21edo: Difference between revisions

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'''21edo''' is an equal division of the octave into 21 steps of 1200c/21 ~= 57.1c each.
'''21edo''' is an equal division of the octave into 21 steps of 1200c/21 ~= 57.1c each.


21edo is unusual from a regular temperament perspective due to its mixture of very accurate (e.g. 23/16, 16/15) and very inaccurate (3/2, 6/5, 7/6) approximations. On the one hand, one could say 21edo has a 5 since its 15/8 is accurate due to error cancellation of the sharp 5 with the flat 3. In root-position major triads, though, the 400c major third sounds much less isodifferential than even in 12edo due to the 3/2 being much flatter, and root-position major and minor triads sound somewhat neogothic as a result and also because of the neominor third; the reverse is true of certain triad inversions, since 0-514-914 is close to being isodifferential (approximately 26:35:44).
21edo is unusual from a regular temperament perspective due to its mixture of very accurate (e.g. 23/16, 16/15) and very inaccurate (3/2, 6/5, 7/6) approximations. On the one hand, one could say 21edo has a 5 since its 15/8 is accurate due to error cancellation of the sharp 5 with the flat 3. In root-position major triads, though, the 400c major third sounds much less isodifferential than even in 12edo due to the 3/2 being much flatter, and root-position major and minor triads sound somewhat neogothic as a result and also because of the neominor third; the reverse is true of certain triad inversions, since 0-514-914 is close to being +1+1 [[DR]] (approximately 23:31:39).


Notable scales:
Notable scales:
* Archylino (2L3m2s) diatonic: 3423432 or 4323432
* Archylino diatonic (2L3m2s): 3423432 or 4323432
* 21edo is the first edo with a [[diasem]] scale: 323132313 (RH) or 313231323 (LH). Diasem provides basic 2.3.7 harmony, though 7/6 and 28/27 are not accurate at all in 21edo.
* Interseptimal diatonic (4L1m2s): 4143414
* 21edo is the first edo with a [[diasem]] scale: 323132313 (RH) or 313231323 (LH). Diasem provides basic 2.3.7 harmony, though 7/6, 28/27, and 9/7 are not accurate at all in 21edo.
* Slentonic (5L6s, sLsLsLsLsLs), interpreted as [[Slendric]][11], generated by stacking the ~8/7 (4\21)
* Slentonic (5L6s, sLsLsLsLsLs), interpreted as [[Slendric]][11], generated by stacking the ~8/7 (4\21)
* Oneirotonic (5L3s, LLsLLsLs), generated by stacking 8\21
* Oneirotonic (5L3s, LLsLLsLs), generated by stacking 8\21
21edo also supports [[Sidewalk]] temperament.
== Basic theory ==
== Basic theory ==
=== Intervals and notation ===
=== Intervals and notation ===
Since 21edo's best fifth is from 7edo, 21edo is notated with CDEFGAB representing 7edo and up/down representing 1\21.
Since 21edo's best fifth is from 7edo, 21edo is notated with CDEFGAB representing 7edo and up/down representing 1\21. This allows all intervals to have a "minor", "neutral/perfect", and "major" variants.
 
{| class="wikitable"
|+
! | Edostep !! | Cents !! | Notation (Ups and downs) !! | Interval name (ups/downs)
!Interval region (ADIN)
|-
|0
|0
|C
|Perfect unison
|Unison
|-
|1
|57.1
|^C
|Up unison
|Farmajor unison
|-
|2
|114.3
|vD
|Down second
|Farminor second
|-
|3
|171.4
|D
|Perfect second
|Neutral second
|-
|4
|228.6
|^D
|Up second
|Farmajor second
|-
|5
|285.7
|vE
|Down third
|Farminor third
|-
|6
|342.9
|E
|Perfect third
|Neutral third
|-
|7
|400
|^E
|Up third
|Farmajor third
|-
|8
|457.1
|vF
|Down fourth
|Farminor fourth
|-
|9
|514.3
|F
|Perfect fourth
|Perfect fourth
|-
|10
|571.4
|^F
|Up fourth
|Farmajor fourth
|-
|11
|628.6
|vG
|Down fifth
|Farminor fifth
|-
|12
|685.7
|G
|Perfect fifth
|Perfect fifth
|-
|13
|742.9
|^G
|Up fifth
|Farmajor fifth
|-
|14
|800
|vA
|Down sixth
|Farminor sixth
|-
|15
|857.1
|A
|Perfect sixth
|Neutral sixth
|-
|16
|914.3
|^A
|Up sixth
|Farmajor sixth
|-
|17
|971.4
|vB
|Down seventh
|Farminor seventh
|-
|18
|1028.6
|B
|Perfect seventh
|Neutral seventh
|-
|19
|1085.7
|^B
|Up seventh
|Farmajor seventh
|-
|20
|1142.9
|vC
|Down octave
|Farminor octave
|-
|21
|1200
|C
|Octave
|Octave
|}


=== Prime harmonic approximations ===
=== Prime harmonic approximations ===
Line 17: Line 157:


=== Erac group ===
=== Erac group ===
As a temperament, 21 may be best described using [[erac]]s: 2.x>3.x<5.7.x<11.x<13.23.29.
As a temperament, 21edo may be described using [[erac]]s: 2.x>3.x<5.7.x<11.x<13.23.29. These specific eracs indicate that the primes are about 1/3 of an edostep off, and that 63edo is an accurate system.


=== Edostep interpretations ===
=== Edostep interpretations ===
Line 29: Line 169:
* 46/45
* 46/45
* 64/63
* 64/63
== Multiples ==
=== 63edo ===
63edo provides a good representation of 2.3.5.7.11.13.23.29.31. Its prime 17 is critically inaccurate.
* The 3 is somewhat sharp, thus supporting [[Parapyth]] temperament, a rank-3 temperament where 32/27 is tuned close to 13/11 and 81/64 is tempered together with 14/11, and where the "spacer" 28/27 is identified with 33/32.
* The 5 is quite flat, thus supporting [[Magic]] temperament where the stack of five 5/4 major thirds becomes one 3/1.
{{Harmonics in ED|63|31|0}}


{{Cat|Edos}}
{{Cat|Edos}}
{{Navbox EDO}}

Latest revision as of 04:10, 7 June 2026

21edo is an equal division of the octave into 21 steps of 1200c/21 ~= 57.1c each.

21edo is unusual from a regular temperament perspective due to its mixture of very accurate (e.g. 23/16, 16/15) and very inaccurate (3/2, 6/5, 7/6) approximations. On the one hand, one could say 21edo has a 5 since its 15/8 is accurate due to error cancellation of the sharp 5 with the flat 3. In root-position major triads, though, the 400c major third sounds much less isodifferential than even in 12edo due to the 3/2 being much flatter, and root-position major and minor triads sound somewhat neogothic as a result and also because of the neominor third; the reverse is true of certain triad inversions, since 0-514-914 is close to being +1+1 DR (approximately 23:31:39).

Notable scales:

  • Archylino diatonic (2L3m2s): 3423432 or 4323432
  • Interseptimal diatonic (4L1m2s): 4143414
  • 21edo is the first edo with a diasem scale: 323132313 (RH) or 313231323 (LH). Diasem provides basic 2.3.7 harmony, though 7/6, 28/27, and 9/7 are not accurate at all in 21edo.
  • Slentonic (5L6s, sLsLsLsLsLs), interpreted as Slendric[11], generated by stacking the ~8/7 (4\21)
  • Oneirotonic (5L3s, LLsLLsLs), generated by stacking 8\21

21edo also supports Sidewalk temperament.

Basic theory

Intervals and notation

Since 21edo's best fifth is from 7edo, 21edo is notated with CDEFGAB representing 7edo and up/down representing 1\21. This allows all intervals to have a "minor", "neutral/perfect", and "major" variants.

Edostep Cents Notation (Ups and downs) Interval name (ups/downs) Interval region (ADIN)
0 0 C Perfect unison Unison
1 57.1 ^C Up unison Farmajor unison
2 114.3 vD Down second Farminor second
3 171.4 D Perfect second Neutral second
4 228.6 ^D Up second Farmajor second
5 285.7 vE Down third Farminor third
6 342.9 E Perfect third Neutral third
7 400 ^E Up third Farmajor third
8 457.1 vF Down fourth Farminor fourth
9 514.3 F Perfect fourth Perfect fourth
10 571.4 ^F Up fourth Farmajor fourth
11 628.6 vG Down fifth Farminor fifth
12 685.7 G Perfect fifth Perfect fifth
13 742.9 ^G Up fifth Farmajor fifth
14 800 vA Down sixth Farminor sixth
15 857.1 A Perfect sixth Neutral sixth
16 914.3 ^A Up sixth Farmajor sixth
17 971.4 vB Down seventh Farminor seventh
18 1028.6 B Perfect seventh Neutral seventh
19 1085.7 ^B Up seventh Farmajor seventh
20 1142.9 vC Down octave Farminor octave
21 1200 C Octave Octave

Prime harmonic approximations

Approximation of prime harmonics in 21edo
Harmonic 2 3 5 7 11 13 17 19 23 29
Error Absolute (¢) 0.0 -16.2 +13.7 +2.6 +20.1 +16.6 +9.3 -11.8 +0.3 -1.0
Relative (%) 0.0 -28.4 +24.0 +4.6 +35.2 +29.1 +16.3 -20.6 +0.5 -1.8
Steps

(reduced)

21

(0)

33

(12)

49

(7)

59

(17)

73

(10)

78

(15)

86

(2)

89

(5)

95

(11)

102

(18)

Erac group

As a temperament, 21edo may be described using eracs: 2.x>3.x<5.7.x<11.x<13.23.29. These specific eracs indicate that the primes are about 1/3 of an edostep off, and that 63edo is an accurate system.

Edostep interpretations

21edo's edostep has the following interpretations in the 2.3.5.7.23.29 subgroup:

  • 24/23
  • 30/29
  • 29/28
  • 49/48
  • 50/49
  • 46/45
  • 64/63

Multiples

63edo

63edo provides a good representation of 2.3.5.7.11.13.23.29.31. Its prime 17 is critically inaccurate.

  • The 3 is somewhat sharp, thus supporting Parapyth temperament, a rank-3 temperament where 32/27 is tuned close to 13/11 and 81/64 is tempered together with 14/11, and where the "spacer" 28/27 is identified with 33/32.
  • The 5 is quite flat, thus supporting Magic temperament where the stack of five 5/4 major thirds becomes one 3/1.


Approximation of prime harmonics in 63edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) 0.0 +2.8 -5.4 +2.6 +1.1 -2.4 +9.3 +7.2 +0.3 -1.0 -2.2
Relative (%) 0.0 +14.7 -28.1 +13.7 +5.6 -12.8 +49.0 +38.1 +1.6 -5.3 -11.4
Steps

(reduced)

63

(0)

100

(37)

146

(20)

177

(51)

218

(29)

233

(44)

258

(6)

268

(16)

285

(33)

306

(54)

312

(60)


View • Talk • EditEqual temperaments
EDOs
Macrotonal 5 • 6 • 7 • 8 • 9 • 10 • 11
12-23 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23
24-35 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35
36-47 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47
48-59 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58
60-71 60 • 63 • 64 • 65 • 67 • 68 • 70
72-83 72 • 77 • 80 • 81
84-95 84 • 87 • 89 • 90 • 93 • 94
Large EDOs 99 • 104 • 106 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665
Nonoctave equal temperaments
Tritave 4 • 9 • 13 • 17 • 26 • 39
Fifth 8 • 9 • 11 • 20
Other


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