23 equal temperament


In music, 23 equal temperament, called 23-TET, 23-EDO, or 23-ET, is the tempered scale derived by dividing the octave into 23 equal steps. Each step represents a frequency ratio of, or 52.174 cents. This system is the largest EDO that has an error of at least 20 cents for the 3rd, 5th, 7th, and 11th 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, as the result of "a cycle of 'blown' fifths" of about 678 cents that may have resulted from overblowing a bamboo pipe. Today, tens of pieces 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, 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 harmonic notation, and melodic notation.
Melodic notation preserves the meaning of sharp and flat, but harmonic structures and interval arithmetic learned from mostly become invalid.

Interval size

Interval name / commentsSize Size MIDI
Octave231200
211095.65
Major sixth 17886.96
"Blown fifth" interval 13678.26
11573.91
Fourth 10521.74
9469.57
8417.39
Major third 7365.22
Minor third 6313.04
5260.87
Large step appearing between B-C or E-F4208.70
"Whole step" between A-B or C-D 3156.52
2104.35
Single step - this is the interval by which and modify pitches152.17