21edo

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)