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[1] 2004 version
Quick reference
Below is a quick guide to the accidentals and commas used in Helmholtz–Ellis notation.
| 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 | 752/729 |
| Harmonic | Just Ratio | Notation (assuming 1/1 is C) | Comments | |
|---|---|---|---|---|
| 2020 version | 2004 version | |||
| 1 | 1/1 | 
|
Default staff notation represents Pythagorean tuning | |
| 3 | 3/2 | 
| ||
| 5 | 5/4 |  
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||
| 7 | 7/4 |   
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||
| 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 |
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||
| 23 | 23/16 | 
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| 25 | 25/16 | 
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||
| 27 | 27/16 |
|
Default staff notation represents Pythagorean tuning | |
| 29 | 29/16 | (todo) | 
|
Definition of the accidental is revised from 145/144 to 261/256 in the 2020 version by Plainsound Music Edition |
| 31 | 31/16 | 
|
(todo) | 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 | (todo) | (todo) | |
| 39 | 39/32 |
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||
| 41 | 41/32 | (todo) | (todo) | |
| 43 | 43/32 | (todo) | (todo) | |
| 45 | 45/32 | 
|
||
| 47 | 47/32 | (todo) | (todo) | |
Helmholtz–Ellis glyphs
 |
Todo: update Update the 29-limit comma. |
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
Combining raise by one 23-limit comma
Combining lower by one 29-limit schisma (old)
Combining raise by one 29-limit schisma (old)
Combining lower by one 31-limit quartertone
Combining raise by one 31-limit quartertone
External links
- HEWM Notation (Helmholtz-Ellis-Wolf-Monzo) – Tonalsoft enyclopedia of microtonal music theory
- Plainsound Harmonic Space Calculator