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, 21edo diasem 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. The bright version occurs in both diasem Locrians. A more xen one can be made by lowering either 343 or 1029 by one step, which exists in some variant of diasem Locrian. Unfortunately, the darker Locrian seventh chord doesn't appear in any unaltered diasem mode in 21edo. To remedy this one could split (add a comma step to) the tritone and the minor 2nd, yielding 11313113313.
Modes
Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. Fourteen of these modes are versions of diatonic modes which come in LH and RH pairs; four are more oneirotonic-like, sharing melodic segments with the oneirotonic scale in 21edo, and they lack either a perfect 4th or a perfect 5th.
| Left-handed modes | Right-handed modes | |
| Modes with both a P4 and a P5 | 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 | |
| Modes lacking either a P4 or a P5 | LLSLMLSLM LH Diasem Lydian |
LSLLMLSLM RH Diasem Lydian |
| LMLLSLMLS LH Diasem Darkened Dorian (Somewhat like Celdorian) |
LLMLSLMLS RH Diasem Darkened Mixo (Somewhat like Illarnekian) | |
| SLMLSLMLL LH Diasem Brightened Aeolian (Somewhat like Hlanithian) |
SLMLSLLML RH Diasem Brightened Dorian (Somewhat like Sardorian) | |
| MLSLMLLSL LH Diasem Locrian |
MLSLMLSLL RH Diasem Locrian |
Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. Modes with LLS are left-handed; modes with LLM are right-handed. Fourteen of these modes are versions of diatonic modes which come in LH and RH pairs; four are less diatonic-like.
Here, "brightened" (resp. "darkened") means that the diatonic scale is overall sharpened (resp. flattened) perfect 5th (resp. the perfect 4th) is missing. These may sound somewhat more like oneirotonic modes.
The modes arranged in cyclic order:
| Left-handed modes | Right-handed modes |
| LSLMLSLML LH Diasem Mixo |
LMLSLMLSL RH Diasem Aeolian |
| SLMLSLMLL LH Diasem Brightened Aeolian |
MLSLMLSLL RH Diasem Locrian |
| LMLSLMLLS LH Diasem Aeolian |
LSLMLSLLM RH Diasem Ionian |
| MLSLMLLSL LH Diasem Locrian |
SLMLSLLML RH Diasem Brightened Dorian |
| LSLMLLSLM LH Diasem Ionian |
LMLSLLMLS RH Diasem Dorian |
| SLMLLSLML LH Diasem Dorian |
MLSLLMLSL RH Diasem Phrygian |
| LMLLSLMLS LH Diasem Darkened Dorian |
LSLLMLSLM RH Diasem Lydian |
| MLLSLMLSL LH Diasem Phrygian |
SLLMLSLML RH Diasem Mixo |
| LLSLMLSLM LH Diasem Lydian |
LLMLSLMLS RH Diasem Darkened Mixo |
Here follow snippets I've written in various diasem modes.
RH Diasem Lydian
LH Diasem Dorian
RH Diasem Dorian
LH Diasem Phrygian
LH Diasem Locrian
RH Diasem Locrian
LH Diasem Pseudo-Celdorian
RH Diasem Pseudo-Illarnekian
Altered diasem modes
- 233231313 Oneiro-Diasem Locrian
- 231322213 Diasem Locrian "#6"
Other diatonic scales
- 0 229 400 629 686 914 1086 1200: 21edo Bright Lydian (not contained in diasem)
Free 21edo
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.