Helmholtz–Ellis notation

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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 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
47 752/729


Odd harmonics in the harmonic series
Harmonic Just Ratio Notation (assuming 1/1 is C) Comments
2020 version 2004 version
1 1/1 HejiC.svg Default staff notation represents Pythagorean tuning
3 3/2 HejiG.svg
5 5/4 Heji17.svgHejiE.svg
7 7/4 Heji37.svgHeji11.svgHejiB.svg
9 9/8 HejiD.svg Default staff notation represents Pythagorean tuning
11 11/8 Heji41.svgHejiF.svg Heji41.svg is not a half sharp. Heji41.svgHeji41.svgF is 7.1¢ flatter than F#.
13 13/8 Heji42.svgHejiA.svg
15 15/8 Heji17.svgHejiB.svg
17 17/16 Heji44.svgHeji25.svgHejiC.svg Heji44.svgHeji12.svgHejiD.svg Definition of the accidental is revised from 256/255 to 2187/2176
in the 2020 version by Plainsound Music Edition
19 19/16 Heji47.svgHeji11.svgHejiE.svg
21 21/16 Heji37.svgHejiF.svg
23 23/16 Heji49.svgHeji25.svgHejiF.svg
25 25/16 Heji23.svgHejiG.svg
27 27/16 HejiA.svg Default staff notation represents Pythagorean tuning
29 29/16 (todo) Heji51.svgHeji12.svgHejiB.svg Definition of the accidental is revised from 145/144 to 261/256
in the 2020 version by Plainsound Music Edition
31 31/16 Heji52.svgHejiC.svg (todo) Definition of the accidental is revised from 1024/1023 to 32/31
in the 2020 version by Plainsound Music Edition
33 33/32 Heji41.svgHejiC.svg
35 35/32 Heji37.svgHeji17.svgHejiD.svg
37 37/32 (todo) (todo)
39 39/32 Heji42.svgHejiE.svg
41 41/32 (todo) (todo)
43 43/32 (todo) (todo)
45 45/32 Heji24.svgHejiF.svg
47 47/32 (todo) (todo)

Helmholtz–Ellis glyphs

Icon-Todo.png Todo: update
Update the 29-limit comma.

External links

See also

Notes