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.

Todo:

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			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		(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
41	41/32	(todo)	(todo)
43	43/32	(todo)	(todo)
45	45/32
47	47/32	(todo)	(todo)

Helmholtz–Ellis glyphs

y 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: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

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External links

• HEWM Notation (Helmholtz-Ellis-Wolf-Monzo) &ndas

External links

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

See also

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