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I ([[User:Inthar]]) view 21edo in two ways:
I ([[User:Inthar|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.  
# 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 intersting MOS scales are [[soft]] [[oneirotonic]] and [[superhard]] [[orwelloid]].
# 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 ==
{| class="wikitable center-all right-3 right-5"
|-
! Degree
! Cents
! colspan="3" | [[Ups and downs notation|Up/down notation]]
! [[User:Inthar/Fox-Raven notation|Fox-Raven]] <br/> [[5L 3s]] <br/> Notation
! Category name
|-
| 0
| 0.00
| 1
| unison
| C
| N
| Unison
|-
| 1
| 57.14
| ^1 <br> vv2
| up unison, <br> double-down 2nd
| C^ <br> Dvv
| N#/Ob
| Subminor 2nd; quartertone; semisecor
|-
| 2
| 114.29
| ^^1 <br> v2
| double-up unison, <br> down 2nd
| C^^ <br> Dv
| O
| Minor 2nd
|-
| 3
| 171.43
| 2
| hepta 2nd
| D
| O#
| Submajor 2nd
|-
| 4
| 228.57
| ^2 <br> vv3
| up 2nd, <br> double-down 3rd
| D^ <br> Evv
| Pb
| Supermajor 2nd
|-
| 5
| 285.71
| ^^2 <br> v3
| double-up 2nd, <br> down 3rd
| D^^ <br> Ev
| P
| Minor 3rd
|-
| 6
| 342.86
| 3
| hepta 3rd
| E
| P#/Qb
| Neutral 3rd; Supraminor 3rd
|-
| 7
| 400.00
| ^3 <br> vv4
| up 3rd, <br> double-down 4th
| E^ <br> Fvv
| Q
| Major 3rd
|-
| 8
| 457.14
| ^^3 <br> v4
| double-up 3rd, <br> down 4th
| E^^ <br> Fv
| Q#
| Naiadic; Subfourth
|-
| 9
| 514.29
| 4
| hepta 4th
| F
| Jb
| Perfect 4th
|-
| 10
| 571.43
| ^4 <br> vv5
| up 4th, <br> double-down 5th
| F^ <br> Gvv
| J
| Narrow Tritone
|-
| 11
| 628.57
| ^^4 <br> v5
| double-up 4th, <br> down 5th
| F^^ <br> Gv
| J#
| Wide Tritone
|-
| 12
| 685.71
| 5
| hepta 5th
| G
| Kb
| Perfect 5th
|-
| 13
| 742.86
| ^5 <br> vv6
| up 5th, <br> double-down 6th
| G^ <br> Avv
| K
| Superfifth
|-
| 14
| 800.00
| ^^5 <br> v6
| double-up 5th, <br> down 6th
| G^^ <br> Av
| K#/Lb
| Minor 6th
|-
| 15
| 857.14
| 6
| hepta 6th
| A
| L
| Neutral 6th; Submajor 6th
|-
| 16
| 914.29
| ^6 <br> vv7
| up 6th, <br> double-down 7th
| A^ <br> Bvv
| L#
| Major 6th
|-
| 17
| 971.43
| ^^6 <br> v7
| double-up 6th, <br> down 7th
| A^^ <br> Bv
| Mb
| Subminor 7th
|-
| 18
| 1028.57
| 7
| hepta 7th
| B
| M
| Supraminor 7th
|-
| 19
| 1085.71
| ^7 <br> vv8
| up 7th, <br> double-down 8ve
| B^ <br> Cvv
| M#
| Major 7th
|-
| 20
| 1142.86
| ^^7 <br> v8
| double-up 7th, <br> down 8ve
| B^^ <br> 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.
 
{| class="wikitable"
|-
| style="text-align:center;" |
| style="text-align:center;" |Left-handed modes
| style="text-align:center;" |Right-handed modes
|-
| rowspan="5"|Modes with both a P4 and a P5
| | LSLMLLSLM <br/>LH Diasem Ionian
| | LSLMLSLLM <br/>RH Diasem Ionian
|-
| | LSLMLSLML <br/>LH Diasem Mixo
| | SLLMLSLML <br/>RH Diasem Mixo
|-
| | SLMLLSLML <br/>LH Diasem Dorian
| | LMLSLLMLS <br/>RH Diasem Dorian
|-
| | LMLSLMLLS <br/>LH Diasem Aeolian
| | LMLSLMLSL <br/>RH Diasem Aeolian
|-
| | MLLSLMLSL <br/>LH Diasem Phrygian
| | MLSLLMLSL <br/>RH Diasem Phrygian
|-
| rowspan=4|Modes lacking either a P4 or a P5
| | LLSLMLSLM <br/>LH Diasem Lydian
| | LSLLMLSLM <br/>RH Diasem Lydian
|-
| | LMLLSLMLS <br/>LH Diasem Darkened Dorian (Somewhat like Celdorian)
| | LLMLSLMLS <br/>RH Diasem Darkened Mixo (Somewhat like Illarnekian)
|-
| | SLMLSLMLL <br/>LH Diasem Brightened Aeolian (Somewhat like Hlanithian)
| | SLMLSLLML <br/>RH Diasem Brightened Dorian (Somewhat like Sardorian)
|-
| | MLSLMLLSL <br/>LH Diasem Locrian
| | MLSLMLSLL <br/>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:
{| class="wikitable"
|-
| style="text-align:center;" |Left-handed modes
| style="text-align:center;" |Right-handed modes
|-
| | LSLMLSLML <br/>LH Diasem Mixo
| | LMLSLMLSL <br/>RH Diasem Aeolian
|-
| | SLMLSLMLL <br/>LH Diasem Brightened Aeolian
| | MLSLMLSLL <br/>RH Diasem Locrian
|-
| | LMLSLMLLS <br/>LH Diasem Aeolian
| | LSLMLSLLM <br/>RH Diasem Ionian
|-
| | MLSLMLLSL <br/>LH Diasem Locrian
| | SLMLSLLML <br/>RH Diasem Brightened Dorian
|-
| | LSLMLLSLM <br/>LH Diasem Ionian
| | LMLSLLMLS <br/>RH Diasem Dorian
|-
| | SLMLLSLML <br/>LH Diasem Dorian
| | MLSLLMLSL <br/>RH Diasem Phrygian
|-
| | LMLLSLMLS <br/>LH Diasem Darkened Dorian
| | LSLLMLSLM <br/>RH Diasem Lydian
|-
| | MLLSLMLSL <br/>LH Diasem Phrygian
| | SLLMLSLML <br/>RH Diasem Mixo
|-
| | LLSLMLSLM <br/>LH Diasem Lydian
| | LLMLSLMLS <br/>RH Diasem Darkened Mixo
|}
 
Here follow snippets I've written in various diasem modes.
===== RH Diasem Lydian =====
[[File:RH Diasem Lydian Example 21edo.mp3]] [[:File:Diasem Lydian Example Score.pdf|Score]]
===== LH Diasem Dorian =====
[[File:LH Diasem Dorian Example 21edo.mp3]]
===== RH Diasem Dorian =====
[[File:RH Diasem Dorian Example 21edo.mp3]] [[:File:RH Diasem Dorian Example Score.pdf|Score]]
===== LH Diasem Phrygian =====
[[File:Diasem Phrygian Example 21edo.mp3]]
 
===== LH Diasem Locrian =====
[[File:LH Diasem Locrian Example 21edo.mp3]] [[:File:Diasem Locrian Example.pdf|Score]]
===== RH Diasem Locrian =====
[[File:RH Diasem Sample 21edo.mp3]]
===== 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 ==
[[File:21edo Fugue Expo.mp3]]
 
== 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.
[[Category:21edo]]
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