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23 tone equal temperament | Music | Commas | 23 tone Equal Modes: | Books | Instruments



23 tone equal temperament


23-tET, or 23-EDO, is a tempered musical system which divides the octave into 23 equal parts of approximately 52.173913 cents, which is also called with the neologism Icositriphony (Icositrifonía). It has good approximations for 5/3, 11/7, 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 just intonation subgroup. If to this subgroup is added the commas of 17-limit 46edo, the larger 17-limit 2*23 subgroup 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th prime edo, following 19edo and coming before 29edo.

Intervals


Ciclo Icositrifonía.png


23edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 23edo "on the fly".

Armodue notation is a nonatonic notation that uses the numbers 1-9 as note names.

Degree*
Cents
Approximate
Ratios **
Major wider
than minor
Major narrower
than minor
Armodue
Notation
Notes
0
0
1/1
P1
D
P1
D
1

1
52.174
33/32, 34/33
A1
D#
d1
Db
2b

2
104.348
17/16, 16/15, 18/17
d2
Eb
A2
E#
1#
Less than 1 cent off 17/16
3
156.522
11/10, 12/11, 35/32
m2
E
M2
E
2


208.696
9/8, 44/39
M2
E#
m2
Eb
3b

5
260.870
7/6, 15/13, 29/25
A2, d3
Ex, Fbb
d2, A3
Ebb, Fx
2#

6
313.043
6/5
m3
Fb
M3
F#
3
Much better 6/5 than 12-edo

365.217
16/13, 21/17, 26/21
M3
F
m3
F
4b

8
417.391
14/11, 33/26
A3
F#
d3
Fb
3#
Practically just 14/11
9
469.565
21/16, 17/13
d4
Gb
A4
G#
4

10·
521.739
23/17, 88/65, 256/189
P4
G
P4
G
5

11
573.913
7/5, 32/23, 46/33
A4
G#
d4
Gb
6b

12
626.087
10/7, 23/16, 33/23
d5
Ab
A5
A#
5#

13·
678.261
34/23, 65/44, 189/128
P5
A
P5
A
6
Great Hornbostel generator
14
730.435
32/21, 26/17
A5
A#
d5
Ab
7b

15
782.609
11/7, 52/33
d6
Bb
A6
B#
6#

16·
834.783
13/8, 34/21, 21/13
m6
B
M6
B
7

17
886.957
5/3
M6
B#
m6
Bb
8b

18
939.130
12/7, 26/15, 50/29
A6, d7
Bx, Cbb
d6, A7
Bbb, Cx
7#

19·
991.304
16/9, 39/22
m7
Cb
M7
C#
8

20
1043.478
11/6, 20/11, 64/35
M7
C
m7
C
9b

21
1095.652
15/8, 17/9, 32/17
A7
C#
d7
Cb
8#

22
1147.826
33/17, 64/33
d8
Db
A8
D#
9

23··
1200
2/1
P8
D
P8
D
1

  • the dots indicate which frets on a 23-edo guitar would have dots.
    • based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible.

The chart below shows some of the Moment of Symmetry (MOS) modes of Mavila available in 23edo, mainly Pentatonic(5-note), anti-diatonic(7-note), 9- and 16-note MOSs. Here the outer ring represents individual step of 23edo itself, while the rings moving inward represent 16, 9, 7 and 5 note MOSs:
23edoMavilaMOS.jpg

23-EDO was proposed by ethnomusicologist Erich von Hornbostel as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.

23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, 11/3, 7/5, 11/7, and 11/5) very well. The lowest harmonics well-approximated by 23-EDO are 13, 17, 21, and 23. See here for more details. Also note that some approximations can be improved by octave stretching.

As with 9-EDO, 16-EDO, and 25-EDO, one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute 4/3's with 5/1 (related to the Armodue system). This means mapping '3/2' to 13 degrees of 23, and results in a 7 notes Anti-diatonic scale of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes Superdiatonic scale (3 3 3 1 3 3 3 3 1). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.

However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a 7-limit temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to 13-EDO and 18-EDO and produces MOS scales of 5 and 8 notes: 5 5 4 5 4 (the "anti-pentatonic") and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/Rapoport/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale.


Kosmorsky's Sephiroth modes

I would argue that the most significant modes of 23 edo are those of the 2 2 2 3 2 2 3 2 2 3 scale (3L 7s fair mosh); This is derived from extending the ~1/3 comma tempered 13th Harmonic, two of which add up to the 21st harmonic and three add up to the 17th harmonic almost perfectly. Interestingly, the chord 8:13:21:34 is a fragment of the fibonacci sequence.

Notated in ascending (standard) form. I have named these 10 modes according to the Sephiroth as follows:
2 2 2 3 2 2 3 2 2 3 - Mode Keter
2 2 3 2 2 3 2 2 3 2 - Chesed
2 3 2 2 3 2 2 3 2 2 - Netzach
3 2 2 3 2 2 3 2 2 2 - Malkuth
2 2 3 2 2 3 2 2 2 3 - Binah
2 3 2 2 3 2 2 2 3 2 - Tiferet
3 2 2 3 2 2 2 3 2 2 - Yesod
2 2 3 2 2 2 3 2 2 3 - Chokmah
2 3 2 2 2 3 2 2 3 2 - Gevurah
3 2 2 2 3 2 2 3 2 2 - Hod


Music

Chromatic canon, by Claudi Meneghin
The Family Supper by Warren Burt
Icositriphonic Heptatonic MOS by Igliashon Jones
His Wandering Kinship with Ashes by Igliashon Jones
Cosmic Chamber by X. J. Scott
Daisies on the Beach by X. J. Scott
Boogie Pieby Aaron Krister Johnson
23 Chilled by Chris Vaisvil
Desert Winds by Carlo Serafini (blog entry)
23 Laments by Carlo Serafini (blog entry)
Doomsday 23 by Carlo Serafini (blog entry)
Barber’s Adagio For Strings in 23ED2 by Carlo Serafini (blog entry)
Nubian Dance by Carlo Serafini (blog entry)
Allegro Moderato by Easley Blackwood
Pentaswing Notes by Andrew Heathwaite
Mindaugas Rex Lithuaniae by Chris Vaisvil (in Superpelog-9 tuning)
Indigorange by Tutim Dennsuul
Wignud by Tutim Dennsuul
Harid by Tutim Dennsuul
A Rest In The Desert play by Gary Mcandrew
Numenoctagon by Jacky Ligon (on spectropolrecords)
23 (album) by Jacky Ligon, Sevish & Tony Dubshot (soundcloud)
23EDO Klavier8 by Shunya Kiyokawa

Commas

23 EDO tempers out the following commas. (Note: This assumes the val < 23 36 53 65 80 85 |.) Also note the discussion above, where there are some commas mentioned that are not in the standard comma list (e.g., 28/27).
Comma
Monzo
Value (Cents)
Name 1
Name 2
Name 3
135/128
| -7 3 1 >
92.18
Major Chroma
Major Limma
Pelogic Comma
15625/15552
| -6 -5 6 >
8.11
Kleisma
Semicomma Majeur

36/35
| 2 2 -1 -1 >
48.77
Septimal Quarter Tone


525/512
| -9 1 2 1 >
43.41
Avicennma
Avicenna's Enharmonic Diesis

4000/3969
| 5 -4 3 -2 >
13.47
Octagar


6144/6125
| 11 1 -3 -2 >
5.36
Porwell


100/99
| 2 -2 2 0 -1 >
17.40
Ptolemisma


441/440
| -3 2 -1 2 -1 >
3.93
Werckisma



23 tone Equal Modes:


10 10 3

9 9 5

8 8 7

7 7 7 2

7 2 7 7

6 6 6 5

6 5 6 6

5 4 5 5 4
3L 2s (father)
5 4 5 4 5

7 1 7 7 1

7 1 7 1 7

5 5 5 5 3
4L 1s (bug)
5 3 5 5 5

4 4 4 4 4 3
5L 1s (Grumpy hexatonic)
4 3 4 4 4 4

5 1 5 1 5 1 5
4L 3s (mish)
3 3 3 5 3 3 3
1L 6s (Happy heptatonic)
4 3 3 3 3 3 4
2L 5s (mavila, anti-diatonic)
3 4 3 3 4 3 3

3 3 4 3 3 3 4

3 3 3 4 3 3 4

3 3 3 4 3 4 3

2 5 2 5 2 5 2
3L 4s (mosh)
4 1 4 4 1 4 4 1
5L 3s (unfair father)
3 3 3 3 3 3 3 2
7L 1s (Grumpy octatonic)
3 2 3 3 3 3 3 3

3 3 3 1 3 3 3 3 1
7L 2s (mavila superdiatonic)
3 3 1 3 3 3 1 3 3

3 2 3 2 3 2 3 2 3
5L 4s (unfair bug)
2 2 2 3 2 2 3 2 2 3
Mode Keter
2 2 3 2 2 3 2 2 3 2
Chesed
2 3 2 2 3 2 2 3 2 2
Netzach
3 2 2 3 2 2 3 2 2 2
Malkuth
2 2 3 2 2 3 2 2 2 3
Binah
2 3 2 2 3 2 2 2 3 2
Tiferet
3 2 2 3 2 2 2 3 2 2
Yesod
2 2 3 2 2 2 3 2 2 3
Chokmah
2 3 2 2 2 3 2 2 3 2
Gevurah
3 2 2 2 3 2 2 3 2 2
Hod
3 1 3 1 3 1 3 1 3 1 3

2 2 2 2 1 2 2 2 1 2 2 2 1
Mode Tishrei
2 2 2 1 2 2 2 1 2 2 2 1 2
Cheshvan
2 2 1 2 2 2 1 2 2 2 1 2 2
Kislev
2 1 2 2 2 1 2 2 2 1 2 2 2
Tevet
1 2 2 2 1 2 2 2 1 2 2 2 2
Shvat
2 2 2 1 2 2 2 1 2 2 2 2 1
Adar minor
2 2 1 2 2 2 1 2 2 2 2 1 2
Adar major
2 1 2 2 2 1 2 2 2 2 1 2 2
Nisan
1 2 2 2 1 2 2 2 2 1 2 2 2
Iyar
2 2 2 1 2 2 2 2 1 2 2 2 1
Sivan
2 2 1 2 2 2 2 1 2 2 2 1 2
Tammuz
2 1 2 2 2 2 1 2 2 2 1 2 2
Av
1 2 2 2 2 1 2 2 2 1 2 2 1
Elul
2 2 1 2 2 1 2 2 1 2 2 1 2 1

2 1 2 2 1 2 2 1 2 2 1 2 2 1

1 1 1 4 1 1 1 1 4 1 1 1 1 4

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1

2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1


Books

Libro_Icositrifónico.PNG

Instruments


Icositriphonic_Guitar.PNG
An Icositriphonic 8-string Guitar. 23-ED2 Guitar by Ron Sword.
Icositriphonic_Bass.JPG
An Icositriphonic P-Bass. 23-ED2 Bass by Tútim Dennsuul Wafiil.
Icositriphonic Gitar 1.png
An Icositriphonic 'Hendrix' TravelGuitar. 23-ED2 'Gitar' by Tútim Dennsuul Wafiil.
Icositritar 1.png
The 'Icositritar', 23-ED2 5-string Acoustic Guitar by Tútim Dennsuul Wafiil.

Teclado Icositrifónico.PNG
An illustrative image of a 23-ED2 Keyboard

Chris Vaisvil made a do it yourself 23 edo electric guitar out of less than $50 of material. Here he is playing it.
playing.jpg
Here is a still shot of the completed instrument.
complette.jpg
This movie is a series of still shots Chris took during the process of making a 23 edo guitar in a stick like form. At the end the guitar is played without effects etc. and the open string tuning is sounded - which starts with a normal E and then adjusted to the 9th / 7th fret unison, like a typical 12edo guitar fashion.