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, which makes it unusual in microtonal music. Perhaps due to this, dozens of songs have been composed in this system.

History

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.

Notation

File:Comparison of 23TET Notations.png

Comparison of 23TET Notations

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 (such as the Circle of Fifths shown in the picture,) 12TET music can be reinterpreted as 23TET Harmonic Notation, so it's also called Conversion Notation.

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

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

Interval size

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.

[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.

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