21edo: Difference between revisions

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== 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. This allows all intervals to have a "minor", "neutral/perfect", and "major" variant.
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"
{| class="wikitable"
|+
|+
! | Edostep !! | Cents !! | Notation (Ups and downs) !! | Interval name
! | Edostep !! | Cents !! | Notation (Ups and downs) !! | Interval name (ups/downs)
!Interval region (ADIN)
|-
|-
|0
|0
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|C
|C
|Perfect unison
|Perfect unison
|Unison
|-
|-
|1
|1
|57.1
|57.1
|^C
|^C
|Inframinor second
|Up unison
|Farmajor unison
|-
|-
|2
|2
|114.3
|114.3
|vD
|vD
|Minor second
|Down second
|Farminor second
|-
|-
|3
|3
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|D
|D
|Perfect second
|Perfect second
|Neutral second
|-
|-
|4
|4
|228.6
|228.6
|^D
|^D
|Major second
|Up second
|Farmajor second
|-
|-
|5
|5
|285.7
|285.7
|vE
|vE
|Minor third
|Down third
|Farminor third
|-
|-
|6
|6
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|E
|E
|Perfect third
|Perfect third
|Neutral third
|-
|-
|7
|7
|400
|400
|^E
|^E
|Major third
|Up third
|Farmajor third
|-
|-
|8
|8
|457.1
|457.1
|^E
|vF
|Minor fourth
|Down fourth
|Farminor fourth
|-
|-
|9
|9
|514.3
|514.3
|F
|F
|Perfect fourth
|Perfect fourth
|Perfect fourth
|-
|-
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|571.4
|571.4
|^F
|^F
|Major fourth
|Up fourth
|Farmajor fourth
|-
|-
|11
|11
|628.6
|628.6
|vG
|vG
|Minor fifth
|Down fifth
|Farminor fifth
|-
|-
|12
|12
|685.7
|685.7
|G
|G
|Perfect fifth
|Perfect fifth
|Perfect fifth
|-
|-
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|742.9
|742.9
|^G
|^G
|Major fifth
|Up fifth
|Farmajor fifth
|-
|-
|14
|14
|800
|800
|vA
|vA
|Minor sixth
|Down sixth
|Farminor sixth
|-
|-
|15
|15
|857.1
|857.1
|A
|A
|Perfect sixth
|Neutral sixth
|Neutral sixth
|-
|-
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|914.3
|914.3
|^A
|^A
|Major sixth
|Up sixth
|Farmajor sixth
|-
|-
|17
|17
|971.4
|971.4
|vB
|vB
|Minor seventh
|Down seventh
|Farminor seventh
|-
|-
|18
|18
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|B
|B
|Perfect seventh
|Perfect seventh
|Neutral seventh
|-
|-
|19
|19
|1085.7
|1085.7
|^B
|^B
|Major seventh
|Up seventh
|Farmajor seventh
|-
|-
|20
|20
|1142.9
|1142.9
|vC
|vC
|Ultramajor seventh
|Down octave
|Farminor octave
|-
|-
|21
|21
|1200
|1200
|C
|C
|Octave
|Octave
|Octave
|}
|}
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=== 63edo ===
=== 63edo ===
63edo provides a good representation of 2.3.5.7.11.13.23.29.31.
63edo provides a good representation of 2.3.5.7.11.13.23.29.31.
* The 3 is somewhat sharp, thus supporting [[Parapyth]] temperament, a rank-3 temperament 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 3 is somewhat sharp, thus supporting [[Parapyth]] temperament, a rank-3 temperament 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.
* The 5 is quite flat, thus supporting [[Magic]] temperament where the stack of five 5/4 major thirds becomes one 3/1.
{{Cat|Edos}}
{{Cat|Edos}}
{{Navbox EDO}}

Latest revision as of 01:16, 14 February 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

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.

  • The 3 is somewhat sharp, thus supporting Parapyth temperament, a rank-3 temperament 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.


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


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