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

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.