User:Inthar/21edo
I (Inthar) view 21edo in two ways:
- 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.
- 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 |
| 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 |
| 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 |
| 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 MLSLMLLSL or 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 most basic and normal sounding Locrian 7th chord is 0-343-629-1029. 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 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
Oneirotonic harmony
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.