# 23edo

(Redirected from 23-edo)
 ← 22edo 23edo 24edo →
Prime factorization 23 (prime)
Step size 52.1739¢
Fifth 13\23 (678.261¢)
Semitones (A1:m2) -1:4 (-52.17¢ : 208.7¢)
Dual sharp fifth 14\23 (730.435¢)
Dual flat fifth 13\23 (678.261¢)
Dual major 2nd 4\23 (208.696¢)
(semiconvergent)
Consistency limit 5
Distinct consistency limit 5
English Wikipedia has an article on:

23 equal divisions of the octave (abbreviated 23edo or 23ed2), also called 23-tone equal temperament (23tet) or 23 equal temperament (23et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 23 equal parts of about 52.2 ¢ each. Each step represents a frequency ratio of 21/23, or the 23rd root of 2.

## Theory

Approximation of odd harmonics in 23edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -23.7 -21.1 +22.5 +4.8 +22.6 -5.7 +7.4 -0.6 +15.5 -1.2 -2.2
Relative (%) -45.4 -40.4 +43.1 +9.2 +43.3 -11.0 +14.2 -1.2 +29.8 -2.3 -4.2
Steps
(reduced)
36
(13)
53
(7)
65
(19)
73
(4)
80
(11)
85
(16)
90
(21)
94
(2)
98
(6)
101
(9)
104
(12)

23edo has good approximations for 5/3, 11/7, 13 and 17, among many others, allowing it to represent the 2.5/3.11/7.13.17 just intonation subgroup. If to this subgroup is added the commas of no-19's 23-limit 46edo, the larger no-19's 23-limit 2*23 subgroup 2.9.15.21.35.33.55.13.17.23 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does no-19's 23-limit 46edo, and may be regarded as a basis for analyzing the harmony of 23edo so far, as approximations to just intervals goes. If one dares to take advantage of this harmony by using 23edo as a period, you get icositritonic, a 23rd-octave temperament, so that the harmony of 23edo is adequately explained by what harmonies you can achieve using only periods and zero generators.

23edo is the 9th prime edo, following 19edo and coming before 29edo.

23edo 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.

23edo 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, 11/5, 11/7) and combinations of them (15/8, 21/16, 33/32, 35/32, 55/32, 77/64) very well. The lowest harmonics well-approximated by 23edo are 9, 13, 15, 17, 21, 23, 31, 33 and 35. See Harmony of 23edo for more details. Also note that some approximations can be improved by octave stretching.

As with 9edo, 16edo, and 25edo, one way to treat 23edo is as a tuning of the 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 antidiatonic scale of 3 3 4 3 3 3 4 (in steps of 23edo), which extends to 9 notes superdiatonic scale (3 3 3 1 3 3 3 3 1). One can notate 23edo using the Armodue system, but just like notating 17edo with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23edo, the "Armodue 6th" is sharper than it is in 16edo, just like the diatonic 5th in 17edo is sharper than in 12edo. In other words, 2b is lower in pitch than 1#, just like how in 17edo Eb is lower than D#.

However, one can also map 3/2 to 14 degrees of 23edo 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 13edo and 18edo 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 13edo "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23edo 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 23edo 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.

## Selected just intervals

The following tables show how 15-odd-limit intervals are represented in 23edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 23edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/7, 14/11 0.117 0.2
5/3, 6/5 2.598 5.0
9/8, 16/9 4.786 9.2
13/8, 16/13 5.745 11.0
11/6, 12/11 5.885 11.3
7/6, 12/7 6.001 11.5
15/8, 16/15 7.383 14.2
11/10, 20/11 8.482 16.3
7/5, 10/7 8.599 16.5
13/9, 18/13 10.531 20.2
15/13, 26/15 13.129 25.2
15/14, 28/15 15.095 28.9
15/11, 22/15 15.212 29.2
13/10, 20/13 15.351 29.4
9/7, 14/9 17.693 33.9
11/9, 18/11 17.809 34.1
13/12, 24/13 17.949 34.4
5/4, 8/5 21.096 40.4
7/4, 8/7 22.478 43.1
11/8, 16/11 22.595 43.3
3/2, 4/3 23.694 45.4
13/11, 22/13 23.834 45.7
13/7, 14/13 23.950 45.9
9/5, 10/9 25.882 49.6
15-odd-limit intervals in 23edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/7, 14/11 0.117 0.2
5/3, 6/5 2.598 5.0
13/8, 16/13 5.745 11.0
13/10, 20/13 15.351 29.4
13/12, 24/13 17.949 34.4
5/4, 8/5 21.096 40.4
7/4, 8/7 22.478 43.1
11/8, 16/11 22.595 43.3
3/2, 4/3 23.694 45.4
9/5, 10/9 26.292 50.4
13/7, 14/13 28.223 54.1
13/11, 22/13 28.340 54.3
15/13, 26/15 39.045 74.8
13/9, 18/13 41.643 79.8
7/5, 10/7 43.575 83.5
11/10, 20/11 43.691 83.7
15/8, 16/15 44.790 85.8
7/6, 12/7 46.173 88.5
11/6, 12/11 46.289 88.7
9/8, 16/9 47.388 90.8
15/14, 28/15 67.269 128.9
15/11, 22/15 67.386 129.2
9/7, 14/9 69.867 133.9
11/9, 18/11 69.983 134.1

## Notation

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 is not E. Chord names are different because C - E - G is not 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 [1]
Major wider
than minor
Major narrower
than minor
Armodue
Notation
Notes
0 0.000 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
4 208.696 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
7 365.217 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# Practically just 11/7
16 834.783 13/8, 34/21, 21/13 m6 B M6 B 7
17 886.957 5/3 M6 B# m6 Bb 8b Much better 5/3 than 12-edo
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# Less than 1 cent off 32/17
22 1147.826 33/17, 64/33 d8 Db A8 D# 9
23 1200.000 2/1 P8 D P8 D 1
1. Based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible.

## Approximation to irrational intervals

### Acoustic π and ϕ

23edo has a very close approximation of acoustic π/2 on 15\23 and a very close approximation of acoustic phi on the step just above (16\23).

Direct approximation
Interval Error (abs, ¢)
π 0.813
π/ϕ 0.879
ϕ 1.692

Not until 72 do we find a better edo in terms of absolute error, and not until 749 do we find one in terms of relative error.

## Regular temperament properties

### Uniform maps

13-limit uniform maps between 22.5 and 23.5
Min. size Max. size Wart notation Map
22.5000 22.5649 23ccdddeeeffff 23 36 52 63 78 83]
22.5649 22.6105 23ccdddeeeff 23 36 52 63 78 84]
22.6105 22.6192 23dddeeeff 23 36 53 63 78 84]
22.6192 22.6916 23deeeff 23 36 53 64 78 84]
22.6916 22.8351 23deff 23 36 53 64 79 84]
22.8351 22.9754 23de 23 36 53 64 79 85]
22.9754 22.9807 23e 23 36 53 65 79 85]
22.9807 23.0289 23 23 36 53 65 80 85]
23.0289 23.0412 23b 23 37 53 65 80 85]
23.0412 23.1054 23bc 23 37 54 65 80 85]
23.1054 23.2697 23bcf 23 37 54 65 80 86]
23.2697 23.3316 23bceef 23 37 54 65 81 86]
23.3316 23.3756 23bcddeef 23 37 54 66 81 86]
23.3756 23.4719 23bcddeefff 23 37 54 66 81 87]
23.4719 23.5000 23bcccddeefff 23 37 55 66 81 87]

### Commas

23edo 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).

Prime
Limit
Ratio Monzo Cents Color name Name(s)
5 135/128 [-7 3 1 92.18 Layobi Major Chroma, Major Limma, Pelogic Comma
5 15625/15552 [-6 -5 6 8.11 Tribiyo Kleisma, Semicomma Majeur
7 36/35 [2 2 -1 -1 48.77 Rugu Septimal Quarter Tone
7 525/512 [-9 1 2 1 43.41 Lazoyoyo Avicennma, Avicenna's Enharmonic Diesis
7 4000/3969 [5 -4 3 -2 13.47 Rurutriyo Octagar
7 6144/6125 [11 1 -3 -2 5.36 Sarurutrigu Porwell
11 100/99 [2 -2 2 0 -1 17.40 Luyoyo Ptolemisma
11 441/440 [-3 2 -1 2 -1 3.93 Luzozogu Werckisma

## Scales

Important mosses include:

• Mavila 2L5s 4334333 (13\23, 1\1)
• Mavila 7L2s 133313333 (13\23, 1\1)
• Sephiroth 3L4s 2525252 (7\23, 1\1)
• Semiquartal 5L4s 332323232 (5\23, 1\1)

The chart below shows some of the mos modes of mavila available in 23edo, mainly Pentatonic (5-note), antidiatonic (7-note), 9- and 16-note mosses. Here the outer ring represents individual step of 23edo itself, while the rings moving inward represent 16, 9, 7 and 5 note mosses:

### 23-tone mos scales

MOS scale Name
10 10 3
9 9 5
8 8 7
7 7 7 2
6 6 6 5
5 4 5 5 4 3L 2s (oneiro-pentatonic)
5 4 5 4 5
7 1 7 7 1
7 1 7 1 7
5 5 5 5 3 4L 1s (bug pentatonic)
4 4 4 4 4 3 5L 1s (machinoid)
5 1 5 1 5 1 5 4L 3s (smitonic)
3 3 3 5 3 3 3 1L 6s (antiarcheotonic)
4 3 3 3 3 3 4
3 3 4 3 3 3 4 2L 5s (mavila, anti-diatonic)
4 3 3 3 3 4 3
2 5 2 5 2 5 2 3L 4s (mosh)
4 1 4 4 1 4 4 1 5L 3s (oneirotonic)
3 3 3 3 3 3 3 2 7L 1s (porcupoid)
3 3 3 1 3 3 3 3 1 7L 2s (mavila superdiatonic)
3 2 3 2 3 2 3 2 3 5L 4s (bug semiquartal)
3 2 2 3 2 2 3 2 2 2 3L 7s (sephiroid)
4 1 1 4 1 1 4 1 1 4 1 4L 7s (kleistonic)
3 1 3 1 3 1 3 1 3 1 3 Palestine 11
3 1 1 3 1 3 1 1 3 1 3 1 1 5L 8s (ateamtonic)
2 2 2 2 1 2 2 2 1 2 2 2 1 10L 3s (luachoid)
2 2 1 2 2 1 2 2 1 2 2 1 2 1 9L 5s (Brittle Titanium)
2 1 2 2 1 2 2 1 2 2 1 2 2 1 Palestine 14
1 1 1 4 1 1 1 1 4 1 1 1 1 4 3L 11s
3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 4L 11s (mynoid)
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 8L 7s
2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 1 7L 9s (mavila chromatic)
2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 Palestine 17
2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 Pathological 5L 13s (ateamtonic[18])
2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 Pathological 4L 15s (mynoid[19]]

### Kosmorsky's Sephiroth modes

Kosmorsky has argued 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

### Miscellaneous

5 5 1 2 5 5 - Antipental blues (approximated from Dwarf17marv)

7 2 4 6 4 - Arcade (approximated from 32afdo)

6 4 1 2 2 6 2 - Blackened skies (approximated from Compton in 72edo)

5 5 3 7 3 - Geode (approximated from 6afdo)

5 4 2 2 4 2 4 - Lost phantom (approximated from Mavila in 30edo)

6 4 2 1 5 1 4 - Lost spirit (approximated from Meantone in 31edo)

5 2 6 6 4 - Mechanical (approximated from 31afdo)

5 4 4 2 8 - Mushroom (approximated from 30afdo)

6 4 3 7 3 - Nightdrive (approximated from Mavila in 30edo)

6 4 1 2 6 4 - Pelagic (approximated from Mavila in 30edo)

2 3 8 2 8 - Approximation of Pelog lima

4 3 6 6 4 - Springwater (approximated from 8afdo)

2 5 2 4 6 4 - Starship (approximated from 68ifdo)

2 4 6 1 10 - Tightrope (this is the original/default tuning)

6 7 4 2 4 - Underpass (approximated from 10afdo)

2 5 6 6 4 - Volcanic (approximated from 16afdo)

## Instruments

23-EDD 8-string Guitar by Ron Sword.

23-EDD Bass by Osmiorisbendi.

23-EDD Baritar by Osmiorisbendi.

23-EDD 5-string Acoustic Guitar by Tútim Dennsuul Wafiil (RIP).

Illustrative 23-EDD Keyboard

Chris Vaisvil made a do it yourself 23 edo electric guitar out of less than \$50 of material. Here he is playing it.

Here is a still shot of the completed instrument.

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.