23 equal temperament
{{short description|Tuning system with no consonant intervals}}
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 {{radic|2|23}}, 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,{{cite web |last1=Monzo |first1=Joe |title=Equal-Temperament |url=http://tonalsoft.com/enc/e/equal-temperament.aspx#edo-table |website=Tonalsoft Encyclopedia of Microtonal Music Theory |publisher=Joe Monzo |accessdate=20 February 2019 |date=2005}} as the result of "a cycle of 'blown' (compressed) fifths"{{cite book |last1=Sethares |first1=William |title=Tuning, Timbre, Spectrum, Scale |date=1998 |publisher=Springer |isbn=9781852337971 |page=211 |url=https://books.google.com/books?id=KChoKKhjOb0C&pg=PA211 |accessdate=20 February 2019}} of about 678 cents that may have resulted from overblowing a bamboo pipe. Today,{{when|date=April 2024}} tens of pieces{{Which|date=August 2020}} 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, {{nobr|12 {{sc|EDO}}}} 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 {{nobr|12 {{sc|EDO}},}} harmonic notation, and melodic notation.
| class="wikitable" style="text-align:center;vertical-align:center;" |
| colspan= 3| Circle of fifths in {{nobr|12 {{sc|EDO}}}} |rowspan=15 style="background:white;"| !colspan= 4| Circle of fifths in |rowspan=15 style="background:white;"| !colspan= 4| Circle of fifths in |
|---|
| Sharp side
! {{small|Enhar- ! Flat side ! Sharp side ! {{small|Enhar- ! Flat side ! {{small|Enhar- ! Flat side ! {{small|Enhar- ! Sharp side ! {{small|Enhar- |
| C
| = | D{{sup|{{music|bb}}}} | C | | D{{sup|{{music|bb}}}} | E{{sup|{{music|x}}}} | C | | D{{sup|{{music|x}}}} | E{{sup|{{music|bb}}}} |
| G
| = | A{{sup|{{music|bb}}}} | G | | A{{sup|{{music|bb}}}} | B{{sup|{{music|x}}}} | G | | A{{sup|{{music|x}}}} | B{{sup|{{music|bb}}}} |
| D
| = | E{{sup|{{music|bb}}}} | D | | E{{sup|{{music|bb}}}} | | D | | E{{sup|{{music|x}}}} | |
| A
| = | B{{sup|{{music|bb}}}} | A | | B{{sup|{{music|bb}}}} | | A | | B{{sup|{{music|x}}}} | |
| E
| = | F{{sup|{{music|b}}}} | E | | F{{sup|{{music|b}}}} | | E | | F{{sup|{{music|#}}}} | |
| B
| = | C{{sup|{{music|b}}}} | B | | C{{sup|{{music|b}}}} | | B | | C{{sup|{{music|#}}}} | |
| F{{sup|{{music|#}}}}
| = | G{{sup|{{music|b}}}} | F{{sup|{{music|#}}}} | | G{{sup|{{music|b}}}} | | F{{sup|{{music|b}}}} | | G{{sup|{{music|#}}}} | |
| C{{sup|{{music|#}}}}
| = | D{{sup|{{music|b}}}} | C{{sup|{{music|#}}}} | | D{{sup|{{music|b}}}} | | C{{sup|{{music|b}}}} | | D{{sup|{{music|#}}}} | |
| G{{sup|{{music|#}}}}
| = | A{{sup|{{music|b}}}} | G{{sup|{{music|#}}}} | | A{{sup|{{music|b}}}} | | G{{sup|{{music|b}}}} | | A{{sup|{{music|#}}}} | |
| D{{sup|{{music|#}}}}
| = | E{{sup|{{music|b}}}} | D{{sup|{{music|#}}}} | | E{{sup|{{music|b}}}} | | D{{sup|{{music|b}}}} | | E{{sup|{{music|#}}}} | |
| A{{sup|{{music|#}}}}
| = | B{{sup|{{music|b}}}} | A{{sup|{{music|#}}}} | | B{{sup|{{music|b}}}} | | A{{sup|{{music|b}}}} | | B{{sup|{{music|#}}}} | |
| E{{sup|{{music|#}}}}
| = | F | E{{sup|{{music|#}}}} | D{{sup|{{music|bb}}}} | F | | E{{sup|{{music|b}}}} | D{{sup|{{music|x}}}} | F | |
| B{{sup|{{music|#}}}}
| = | C | B{{sup|{{music|#}}}} | A{{sup|{{music|bb}}}} | C | | B{{sup|{{music|b}}}} | A{{sup|{{music|x}}}} | C | |
Melodic notation preserves the meaning of sharp and flat, but harmonic structures and interval arithmetic learned from {{nobr|12 {{sc|edo}}}} mostly become invalid.
Interval size
| class="wikitable sortable" style="text-align: center"
!Interval name / comments !Size (steps) !Size (cents) !MIDI |
| Octave
|23 |1200 | |
| |21
|1095.65 |{{audio|21 steps in 23-et on C.mid|Play}} |
| Major sixth (3 cents sharp of 5/3)
|17 |886.96 |{{audio|17 steps in 23-et on C.mid|Play}} |
| "Blown fifth" interval (24 cents flat of a 3/2 perfect fifth)
|13 |678.26 |{{audio|13 steps in 23-et on C.mid|Play}} |
| |11
|573.91 |{{audio|11 steps in 23-et on C.mid|Play}} |
| Fourth (octave inversion of "blown fifth")
|10 |521.74 |{{audio|10 steps in 23-et on C.mid|Play}} |
| |{{0}}9
|469.57 |{{audio|9 steps in 23-et on C.mid|Play}} |
| |{{0}}8
|417.39 |{{audio|8 steps in 23-et on C.mid|Play}} |
| Major third (21 cents flat of 5/4)
|{{0}}7 |365.22 |{{audio|7 steps in 23-et on C.mid|Play}} |
| Minor third (3 cents flat of 6/5)
|{{0}}6 |313.04 |{{audio|6 steps in 23-et on C.mid|Play}} |
| |{{0}}5
|260.87 |{{audio|5 steps in 23-et on C.mid|Play}} |
| Large step appearing between B-C or E-F
|{{0}}4 |208.70 |{{audio|4 steps in 23-et on C.mid|Play}} |
| "Whole step" between A-B or C-D (actually smaller than the step from B-C)
|{{0}}3 |156.52 |{{audio|3 steps in 23-et on C.mid|Play}} |
| |{{0}}2
|104.35 |{{audio|2 steps in 23-et on C.mid|Play}} |
| Single step - this is the interval by which {{music|sharp}} and {{music|flat}} modify pitches
|{{0}}1 |{{0}}52.17 |{{audio|1 step in 23-et on C.mid|Play}} |
Scale diagram
| class="wikitable" style="text-align:center;"
| bgcolor="#ffeeee" |Step (cents) | | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | colspan="2" bgcolor="#ffeeee" |52 | |
| bgcolor="#fffbee" |Melodic Notation note name
| colspan="2" bgcolor="#fffbee" |A | colspan="2" bgcolor="#fffbee" |A{{music|sharp}} | colspan="2" bgcolor="#fffbee" |B{{music|flat}} | colspan="2" bgcolor="#fffbee" |B | colspan="2" bgcolor="#fffbee" |B{{music|sharp}} | colspan="2" bgcolor="#fffbee" |B{{music|doublesharp}} | colspan="2" bgcolor="#fffbee" |C{{music|flat}} | colspan="2" bgcolor="#fffbee" |C | colspan="2" bgcolor="#fffbee" |C{{music|sharp}} | colspan="2" bgcolor="#fffbee" |D{{music|flat}} | colspan="2" bgcolor="#fffbee" |D | colspan="2" bgcolor="#fffbee" |D{{music|sharp}} | colspan="2" bgcolor="#fffbee" |E{{music|flat}} | colspan="2" bgcolor="#fffbee" |E | colspan="2" bgcolor="#fffbee" |E{{music|sharp}} | colspan="2" bgcolor="#fffbee" |E{{music|doublesharp}} | colspan="2" bgcolor="#fffbee" |F{{music|flat}} | colspan="2" bgcolor="#fffbee" |F | colspan="2" bgcolor="#fffbee" |F{{music|sharp}} | colspan="2" bgcolor="#fffbee" |G{{music|flat}} | colspan="2" bgcolor="#fffbee" |G | colspan="2" bgcolor="#fffbee" |G{{music|sharp}} | colspan="2" bgcolor="#fffbee" |A{{music|flat}} | colspan="2" bgcolor="#fffbee" |A |
| bgcolor="#fffbee" |Harmonic Notation note name
| colspan="2" bgcolor="#fffbee" |A | colspan="2" bgcolor="#fffbee" |A{{music|flat}} | colspan="2" bgcolor="#fffbee" |B{{music|sharp}} | colspan="2" bgcolor="#fffbee" |B | colspan="2" bgcolor="#fffbee" |B{{music|flat}} | colspan="2" bgcolor="#fffbee" |B{{music|doubleflat}} | colspan="2" bgcolor="#fffbee" |C{{music|sharp}} | colspan="2" bgcolor="#fffbee" |C | colspan="2" bgcolor="#fffbee" |C{{music|flat}} | colspan="2" bgcolor="#fffbee" |D{{music|sharp}} | colspan="2" bgcolor="#fffbee" |D | colspan="2" bgcolor="#fffbee" |D{{music|flat}} | colspan="2" bgcolor="#fffbee" |E{{music|sharp}} | colspan="2" bgcolor="#fffbee" |E | colspan="2" bgcolor="#fffbee" |E{{music|flat}} | colspan="2" bgcolor="#fffbee" |E{{music|doubleflat}} | colspan="2" bgcolor="#fffbee" |F{{music|sharp}} | colspan="2" bgcolor="#fffbee" |F | colspan="2" bgcolor="#fffbee" |F{{music|flat}} | colspan="2" bgcolor="#fffbee" |G{{music|sharp}} | colspan="2" bgcolor="#fffbee" |G | colspan="2" bgcolor="#fffbee" |G{{music|flat}} | colspan="2" bgcolor="#fffbee" |A{{music|sharp}} | colspan="2" bgcolor="#fffbee" |A |
| bgcolor="#eeeeff" |Interval (cents)
| colspan="2" bgcolor="#eeeeff" |0 | colspan="2" bgcolor="#eeeeff" |52 | colspan="2" bgcolor="#eeeeff" |104 | colspan="2" bgcolor="#eeeeff" |157 | colspan="2" bgcolor="#eeeeff" |209 | colspan="2" bgcolor="#eeeeff" |261 | colspan="2" bgcolor="#eeeeff" |313 | colspan="2" bgcolor="#eeeeff" |365 | colspan="2" bgcolor="#eeeeff" |417 | colspan="2" bgcolor="#eeeeff" |470 | colspan="2" bgcolor="#eeeeff" |522 | colspan="2" bgcolor="#eeeeff" |574 | colspan="2" bgcolor="#eeeeff" |626 | colspan="2" bgcolor="#eeeeff" |678 | colspan="2" bgcolor="#eeeeff" |730 | colspan="2" bgcolor="#eeeeff" |783 | colspan="2" bgcolor="#eeeeff" |835 | colspan="2" bgcolor="#eeeeff" |887 | colspan="2" bgcolor="#eeeeff" |939 | colspan="2" bgcolor="#eeeeff" |991 | colspan="2" bgcolor="#eeeeff" |1043 | colspan="2" bgcolor="#eeeeff" |1096 | colspan="2" bgcolor="#eeeeff" |1148 | colspan="2" bgcolor="#eeeeff" |1200 |
=Modes=
{{Missing information|section|23EDO modes|date=February 2019}}
See also
References
{{Reflist}}
{{Microtonal music}}
{{Musical tuning}}