In music,23 equal temperament, called 23-TET, 23-EDO ("Equal Division of the Octave"), or 23-ET, is thetempered scale derived by dividing the octave into 23 equal steps (equal frequency ratios). Each step represents a frequency ratio of23√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. The lack of approximation to simple intervals makes the scale notable among thoseseeking to break free from conventional harmony rules.
23-EDO was advocated by ethnomusicologistErich 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 fromoverblowing a bamboo pipe. Today,[when?] tens of pieces[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, calledmelodic notation andharmonic notation.
Harmonic notation preserves harmonic structures and interval arithmetic, but sharp and flat have reversed meanings. Because it preserves harmonic structures,12EDO music can be reinterpreted as 23 EDO harmonic notation, so it is also calledconversion notation.
An example of these harmonic structures is the circle of fifths below, shown in12EDO, harmonic notation, and melodic notation.
| Circle of fifths in 12EDO | Circle of fifths in 23EDO harmonic notation | Circle of fifths in 23EDO melodic notation | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sharp side | Enhar- monic? | Flat side | Sharp side | Enhar- monic | Flat side | Enhar- monic | Flat side | Enhar- monic | Sharp side | Enhar- monic | ||
| 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 learned from12EDO mostly become invalid.
| Interval name / comments | Size (steps) | Size (cents) | MIDI |
|---|---|---|---|
| Octave | 23 | 1200 | |
| 21 | 1095.65 | Playⓘ | |
| Major sixth (3 cents sharp of 5/3) | 17 | 886.96 | Playⓘ |
| "Blown fifth" interval (24 cents flat of a 3/2perfect fifth) | 13 | 678.26 | Playⓘ |
| 11 | 573.91 | Playⓘ | |
| Fourth (octave inversion of "blown fifth") | 10 | 521.74 | Playⓘ |
| 9 | 469.57 | Playⓘ | |
| 8 | 417.39 | Playⓘ | |
| Major third (21 cents flat of 5/4) | 7 | 365.22 | Playⓘ |
| Minor third (3 cents flat of 6/5) | 6 | 313.04 | Playⓘ |
| 5 | 260.87 | Playⓘ | |
| Large step appearing between B-C or E-F | 4 | 208.70 | Playⓘ |
| "Whole step" between A-B or C-D (actually smaller than the step from B-C) | 3 | 156.52 | Playⓘ |
| 2 | 104.35 | Playⓘ | |
| Single step - this is the interval by which♯ and♭ modify pitches | 1 | 52.17 | Playⓘ |
| 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 | ||||||||||||||||||||||||
 | This sectionis missing information about 23EDO modes. Please expand the section to include this information. Further details may exist on thetalk page.(February 2019) |
