Helmholtz–Ellis notation

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

• 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
• Extended Helmholtz-Ellis JI Pitch Notation by Marc Sabat and Wolfgang von Schweinitz from Plainsound Music Edition - deprecated 2004 version

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

Odd harmonics in the harmonic series

Harmonic	Just Ratio	Notation (assuming 1/1 is C)			Comments
		2020 version (update 03.2025)	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			-	Pre-March 2025 notation, where represents 752/729, in the update March 2025 of version 2022, represents 48/47

Helmholtz-Ellis glyphs

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

External links

• HEWM Notation (Helmholtz-Ellis-Wolf-Monzo) - Tonalsoft enyclopedia of microtonal music theory
• Plainsound Harmonic Space Calculator
• Glyphs in smufl

See also

• Color notation
• Functional Just System (FJS)
• Ben Johnston's notation
• Category:SVG