User:Inthar/21edo

I (Inthar) view 21edo in two ways:

1. Viewed more traditionally, it's a "color system" with traditional triadic harmony, 7edo circles of fifths, and many interesting colors for intervals. One possible scalar realization of this is the MV3 scale diasem with step ratio L:M:S = 3:2:1.
2. As a MOS system, its most interesting MOS scales are soft oneirotonic and perhaps miracle and orwelloid. (The other mos scales are variants of 12edo multimosses.)

Intervals

Degree	Cents	Up/down notation			Fox-Raven 5L 3s Notation	Category name
0	0.00	1	unison	C	N	Unison
1	57.14	^1 vv2	up unison, double-down 2nd	C^ Dvv	N#/Ob	Subminor 2nd; quartertone; semisecor
2	114.29	^^1 v2	double-up unison, down 2nd	C^^ Dv	O	Minor 2nd
3	171.43	2	hepta 2nd	D	O#	Submajor 2nd
4	228.57	^2 vv3	up 2nd, double-down 3rd	D^ Evv	Pb	Supermajor 2nd
5	285.71	^^2 v3	double-up 2nd, down 3rd	D^^ Ev	P	Minor 3rd
6	342.86	3	hepta 3rd	E	P#/Qb	Neutral 3rd; Supraminor 3rd
7	400.00	^3 vv4	up 3rd, double-down 4th	E^ Fvv	Q	Major 3rd
8	457.14	^^3 v4	double-up 3rd, down 4th	E^^ Fv	Q#	Naiadic; Subfourth
9	514.29	4	hepta 4th	F	Jb	Perfect 4th
10	571.43	^4 vv5	up 4th, double-down 5th	F^ Gvv	J	Narrow Tritone
11	628.57	^^4 v5	double-up 4th, down 5th	F^^ Gv	J#	Wide Tritone
12	685.71	5	hepta 5th	G	Kb	Perfect 5th
13	742.86	^5 vv6	up 5th, double-down 6th	G^ Avv	K	Superfifth
14	800.00	^^5 v6	double-up 5th, down 6th	G^^ Av	K#/Lb	Minor 6th
15	857.14	6	hepta 6th	A	L	Neutral 6th; Submajor 6th
16	914.29	^6 vv7	up 6th, double-down 7th	A^ Bvv	L#	Major 6th
17	971.43	^^6 v7	double-up 6th, down 7th	A^^ Bv	Mb	Subminor 7th
18	1028.57	7	hepta 7th	B	M	Supraminor 7th
19	1085.71	^7 vv8	up 7th, double-down 8ve	B^ Cvv	M#	Major 7th
20	1142.86	^^7 v8	double-up 7th, down 8ve	B^^ Cv	Nb	Supermajor 7th
21	1200.00	8	8ve	C	N	Octave

Traditional harmony

• Third flavors: 286 (min3, soft minor), 343 (neu3, supraminor-ish neutral), 400 (maj3, neogothic-ish major)
• Seconds: 114 (min2), 171 (submaj2), 229 (supmaj2), 171 and 228 work better as major seconds than 15edo's 160 and 240.
• Sixths: 800 (min6), 857 (neu6), 914 (maj6)
• Sevenths: 971 (submin7), 1029 (supmin7), 1086 (maj7), 1143 (supmaj7)
• Tritones: 571 (subtri), 628 (suptri)
• Fourths: 514 (perfect~acute 4th), 457 (sub4th, sounds more like a sub4th than a naiadic)
• More dissonant extensions (might want to use nejification for these?): 57 (submin2), 743 (sup5)
• 1143 can be used internally but not 1257.

Diasem harmony

Diasem is an MV3 scale with step pattern either (left-handed) MLSLMLLSL or (right-handed) MLSLMLSLL which are non-superimposable mirror images. As its name suggests, it has qualities intermediate that of 19edo's diatonic mos and 23edo's semiquartal mos. 21edo diasem also has a non-chiral alteration 231323223, which, however, is only MV3 because of the specific step ratio 3:2:1 it has in 21edo.

The basic Locrian 7th chord has a brighter version 0-343-629-1029 and a darker version 0-286-571-971. A more xen one can be made by lowering either 343 or 1029 by one step, which exists in some variant of diasem Locrian.

The (right-handed) diasem Lydian 313323132 would benefit from the addition of the 629 cent tritone, yielding 3133113132.

Other diatonic scales

• 0 229 400 629 686 914 1086 1200: 21edo Bright Lydian
• 0 171 400 571 686 857 1086 1200: 21edo Dark Lydian

Modulations

5ths and 4ths

• hepta5 / hepta4
• sup5 / sub4

3rds and 6ths

• maj3 / min6
• hepta3 / hepta6
• min3 / maj6

2nds and 7ths

• submaj2 / supmin7
• supmaj2 / submin7

Dissonances

• minT / majT (571/628 tritone)
• min2/maj7, submin2/supmaj7

The important identites in "5-limit" harmony are 7\*hepta interval = unison, submaj2 + supmaj2 = maj3, and 3\*maj3 = unison. it's also important that 2\*min2 = supmaj2. 3*sub4 = hepta2

Miracle is a good mapping.