma
+|+ style="font-size: 105%;" | Formal commas
 |-
-| [[5-limit|5]]
+! Prime !! Formal comma
-| [[81/80]]
 |-
-| [[7-limit|7]]
+| [[5-limit|5]] || [[81/80]]
-| [[64/63]]
 |-
-| [[11-limit|11]]
+| [[7-limit|7]] || [[64/63]]
-| [[33/32]]
 |-
-| [[13-limit|13]]
+| [[11-limit|11]] || [[33/32]]
-| [[27/26]]
 |-
-| [[17-limit|17]]
+| [[13-limit|13]] || [[27/26]]
-| [[2187/2176]]
 |-
-| [[19-limit|19]]
+| [[17-limit|17]] || [[2187/2176]]
-| [[513/512]]
 |-
-| [[23-limit|23]]
+| [[19-limit|19]] || [[513/512]]
-| [[736/729]]
 |-
-| [[29-limit|29]]
+| [[23-limit|23]] || [[736/729]]
-| [[261/256]]
 |-
-| [[31-limit|31]]
+| [[29-limit|29]] || [[261/256]]
-| [[32/31]]
 |-
-| [[37-limit|37]]
+| [[31-limit|31]] || [[32/31]]
-| [[37/36]]
 |-
-| [[41-limit|41]]
+| [[37-limit|37]] || [[37/36]]
-| [[82/81]]
 |-
-| [[43-limit|43]]
+| [[41-limit|41]] || [[82/81]]
-| [[129/128]]
 |-
-| [[47-limit|47]]
+| [[43-limit|43]] || [[129/128]]
-| [[752/729]]
+|-
+| [[47-limit|47]] || [[48/47]]<ref group="note" name="v2025">In the recent March 2025 revision of HEJI, the 47 comma was changed from [[752/729]] to [[48/47]], which has yet to gain popular support. See [https://heji.plainsound.org The Helmholtz–Ellis JI Pitch Notation (HEJI)].</ref>
 |}
-=== Prime harmonics ===
+{| class="wikitable sortable center-all left-6" style="margin: auto auto auto auto;"
-{| class="wikitable sortable center-all left-5"
+|+ style="font-size: 105%;" | Odd harmonics in the harmonic series
-! rowspan="2" |Prime
-! rowspan="2" |Just Ratio
-! colspan="2" |Notation (assuming 1/1 is C)
-! rowspan="2" |Comments
 |-
-! 2020 version
+! rowspan="2" | Harmonic
+! rowspan="2" | [[Octave reduced]] ratio
+! colspan="3" | Notation (assuming 1/1 is C)
+! rowspan="2" | Comments
+|-
+! New 2020 version<ref group="note" name="v2025"/>
+! Old 2020 version
 ! 2004 version
 |-
 | 1
 | 1/1
-| [[File:HejiC.svg|48x48px]]
+| colspan="3" | [[File:HejiC.svg|48x48px]]
-| [[File:HejiC.svg|48x48px]]
+|
-| Default staff notation represents Pythagorean tuning
 |-
 | 3
 | 3/2
-| [[File:HejiG.svg|48x48px]]
+| colspan="3" | [[File:HejiG.svg|48x48px]]
-| [[File:HejiG.svg|48x48px]]
+|
-| Default staff notation represents Pythagorean tuning
 |-
 | 5
 | 5/4
-| [[File:Heji17.svg|48x48px]][[File:HejiE.svg|48x48px]]
+| colspan="3" | [[File:Heji17.svg|54x54px]][[File:HejiE.svg|48x48px]]
-| [[File:Heji17.svg|48x48px]][[File:HejiE.svg|48x48px]]
+|
-|
 |-
 | 7
 | 7/4
-| [[File:Heji37.svg|48x48px]][[File:Heji11.svg|48x48px]][[File:HejiB.svg|48x48px]]
+| colspan="3" | [[File:Heji37.svg|54x54px]][[File:Heji11.svg|54x54px]][[File:HejiB.svg|48x48px]]
-| [[File:Heji37.svg|48x48px]][[File:Heji11.svg|48x48px]][[File:HejiB.svg|48x48px]]
+|
-|
+|-
+| 9
+| 9/8
+| colspan="3" | [[File:HejiD.svg|48x48px]]
+| style="text-align: left"; |
 |-
 | 11
 | 11/8
-| [[File:Heji41.svg|48x48px]][[File:HejiF.svg|48x48px]]
+| colspan="3" | [[File:Heji41.svg|54x54px]][[File:HejiF.svg|48x48px]]
-| [[File:Heji41.svg|48x48px]][[File:HejiF.svg|48x48px]]
+| style="text-align: left"; | [[File:Heji41.svg|54x54px]] is ''not'' a half-sharp, as [[File:Heji41.svg|54x54px]][[File:Heji41.svg|54x54px]]F is [[243/242]] flatter than F♯.
-|
 |-
 | 13
 | 13/8
-| [[File:Heji42.svg|48x48px]][[File:HejiA.svg|48x48px]]
+| colspan="3" | [[File:Heji42.svg|54x54px]][[File:HejiA.svg|48x48px]]
-| [[File:Heji42.svg|48x48px]][[File:HejiA.svg|48x48px]]
+|
-|
+|-
+| 15
+| 15/8
+| colspan="3" | [[File:Heji17.svg|54x54px]][[File:HejiB.svg|48x48px]]
+|
 |-
 | 17
 | 17/16
-| [[File:Heji44.svg|48x48px]][[File:Heji25.svg|48x48px]][[File:HejiC.svg|48x48px]]
+| colspan="2" | [[File:Heji44.svg|54x54px]][[File:Heji25.svg|54x54px]][[File:HejiC.svg|48x48px]]
-| [[File:Heji44.svg|48x48px]][[File:Heji12.svg|48x48px]][[File:HejiD.svg|48x48px]]
+| [[File:Heji44.svg|54x54px]][[File:Heji12.svg|54x54px]][[File:HejiD.svg|48x48px]]
-| Definition of the accidental is revised from 256/255 to 2187/2176<br>in the 2020 version by Plainsound Music Edition
+| style="text-align: left"; | Definition of the accidental is revised from 256/255 to 2187/2176 in the 2020 version by Plainsound Music Edition
 |-
 | 19
 | 19/16
-| [[File:Heji47.svg|48x48px]][[File:Heji11.svg|48x48px]][[File:HejiE.svg|48x48px]]
+| colspan="3" | [[File:Heji47.svg|54x54px]][[File:Heji11.svg|54x54px]][[File:HejiE.svg|48x48px]]
-| [[File:Heji47.svg|48x48px]][[File:Heji11.svg|48x48px]][[File:HejiE.svg|48x48px]]
+|
-|
+|-
+| 21
+| 21/16
+| colspan="3" | [[File:Heji37.svg|54x54px]][[File:HejiF.svg|48x48px]]
+|
 |-
 | 23
 | 23/16
-| [[File:Heji49.svg|48x48px]][[File:Heji25.svg|48x48px]][[File:HejiF.svg|48x48px]]
+| colspan="3" | [[File:Heji49.svg|54x54px]][[File:Heji25.svg|48x48px]][[File:HejiF.svg|48x48px]]
-| [[File:Heji49.svg|48x48px]][[File:Heji25.svg|48x48px]][[File:HejiF.svg|48x48px]]
+|
-|
+|-
+| 25
+| 25/16
+| colspan="3" | [[File:Heji23.svg|54x54px]][[File:HejiG.svg|48x48px]]
+|
+|-
+| 27
+| 27/16
+| colspan="3" | [[File:HejiA.svg|48x48px]]
+|
 |-
 | 29
 | 29/16
-| (todo)
+| colspan="2" | [[File:HejiaccidentalCombiningRaise29LimitComma.svg|48x48px]][[File:Heji11.svg|54x54px]][[File:HejiB.svg|48x48px]]
-| [[File:Heji51.svg|48x48px]][[File:Heji12.svg|48x48px]][[File:HejiB.svg|48x48px]]
+| [[File:Heji51.svg|54x54px]][[File:Heji12.svg|54x54px]][[File:HejiB.svg|48x48px]]
-| Definition of the accidental is revised from 145/144 to 261/256<br>in the 2020 version by Plainsound Music Edition
+| style="text-align: left"; | Definition of the accidental is revised from 145/144 to 261/256 in the 2020 version by Plainsound Music Edition
 |-
 | 31
 | 31/16
-| [[File:Heji52.svg|48x48px]][[File:HejiC.svg|48x48px]]
+| colspan="2" | [[File:Heji52.svg|54x54px]][[File:HejiC.svg|48x48px]]
-| (todo)
+| [[File:HejiaccidentalCombiningLower31SchismaOld.svg|48x48px]][[File:Heji40.svg|54x54px]][[File:HejiC.svg|48x48px]]
-| Definition of the accidental is revised from 1024/1023 to 32/31<br>in the 2020 version by Plainsound Music Edition
+| style="text-align: left"; | Definition of the accidental is revised from 1024/1023 to 32/31 in the 2020 version by Plainsound Music Edition
+|-
+| 33
+| 33/32
+| colspan="3" | [[File:Heji41.svg|54x54px]][[File:HejiC.svg|48x48px]]
+|
+|-
+| 35
+| 35/32
+| colspan="3" | [[File:Heji37.svg|54x54px]][[File:Heji17.svg|54x54px]][[File:HejiD.svg|48x48px]]
+|
 |-
 | 37
 | 37/32
-| (todo)
+| colspan="2" | [[File:HejiaccidentalCombiningRaise37Quartertone.svg|48x48px]][[File:HejiD.svg|48x48px]]
-| (todo)
+| -
 |
+|-
+| 39
+| 39/32
+| colspan="3" | [[File:Heji42.svg|54x54px]][[File:HejiE.svg|48x48px]]
+|
 |-
 | 41
 | 41/32
-| (todo)
+| colspan="2" | [[File:HejiaccidentalCombiningRaise41Comma.svg|48x48px]][[File:HejiE.svg|48x48px]]
-| (todo)
+| -
 |
 |-
 | 43
 | 43/32
-| (todo)
+| colspan="2" | [[File:HejiaccidentalCombiningRaise43Comma.svg|48x48px]][[File:HejiF.svg|48x48px]]
-| (todo)
+| -
 |
+|-
+| 45
+| 45/32
+| colspan="3" | [[File:Heji24.svg|54x54px]][[File:HejiF.svg|48x48px]]
+|
 |-
 | 47
 | 47/32
-| (todo)
+| [[File:HejiLowerByOne47limitSixthtone.png|24x36px]][[File:HejiG.svg|48x48px]]
-| (todo)
+| [[File:HejiaccidentalCombiningRaise47Quartertone.svg|48x48px]][[File:Heji25.svg|54x54px]][[File:HejiF.svg|48x48px]]
-|
+| -
+| In pre–March-2025 notation, [[File:HejiaccidentalCombiningRaise47Quartertone.svg|36x36px]] represents [[752/729]]; in the update March 2025 of version 2020, [[File:HejiRaiseByOne47limitSixthtone.png|24x36px]] represents [[48/47]].
 |}
-== Helmholtz-Ellis glyphs ==
+== Helmholtz–Ellis glyphs ==
-{{Todo|update|inline=1|comment=Update the 29-limit comma. }}
+<div style="margin: auto auto auto auto;">
+<gallery>
+File:Heji1.svg | Double flat lowered by three syntonic commas
+File:Heji2.svg | Double flat lowered by two syntonic commas
+File:Heji3.svg | Double flat lowered by one syntonic comma
+File:Heji4.svg | Double flat
+File:Heji5.svg | Double flat raised by one syntonic comma
+File:Heji6.svg | Double flat raised by two syntonic commas
+File:Heji7.svg | Double flat raised by three syntonic commas
+File:Heji8.svg | Flat lowered by three syntonic commas
+File:Heji9.svg | Flat lowered by two syntonic commas
+File:Heji10.svg | Flat lowered by one syntonic comma
+File:Heji11.svg | Flat
+File:Heji12.svg | Flat raised by one syntonic comma
+File:Heji13.svg | Flat raised by two syntonic commas
+File:Heji14.svg | Flat raised by three syntonic commas
+File:Heji15.svg | Natural lowered by three syntonic commas
+File:Heji16.svg | Natural lowered by two syntonic commas
+File:Heji17.svg | Natural lowered by one syntonic comma
+File:Heji18.svg | Natural
+File:Heji19.svg | Natural raised by one syntonic comma
+File:Heji20.svg | Natural raised by two syntonic commas
+File:Heji21.svg | Natural raised by three syntonic commas
+File:Heji22.svg | Sharp lowered by three syntonic commas
+File:Heji23.svg | Sharp lowered by two syntonic commas
+File:Heji24.svg | Sharp lowered by one syntonic comma
+File:Heji25.svg | Sharp
+File:Heji26.svg | Sharp raised by one syntonic comma
+File:Heji27.svg | Sharp raised by two syntonic commas
+File:Heji28.svg | Sharp raised by three syntonic commas
+File:Heji29.svg | Double sharp lowered by three syntonic commas
+File:Heji30.svg | Double sharp lowered by two syntonic commas
+File:Heji31.svg | Double sharp lowered by one syntonic comma
+File:Heji32.svg | Double sharp
+File:Heji33.svg | Double sharp raised by one syntonic comma
+File:Heji34.svg | Double sharp raised by two syntonic commas
+File:Heji35.svg | Double sharp raised by three syntonic commas
+File:Heji36.svg | Lower by two septimal commas
+File:Heji37.svg | Lower by one septimal comma
+File:Heji38.svg | Raise by one septimal comma
+File:Heji39.svg | Raise by two septimal commas
+File:Heji40.svg | Lower by one undecimal quartertone
+File:Heji41.svg | Raise by one undecimal quartertone
+File:Heji42.svg | Lower by one tridecimal third tone
+File:Heji43.svg | Raise by one tridecimal third tone
+File:Heji44.svg | Combining lower by one 17-limit schisma
+File:Heji45.svg | Combining raise by one 17-limit schisma
+File:Heji46.svg | Combining lower by one 19-limit schisma
+File:Heji47.svg | Combining raise by one 19-limit schisma
+File:Heji48.svg | Combining lower by one 23-limit comma
+File:Heji49.svg | Combining raise by one 23-limit comma
+File:Heji accidentalCombiningLower29LimitComma.svg | Combining lower by one 29-limit comma
+File:HejiaccidentalCombiningRaise29LimitComma.svg | Combining raise by one 29-limit comma
+File:Heji52.svg | Combining lower by one 31-limit quartertone
+File:Heji53.svg | Combining raise by one 31-limit quartertone
+File:HejiaccidentalCombiningLower37Quartertone.svg | Combining lower by one 37-limit quartertone
+File:HejiaccidentalCombiningRaise37Quartertone.svg | Combining raise by one 37-limit quartertone
+File:HejiaccidentalCombiningLower41Comma.svg | Combining lower by one 41-limit comma
+File:HejiaccidentalCombiningRaise41Comma.svg | Combining raise by one 41-limit comma
+File:HejiaccidentalCombiningLower43Comma.svg | Combining lower by one 43-limit comma
+File:HejiaccidentalCombiningRaise43Comma.svg | Combining raise by one 43-limit comma
+File:HejiRaiseByOne47limitSixthtone.png | Raise by one 47-limit sixthtone (update 03.2025)
+File:HejiLowerByOne47limitSixthtone.png | Lower by one 47-limit sixthtone (update 03.2025)
+</gallery>
+</div>
+=== Old/deprecated symbols ===
+<div style="margin: auto auto auto auto;">
 <gallery>
-File:Heji1.svg|Double flat lowered by three syntonic commas
+File:Heji50.svg | Combining lower by one 29-limit schisma (old)
-File:Heji2.svg|Double flat lowered by two syntonic commas
+File:Heji51.svg | Combining raise by one 29-limit schisma (old)
-File:Heji3.svg|Double flat lowered by one syntonic comma
+File:HejiaccidentalCombiningLower31SchismaOld.svg | Combining lower by one 31-limit schisma (old)
-File:Heji4.svg|Double flat
+File:HejiaccidentalCombiningRaise31SchismaOld.svg | Combining raise by one 31-limit schisma (old)
-File:Heji5.svg|Double flat raised by one syntonic comma
+File:HejiaccidentalCombiningLower47Quartertone.svg | Combining lower by one 47-limit quartertone (old)
-File:Heji6.svg|Double flat raised by two syntonic commas
+File:HejiaccidentalCombiningRaise47Quartertone.svg | Combining raise by one 47-limit quartertone (old)
-File:Heji7.svg|Double flat raised by three syntonic commas
-File:Heji8.svg|Flat lowered by three syntonic commas
-File:Heji9.svg|Flat lowered by two syntonic commas
-File:Heji10.svg|Flat lowered by one syntonic comma
-File:Heji11.svg|Flat
-File:Heji12.svg|Flat raised by one syntonic comma
-File:Heji13.svg|Flat raised by two syntonic commas
-File:Heji14.svg|Flat raised by three syntonic commas
-File:Heji15.svg|Natural lowered by three syntonic commas
-File:Heji16.svg|Natural lowered by two syntonic commas
-File:Heji17.svg|Natural lowered by one syntonic comma
-File:Heji18.svg|Natural
-File:Heji19.svg|Natural raised by one syntonic comma
-File:Heji20.svg|Natural raised by two syntonic commas
-File:Heji21.svg|Natural raised by three syntonic commas
-File:Heji22.svg|Sharp lowered by three syntonic commas
-File:Heji23.svg|Sharp lowered by two syntonic commas
-File:Heji24.svg|Sharp lowered by one syntonic comma
-File:Heji25.svg|Sharp
-File:Heji26.svg|Sharp raised by one syntonic comma
-File:Heji27.svg|Sharp raised by two syntonic commas
-File:Heji28.svg|Sharp raised by three syntonic commas
-File:Heji29.svg|Double sharp lowered by three syntonic commas
-File:Heji30.svg|Double sharp lowered by two syntonic commas
-File:Heji31.svg|Double sharp lowered by one syntonic comma
-File:Heji32.svg|Double sharp
-File:Heji33.svg|Double sharp raised by one syntonic comma
-File:Heji34.svg|Double sharp raised by two syntonic commas
-File:Heji35.svg|Double sharp raised by three syntonic commas
-File:Heji36.svg|Lower by two septimal commas
-File:Heji37.svg|Lower by one septimal comma
-File:Heji38.svg|Raise by one septimal comma
-File:Heji39.svg|Raise by two septimal commas
-File:Heji40.svg|Lower by one undecimal quartertone
-File:Heji41.svg|Raise by one undecimal quartertone
-File:Heji42.svg|Lower by one tridecimal third tone
-File:Heji43.svg|Raise by one tridecimal third tone
-File:Heji44.svg|Combining lower by one 17-limit schisma
-File:Heji45.svg|Combining raise by one 17-limit schisma
-File:Heji46.svg|Combining lower by one 19-limit schisma
-File:Heji47.svg|Combining raise by one 19-limit schisma
-File:Heji48.svg|Combining lower by one 23-limit comma
-File:Heji49.svg|Combining raise by one 23-limit comma
-File:Heji50.svg|Combining lower by one 29-limit schisma (old)
-File:Heji51.svg|Combining raise by one 29-limit schisma (old)
-File:Heji52.svg|Combining lower by one 31-limit quartertone
-File:Heji53.svg|Combining raise by one 31-limit quartertone
 </gallery>
+</div>
 == External links ==
 * [http://tonalsoft.com/enc/h/hewm.aspx HEWM Notation (Helmholtz-Ellis-Wolf-Monzo)] – Tonalsoft enyclopedia of microtonal music theory
-* [https://www.plainsound.org/HEJI/ Plainsound Harmonic Space Calculator]
+* [https://hejicalc.plainsound.org/ Plainsound Harmonic Space Calculator]
+* [https://w3c.github.io/smufl/latest/tables/extended-helmholtz-ellis-accidentals-just-intonation-supplement.html Glyphs in SMuFL]
 == See also ==
-* [[Functional Just System]] (FJS) – a logical notation system for the entirety of just intonation
+* [[Color notation]]
+* [[Functional Just System]] (FJS)
 * [[Ben Johnston's notation]]
+* [[:Category: SVG]]
 == Notes ==
+<references group="note"/>
+== References ==
 <references/>
-[[Category:Helmholtz-Ellis notation| ]] <!-- main article -->
+{{Navbox notation}}
+[[Category:Helmholtz–Ellis notation| ]] <!-- Main article -->
 [[Category:Notation]]
 [[Category:Just intonation]]