23 equal temperament

In music, 23 equal temperament, called 23-TET, 23-EDO ("Equal Division of the Octave"), or 23-ET, is the tempered scale derived by dividing the octave into 23 equal steps (equal frequency ratios). Each step represents a frequency ratio of ²³√2, or 52.174 cents. This system is the largest EDO that has an error of at least 20 cents for the 3rd (3:2), 5th (5:4), 7th (7:4), and 11th (11:8) harmonics. The lack of approximation to simple intervals makes the scale notable among those seeking to break free from conventional harmony rules.

History and use

23-EDO was advocated by ethnomusicologist Erich von Hornbostel in the 1920s,[1] as the result of "a cycle of 'blown' (compressed) fifths"[2] of about 678 cents that may have resulted from "overblowing" a bamboo pipe. Today, tens of songs have been composed in this system.

Notation

There are two ways to notate the 23-tone system with the traditional letter names and system of sharps and flats, called melodic notation and harmonic notation.

Harmonic notation preserves harmonic structures and interval arithmetic, but sharp and flat have reversed meanings. Because it preserves harmonic structures, 12-EDO music can be reinterpreted as 23-EDO harmonic notation, so it is also called conversion notation.

An example of these harmonic structures is the Circle of Fifths below (shown in 12-EDO, harmonic notation, and melodic notation.)

Circle of Fifths in 12-EDO			Circle of Fifths in 23-EDO Harmonic Notation				Circle of Fifths in 23-EDO Melodic Notation
Sharp Side	Enharmonicity	Flat Side	Sharp Side	Enharmonicity	Flat Side	Enharmonicity	Flat Side	Enharmonicity	Sharp Side	Enharmonicity
C	=	D	C		D	E	C		D	E
G	=	A	G		A	B	G		A	B
D	=	E	D		E		D		E
A	=	B	A		B		A		B
E	=	F♭	E		F♭		E		F♯
B	=	C♭	B		C♭		B		C♯
F♯	=	G♭	F♯		G♭		F♭		G♯
C♯	=	D♭	C♯		D♭		C♭		D♯
G♯	=	A♭	G♯		A♭		G♭		A♯
D♯	=	E♭	D♯		E♭		D♭		E♯
A♯	=	B♭	A♯		B♭		A♭		B♯
E♯	=	F	E♯	D	F		E♭	D	F
B♯	=	C	B♯	A	C		B♭	A	C

Melodic notation preserves the meaning of sharp and flat, but harmonic structures and interval arithmetic no longer work.

Interval size

Interval name / comments	Size (steps)	Size (cents)	MIDI
Octave	23	1200
	21	1095.65	Play^ⓘ
Major sixth (3 cents sharp of 5/3)	17	886.96	Play^ⓘ
"Blown fifth" interval (24 cents flat of a 3/2 perfect fifth)	13	678.26	Play^ⓘ
	11	573.91	Play^ⓘ
Fourth (octave inversion of "blown fifth")	10	521.74	Play^ⓘ
	09	469.57	Play^ⓘ
	08	417.39	Play^ⓘ
Major third (21 cents flat of 5/4)	07	365.22	Play^ⓘ
Minor third (3 cents flat of 6/5)	06	313.04	Play^ⓘ
	05	260.87	Play^ⓘ
Large step appearing between B-C or E-F	04	208.70	Play^ⓘ
"Whole step" between A-B or C-D (actually smaller than the step from B-C)	03	156.52	Play^ⓘ
	02	104.35	Play^ⓘ
Single step - this is the interval by which ♯ and ♭ modify pitches	01	052.17	Play^ⓘ

Scale diagram

Step (cents)		52		52		52		52		52		52		52		52		52		52		52		52		52		52		52		52		52		52		52		52		52		52		52
Melodic Notation note name	A		A♯		B♭		B		B♯		B C		C♭		C		C♯		D♭		D		D♯		E♭		E		E♯		E F		F♭		F		F♯		G♭		G		G♯		A♭		A
Harmonic Notation note name	A		A♭		B♯		B		B♭		B C		C♯		C		C♭		D♯		D		D♭		E♯		E		E♭		E F		F♯		F		F♭		G♯		G		G♭		A♯		A
Interval (cents)	0		52		104		157		209		261		313		365		417		470		522		574		626		678		730		783		835		887		939		991		1043		1096		1148		1200

Modes

References

Monzo, Joe (2005). "Equal-Temperament". Tonalsoft Encyclopedia of Microtonal Music Theory. Joe Monzo. Retrieved 20 February 2019.
Sethares, William (1998). Tuning, Timbre, Spectrum, Scale. Springer. p. 211. ISBN 9781852337971. Retrieved 20 February 2019.

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[1] Monzo, Joe (2005). "Equal-Temperament". Tonalsoft Encyclopedia of Microtonal Music Theory. Joe Monzo. Retrieved 20 February 2019.

[2] Sethares, William (1998). Tuning, Timbre, Spectrum, Scale. Springer. p. 211. ISBN 9781852337971. Retrieved 20 February 2019.