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
interval name size (steps) size (cents) midi just ratio just (cents) midi error
octave 23 1200 2:1 1200 0
major seventh 21 1095.65 Play  15:80 1088.27 Play  +07.38
major sixth 17 886.96 Play  5:3 884.36 Play  +02.60
augmented fourth 11 573.91 Play  25:18 568.72 Play  +05.19
doubly augmented third 10 521.74 Play  3125:2304 527.66 −05.92
augmented third 09 469.57 Play  125:960 456.99 Play  +12.58
diminished fourth 08 417.39 Play  32:25 427.37 Play  −09.98
doubly diminished fourth 07 365.22 Play  768:625 356.70 +08.52
minor third 06 313.04 Play  6:5 315.64 Play  −02.60
augmented second 05 260.87 Play  125:108 253.08 Play  +07.79
whole tone, major tone 04 208.70 Play  9:8 203.91 Play  +04.79
diminished second + septimal diatonic semitone 03 156.52 Play  192:175 160.50 −03.98
septimal diatonic semitone 02 104.35 Play  15:14 119.44 Play  −15.09
diatonic semitone, just 02 104.35 16:15 111.73 Play  −07.38
diminished second 01 052.17 Play  128:125 041.06 Play  +11.11

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

See also

References
  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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