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


== Introductory materials ==
== 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 ==
== 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>
|}
|}


=== Odd harmonics ===
{| class="wikitable sortable center-all left-6" style="margin: auto auto auto auto;"
{| class="wikitable sortable center-all"
|+ style="font-size: 105%;" | Odd harmonics in the harmonic series
|-
! rowspan="2" | Harmonic
! rowspan="2" | Harmonic
! rowspan="2" | Just Ratio
! rowspan="2" | [[Octave reduced]] ratio
! colspan="2" | Notation with C as root
! colspan="3" | Notation (assuming 1/1 is C)
! colspan="2" | Notation with A as root
! rowspan="2" | Comments
! rowspan="2" | Comments
|-
|-
! 2020 version
! New 2020 version<ref group="note" name="v2025"/>
! 2004 version
! Old 2020 version
! 2020 version
! 2004 version
! 2004 version
|-
|-
| 1
| 1
| 1/1
| 1/1
| colspan="2" | [[File:HejiC.svg|48x48px]]
| colspan="3" | [[File:HejiC.svg|48x48px]]
| colspan="2" | [[File:HejiA.svg|48x48px]]
|  
| rowspan="2" | Default staff notation represents Pythagorean tuning
|-
|-
| 3
| 3
| 3/2
| 3/2
| colspan="2" | [[File:HejiG.svg|48x48px]]
| colspan="3" | [[File:HejiG.svg|48x48px]]
| colspan="2" | [[File:HejiE.svg|48x48px]]
|  
|-
|-
| 5
| 5
| 5/4
| 5/4
| colspan="2" | [[File:Heji17.svg|54x54px]][[File:HejiE.svg|48x48px]]
| colspan="3" | [[File:Heji17.svg|54x54px]][[File:HejiE.svg|48x48px]]
| colspan="2" | [[File:Heji24.svg|54x54px]][[File:HejiC.svg|48x48px]]
|  
|  
|-
|-
| 7
| 7
| 7/4
| 7/4
| colspan="2" | [[File:Heji37.svg|54x54px]][[File:Heji11.svg|54x54px]][[File:HejiB.svg|48x48px]]
| colspan="3" | [[File:Heji37.svg|54x54px]][[File:Heji11.svg|54x54px]][[File:HejiB.svg|48x48px]]
| colspan="2" | [[File:Heji37.svg|54x54px]][[File:HejiG.svg|48x48px]]
|  
|  
|-
|-
| 9
| 9
| 9/8
| 9/8
| colspan="2" | [[File:HejiD.svg|48x48px]]
| colspan="3" | [[File:HejiD.svg|48x48px]]
| colspan="2" | [[File:HejiB.svg|48x48px]]
| style="text-align: left"; |  
| Default staff notation represents Pythagorean tuning
|-
|-
| 11
| 11
| 11/8
| 11/8
| colspan="2" | [[File:Heji41.svg|54x54px]][[File:HejiF.svg|48x48px]]
| colspan="3" | [[File:Heji41.svg|54x54px]][[File:HejiF.svg|48x48px]]
| colspan="2" | [[File:Heji41.svg|54x54px]][[File:HejiD.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
| 13/8
| 13/8
| colspan="2" | [[File:Heji42.svg|54x54px]][[File:HejiA.svg|48x48px]]
| colspan="3" | [[File:Heji42.svg|54x54px]][[File:HejiA.svg|48x48px]]
| colspan="2" | [[File:Heji42.svg|54x54px]][[File:Heji25.svg|54x54px]][[File:HejiF.svg|48x48px]]
|  
|  
|-
|-
| 15
| 15
| 15/8
| 15/8
| colspan="2" | [[File:Heji17.svg|54x54px]][[File:HejiB.svg|48x48px]]
| colspan="3" | [[File:Heji17.svg|54x54px]][[File:HejiB.svg|48x48px]]
| colspan="2" | [[File:Heji24.svg|54x54px]][[File:HejiG.svg|48x48px]]
|  
|  
|-
|-
| 17
| 17
| 17/16
| 17/16
| [[File:Heji44.svg|54x54px]][[File:Heji25.svg|54x54px]][[File:HejiC.svg|48x48px]]
| colspan="2" | [[File:Heji44.svg|54x54px]][[File:Heji25.svg|54x54px]][[File:HejiC.svg|48x48px]]
| [[File:Heji44.svg|54x54px]][[File:Heji12.svg|54x54px]][[File:HejiD.svg|48x48px]]
| [[File:Heji44.svg|54x54px]][[File:Heji12.svg|54x54px]][[File:HejiD.svg|48x48px]]
| [[File:Heji44.svg|54x54px]][[File:Heji25.svg|54x54px]][[File:HejiA.svg|48x48px]]
| style="text-align: left"; | Definition of the accidental is revised from 256/255 to 2187/2176 in the 2020 version by Plainsound Music Edition
| [[File:Heji44.svg|54x54px]][[File:Heji12.svg|54x54px]][[File:HejiB.svg|48x48px]]
| Definition of the accidental is revised from 256/255 to 2187/2176<br>in the 2020 version by Plainsound Music Edition
|-
|-
| 19
| 19
| 19/16
| 19/16
| colspan="2" | [[File:Heji47.svg|54x54px]][[File:Heji11.svg|54x54px]][[File:HejiE.svg|48x48px]]
| colspan="3" | [[File:Heji47.svg|54x54px]][[File:Heji11.svg|54x54px]][[File:HejiE.svg|48x48px]]
| colspan="2" | [[File:Heji47.svg|54x54px]][[File:HejiC.svg|48x48px]]
|  
|  
|-
|-
| 21
| 21
| 21/16
| 21/16
| colspan="2" | [[File:Heji37.svg|54x54px]][[File:HejiF.svg|48x48px]]
| colspan="3" | [[File:Heji37.svg|54x54px]][[File:HejiF.svg|48x48px]]
| colspan="2" | [[File:Heji37.svg|54x54px]][[File:HejiD.svg|48x48px]]
|  
|  
|-
|-
| 23
| 23
| 23/16
| 23/16
| colspan="2" | [[File:Heji49.svg|54x54px]][[File:Heji25.svg|54x54px]][[File:HejiF.svg|48x48px]]
| colspan="3" | [[File:Heji49.svg|54x54px]][[File:Heji25.svg|48x48px]][[File:HejiF.svg|48x48px]]
| colspan="2" | [[File:Heji49.svg|54x54px]][[File:Heji25.svg|54x54px]][[File:HejiD.svg|48x48px]]
|  
|  
|-
|-
| 25
| 25
| 25/16
| 25/16
| colspan="2" | [[File:Heji23.svg|54x54px]][[File:HejiG.svg|48x48px]]
| colspan="3" | [[File:Heji23.svg|54x54px]][[File:HejiG.svg|48x48px]]
| colspan="2" | [[File:Heji23.svg|54x54px]][[File:HejiE.svg|48x48px]]
|  
|  
|-
|-
| 27
| 27
| 27/16
| 27/16
| colspan="2" | [[File:HejiA.svg|48x48px]]
| colspan="3" | [[File:HejiA.svg|48x48px]]
| colspan="2" | [[File:Heji25.svg|54x54px]][[File:HejiF.svg|48x48px]]
|  
| Default staff notation represents Pythagorean tuning
|-
|-
| 29
| 29
| 29/16
| 29/16
| (todo)
| colspan="2" | [[File:HejiaccidentalCombiningRaise29LimitComma.svg|48x48px]][[File:Heji11.svg|54x54px]][[File:HejiB.svg|48x48px]]
| [[File:Heji51.svg|54x54px]][[File:Heji12.svg|54x54px]][[File:HejiB.svg|48x48px]]
| [[File:Heji51.svg|54x54px]][[File:Heji12.svg|54x54px]][[File:HejiB.svg|48x48px]]
| (todo)
| style="text-align: left"; | Definition of the accidental is revised from 145/144 to 261/256 in the 2020 version by Plainsound Music Edition
| [[File:Heji51.svg|54x54px]][[File:Heji19.svg|54x54px]][[File:HejiG.svg|48x48px]]
| Definition of the accidental is revised from 145/144 to 261/256<br>in the 2020 version by Plainsound Music Edition
|-
|-
| 31
| 31
| 31/16
| 31/16
| [[File:Heji52.svg|54x54px]][[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]]
| [[File:Heji52.svg|54x54px]][[File:HejiA.svg|48x48px]]
| style="text-align: left"; | Definition of the accidental is revised from 1024/1023 to 32/31 in the 2020 version by Plainsound Music Edition
| (todo)
| Definition of the accidental is revised from 1024/1023 to 32/31<br>in the 2020 version by Plainsound Music Edition
|-
|-
| 33
| 33
| 33/32
| 33/32
| colspan="2" | [[File:Heji41.svg|54x54px]][[File:HejiC.svg|48x48px]]
| colspan="3" | [[File:Heji41.svg|54x54px]][[File:HejiC.svg|48x48px]]
| colspan="2" | [[File:Heji41.svg|54x54px]][[File:HejiA.svg|48x48px]]
|  
|  
|-
|-
| 35
| 35
| 35/32
| 35/32
| colspan="2" | [[File:Heji37.svg|54x54px]][[File:Heji17.svg|54x54px]][[File:HejiD.svg|48x48px]]
| colspan="3" | [[File:Heji37.svg|54x54px]][[File:Heji17.svg|54x54px]][[File:HejiD.svg|48x48px]]
| colspan="2" | [[File:Heji37.svg|54x54px]][[File:Heji17.svg|54x54px]][[File:HejiB.svg|48x48px]]
|  
|-
|-
| 37
| 37
| 37/32
| 37/32
| (todo)
| colspan="2" | [[File:HejiaccidentalCombiningRaise37Quartertone.svg|48x48px]][[File:HejiD.svg|48x48px]]
| (todo)
| -
| (todo)
|  
| (todo)
|
|-
|-
| 39
| 39
| 39/32
| 39/32
| colspan="2" | [[File:Heji42.svg|54x54px]][[File:HejiE.svg|48x48px]]
| colspan="3" | [[File:Heji42.svg|54x54px]][[File:HejiE.svg|48x48px]]
| colspan="2" | [[File:Heji42.svg|54x54px]][[File:Heji25.svg|54x54px]][[File:HejiC.svg|48x48px]]
|  
|  
|-
|-
| 41
| 41
| 41/32
| 41/32
| (todo)
| colspan="2" | [[File:HejiaccidentalCombiningRaise41Comma.svg|48x48px]][[File:HejiE.svg|48x48px]]
| (todo)
| -
| (todo)
|  
| (todo)
|
|-
|-
| 43
| 43
| 43/32
| 43/32
| (todo)
| colspan="2" | [[File:HejiaccidentalCombiningRaise43Comma.svg|48x48px]][[File:HejiF.svg|48x48px]]
| (todo)
| -
| (todo)
|  
| (todo)
|
|-
|-
| 45
| 45
| 45/32
| 45/32
| colspan="2" | [[File:Heji24.svg|54x54px]][[File:HejiF.svg|48x48px]]
| colspan="3" | [[File:Heji24.svg|54x54px]][[File:HejiF.svg|48x48px]]
| colspan="2" | [[File:Heji24.svg|54x54px]][[File:HejiD.svg|48x48px]]
|  
|  
|-
|-
| 47
| 47
| 47/32
| 47/32
| (todo)
| [[File:HejiLowerByOne47limitSixthtone.png|24x36px]][[File:HejiG.svg|48x48px]]
| (todo)
| [[File:HejiaccidentalCombiningRaise47Quartertone.svg|48x48px]][[File:Heji25.svg|54x54px]][[File:HejiF.svg|48x48px]]
| (todo)
| -
| (todo)
| 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>
<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>
</gallery>
</div>


== External links ==
== External links ==
* [http://tonalsoft.com/enc/h/hewm.aspx HEWM Notation (Helmholtz-Ellis-Wolf-Monzo)] – Tonalsoft enyclopedia of microtonal music theory
* [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 ==
== 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]]
* [[Ben Johnston's notation]]
* [[:Category: SVG]]


== Notes ==
== Notes ==
<references group="note"/>
== References ==
<references/>
<references/>


[[Category:Helmholtz-Ellis notation| ]] <!-- main article -->
{{Navbox notation}}
 
[[Category:Helmholtz–Ellis notation| ]] <!-- Main article -->
[[Category:Notation]]
[[Category:Notation]]
[[Category:Just intonation]]
[[Category:Just intonation]]

Latest revision as of 17:24, 27 December 2025

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 accidentals defined by formal commas 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

Quick reference

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

Formal commas
Prime Formal comma
5 81/80
7 64/63
11 33/32
13 27/26
17 2187/2176
19 513/512
23 736/729
29 261/256
31 32/31
37 37/36
41 82/81
43 129/128
47 48/47[note 1]
Odd harmonics in the harmonic series
Harmonic Octave reduced ratio Notation (assuming 1/1 is C) Comments
New 2020 version[note 1] Old 2020 version 2004 version
1 1/1
3 3/2
5 5/4
7 7/4
9 9/8
11 11/8 is not a half-sharp, as F is 243/242 flatter than F♯.
13 13/8
15 15/8
17 17/16 Definition of the accidental is revised from 256/255 to 2187/2176 in the 2020 version by Plainsound Music Edition
19 19/16
21 21/16
23 23/16
25 25/16
27 27/16
29 29/16 Definition of the accidental is revised from 145/144 to 261/256 in the 2020 version by Plainsound Music Edition
31 31/16 Definition of the accidental is revised from 1024/1023 to 32/31 in the 2020 version by Plainsound Music Edition
33 33/32
35 35/32
37 37/32 -
39 39/32
41 41/32 -
43 43/32 -
45 45/32
47 47/32 - In pre–March-2025 notation, represents 752/729; in the update March 2025 of version 2020, represents 48/47.

Helmholtz–Ellis glyphs

  • Double flat lowered by three syntonic commas
    Double flat lowered by three syntonic commas
  • Double flat lowered by two syntonic commas
    Double flat lowered by two syntonic commas
  • Double flat lowered by one syntonic comma
    Double flat lowered by one syntonic comma
  • Double flat
    Double flat
  • Double flat raised by one syntonic comma
    Double flat raised by one syntonic comma
  • Double flat raised by two syntonic commas
    Double flat raised by two syntonic commas
  • Double flat raised by three syntonic commas
    Double flat raised by three syntonic commas
  • Flat lowered by three syntonic commas
    Flat lowered by three syntonic commas
  • Flat lowered by two syntonic commas
    Flat lowered by two syntonic commas
  • Flat lowered by one syntonic comma
    Flat lowered by one syntonic comma
  • Flat
    Flat
  • Flat raised by one syntonic comma
    Flat raised by one syntonic comma
  • Flat raised by two syntonic commas
    Flat raised by two syntonic commas
  • Flat raised by three syntonic commas
    Flat raised by three syntonic commas
  • Natural lowered by three syntonic commas
    Natural lowered by three syntonic commas
  • Natural lowered by two syntonic commas
    Natural lowered by two syntonic commas
  • Natural lowered by one syntonic comma
    Natural lowered by one syntonic comma
  • Natural
    Natural
  • Natural raised by one syntonic comma
    Natural raised by one syntonic comma
  • Natural raised by two syntonic commas
    Natural raised by two syntonic commas
  • Natural raised by three syntonic commas
    Natural raised by three syntonic commas
  • Sharp lowered by three syntonic commas
    Sharp lowered by three syntonic commas
  • Sharp lowered by two syntonic commas
    Sharp lowered by two syntonic commas
  • Sharp lowered by one syntonic comma
    Sharp lowered by one syntonic comma
  • Sharp
    Sharp
  • Sharp raised by one syntonic comma
    Sharp raised by one syntonic comma
  • Sharp raised by two syntonic commas
    Sharp raised by two syntonic commas
  • Sharp raised by three syntonic commas
    Sharp raised by three syntonic commas
  • Double sharp lowered by three syntonic commas
    Double sharp lowered by three syntonic commas
  • Double sharp lowered by two syntonic commas
    Double sharp lowered by two syntonic commas
  • Double sharp lowered by one syntonic comma
    Double sharp lowered by one syntonic comma
  • Double sharp
    Double sharp
  • Double sharp raised by one syntonic comma
    Double sharp raised by one syntonic comma
  • Double sharp raised by two syntonic commas
    Double sharp raised by two syntonic commas
  • Double sharp raised by three syntonic commas
    Double sharp raised by three syntonic commas
  • Lower by two septimal commas
    Lower by two septimal commas
  • Lower by one septimal comma
    Lower by one septimal comma
  • Raise by one septimal comma
    Raise by one septimal comma
  • Raise by two septimal commas
    Raise by two septimal commas
  • Lower by one undecimal quartertone
    Lower by one undecimal quartertone
  • Raise by one undecimal quartertone
    Raise by one undecimal quartertone
  • Lower by one tridecimal third tone
    Lower by one tridecimal third tone
  • Raise by one tridecimal third tone
    Raise by one tridecimal third tone
  • Combining lower by one 17-limit schisma
    Combining lower by one 17-limit schisma
  • Combining raise by one 17-limit schisma
    Combining raise by one 17-limit schisma
  • Combining lower by one 19-limit schisma
    Combining lower by one 19-limit schisma
  • Combining raise by one 19-limit schisma
    Combining raise by one 19-limit schisma
  • Combining lower by one 23-limit comma
    Combining lower by one 23-limit comma
  • Combining raise by one 23-limit comma
    Combining raise by one 23-limit comma
  • Combining lower by one 29-limit comma
    Combining lower by one 29-limit comma
  • Combining raise by one 29-limit comma
    Combining raise by one 29-limit comma
  • Combining lower by one 31-limit quartertone
    Combining lower by one 31-limit quartertone
  • Combining raise by one 31-limit quartertone
    Combining raise by one 31-limit quartertone
  • Combining lower by one 37-limit quartertone
    Combining lower by one 37-limit quartertone
  • Combining raise by one 37-limit quartertone
    Combining raise by one 37-limit quartertone
  • Combining lower by one 41-limit comma
    Combining lower by one 41-limit comma
  • Combining raise by one 41-limit comma
    Combining raise by one 41-limit comma
  • Combining lower by one 43-limit comma
    Combining lower by one 43-limit comma
  • Combining raise by one 43-limit comma
    Combining raise by one 43-limit comma
  • Raise by one 47-limit sixthtone (update 03.2025)
    Raise by one 47-limit sixthtone (update 03.2025)
  • Lower by one 47-limit sixthtone (update 03.2025)
    Lower by one 47-limit sixthtone (update 03.2025)

Old/deprecated symbols

  • Combining lower by one 29-limit schisma (old)
    Combining lower by one 29-limit schisma (old)
  • Combining raise by one 29-limit schisma (old)
    Combining raise by one 29-limit schisma (old)
  • Combining lower by one 31-limit schisma (old)
    Combining lower by one 31-limit schisma (old)
  • Combining raise by one 31-limit schisma (old)
    Combining raise by one 31-limit schisma (old)
  • Combining lower by one 47-limit quartertone (old)
    Combining lower by one 47-limit quartertone (old)
  • Combining raise by one 47-limit quartertone (old)
    Combining raise by one 47-limit quartertone (old)

External links

See also

Notes

  1. ↑ 1.0 1.1 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 The Helmholtz–Ellis JI Pitch Notation (HEJI).

References

  1. ↑ Marc Sabat : Music & Writings.


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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.
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