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]]. It consists of a set of [[nominal-accidental chain|accidentals]] defined by [[formal comma]]s for 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]].
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== Introductory materials ==
== Introductory materials ==
* [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
* [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://masa.plainsound.org/work.html#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 ==
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| [[43-limit|43]] || [[129/128]]
| [[43-limit|43]] || [[129/128]]
|-
|-
| [[47-limit|47]] || [[48/47]]<ref>The recent March 2025 revision of HEJI changed the 47 comma 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>
| [[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>
|}
|}


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|-
|-
! rowspan="2" | Harmonic
! rowspan="2" | Harmonic
! rowspan="2" | [[Octave reduced]] Ratio
! rowspan="2" | [[Octave reduced]] ratio
! colspan="3" | Notation (assuming 1/1 is C)
! colspan="3" | Notation (assuming 1/1 is C)
! rowspan="2" | Comments
! rowspan="2" | Comments
|-
|-
! 2020 version (update 03.2025)
! New 2020 version<ref group="note" name="v2025"/>
! 2020 version
! Old 2020 version
! 2004 version
! 2004 version
|-
|-
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== Notes ==
== Notes ==
<references />
<references group="note"/>
 
== References ==
<references/>


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

Revision as of 17:18, 27 December 2025

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 accidentals defined by formal commas for 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 Default staff notation represents Pythagorean tuning
3 3/2
5 5/4
7 7/4
9 9/8 Default staff notation represents Pythagorean tuning
11 11/8 is not a half-sharp, as F is 7.1¢ 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 Default staff notation represents Pythagorean tuning
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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