# 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.

## 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, dozens 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's 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

## 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

## See also

## 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.