23edo

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 2^1/23, 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 2.9.15.21.33.13.17.23 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

Armodue Notation

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	Mavila comma, major chroma
5	15625/15552	[-6 -5 6⟩	8.11	Tribiyo	Kleisma, semicomma majeur
7	36/35	[2 2 -1 -1⟩	48.77	Rugu	Mint comma, septimal quartertone
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 comma
7	6144/6125	[11 1 -3 -2⟩	5.36	Sarurutrigu	Porwell comma
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.

Lumatone

See: Lumatone mapping for 23edo

Music

Main article: 23edo/Music

See also: Category:23edo tracks

Further reading

• Sword, Ron. Icosikaitriphonic Scales for Guitar: A Repository of Theory, Reference Materials, and Scale Charts for Xentonal Families. 2010.