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; 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.
Mode
Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. 14 of these modes are versions of diatonic modes which come in LH and RH pairs; four are more oneirotonic-like.
| Left-handed modes | Right-handed modes | |
| Diatonic-like modes | LLSLMLSLM LH Diasem Lydian | LSLLMLSLM RH Diasem Lydian |
| LSLMLLSLM LH Diasem Ionian | LSLMLSLLM RH Diasem Ionian | |
| LSLMLSLML LH Diasem Mixo | SLLMLSLML RH Diasem Mixo | |
| SLMLLSLML LH Diasem Dorian | LMLSLLMLS RH Diasem Dorian | |
| LMLSLMLLS LH Diasem Aeolian | LMLSLMLSL RH Diasem Aeolian | |
| MLLSLMLSL LH Diasem Phrygian | MLSLLMLSL RH Diasem Phrygian | |
| MLSLMLLSL LH Diasem Locrian | MLSLMLSLL RH Diasem Locrian | |
| The odd ones out | SLMLSLMLL LH Diasem Pseudo-Hlanithian | LLMLSLMLS RH Diasem Pseudo-Illarnekian |
| LMLLSLMLS LH Diasem Pseudo-Celdorian | SLMLSLLML RH Diasem Pseudo-Sardorian |
Here follow snippets I've written in various diasem modes.
RH Diasem Lydian
RH Diasem Dorian
LH Diasem Locrian
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.