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 232, 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, dozens of songs[which?] have been composed in this system.


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 = Ddouble flat C Ddouble flat Edouble sharp C Ddouble sharp Edouble flat
G = Adouble flat G Adouble flat Bdouble sharp G Adouble sharp Bdouble flat
D = Edouble flat D Edouble flat D Edouble sharp
A = Bdouble flat A Bdouble flat A Bdouble sharp
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 Ddouble flat F E Ddouble sharp F
B = C B Adouble flat C B Adouble sharp 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 audio speaker iconPlay 
Major sixth (3 cents sharp of 5/3) 17 886.96 audio speaker iconPlay 
"Blown fifth" interval (24 cents flat of a 3/2 perfect fifth) 13 678.26 audio speaker iconPlay 
11 573.91 audio speaker iconPlay 
Fourth (octave inversion of "blown fifth") 10 521.74 audio speaker iconPlay 
09 469.57 audio speaker iconPlay 
08 417.39 audio speaker iconPlay 
Major third (21 cents flat of 5/4) 07 365.22 audio speaker iconPlay 
Minor third (3 cents flat of 6/5) 06 313.04 audio speaker iconPlay 
05 260.87 audio speaker iconPlay 
Large step appearing between B-C or E-F 04 208.70 audio speaker iconPlay 
"Whole step" between A-B or C-D (actually smaller than the step from B-C) 03 156.52 audio speaker iconPlay 
02 104.35 audio speaker iconPlay 
Single step - this is the interval by which and modify pitches 01 052.17 audio speaker iconPlay 

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 Bdouble sharp
Cdouble flat
C C C D D D E E E Edouble sharp
Fdouble flat
Harmonic Notation note name A A B B B Bdouble flat
Cdouble sharp
C C C D D D E E E Edouble flat
Fdouble sharp
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


See also


  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.