User:Inthar/21edo: Difference between revisions

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I ([[User:Inthar|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 interesting 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 ==
== Intervals ==
Line 8: Line 8:
! Degree
! Degree
! Cents
! Cents
! colspan="3" | [[Ups and Downs Notation|Up/down notation]]
! colspan="3" | [[Ups and downs notation|Up/down notation]]
! Fox-Raven <br/> 5L3s Octotonic <br/> Notation
! [[User:Inthar/Fox-Raven notation|Fox-Raven]] <br/> [[5L 3s]] <br/> Notation
! Category name
! Category name
|-
|-
Line 24: Line 24:
| ^1 <br> vv2
| ^1 <br> vv2
| up unison, <br> double-down 2nd
| up unison, <br> double-down 2nd
| ^C <br> vvD
| C^ <br> Dvv
| N#/Ob
| N#/Ob
| Subminor 2nd
| Subminor 2nd; quartertone; semisecor
|-
|-
| 2
| 2
Line 32: Line 32:
| ^^1 <br> v2
| ^^1 <br> v2
| double-up unison, <br> down 2nd
| double-up unison, <br> down 2nd
| ^^C <br> vD
| C^^ <br> Dv
| O
| O
| Minor 2nd
| Minor 2nd
Line 48: Line 48:
| ^2 <br> vv3
| ^2 <br> vv3
| up 2nd, <br> double-down 3rd
| up 2nd, <br> double-down 3rd
| ^D <br> vvE
| D^ <br> Evv
| Pb
| Pb
| Supermajor 2nd
| Supermajor 2nd
Line 56: Line 56:
| ^^2 <br> v3
| ^^2 <br> v3
| double-up 2nd, <br> down 3rd
| double-up 2nd, <br> down 3rd
| ^^D <br> vE
| D^^ <br> Ev
| P
| P
| Minor 3rd
| Minor 3rd
Line 66: Line 66:
| E
| E
| P#/Qb
| P#/Qb
| Neutral 3rd
| Neutral 3rd; Supraminor 3rd
|-
|-
| 7
| 7
Line 72: Line 72:
| ^3 <br> vv4
| ^3 <br> vv4
| up 3rd, <br> double-down 4th
| up 3rd, <br> double-down 4th
| ^E <br> vvF
| E^ <br> Fvv
| Q
| Q
| Major 3rd
| Major 3rd
Line 80: Line 80:
| ^^3 <br> v4
| ^^3 <br> v4
| double-up 3rd, <br> down 4th
| double-up 3rd, <br> down 4th
| ^^E <br> vF
| E^^ <br> Fv
| Q#
| Q#
| Naiadic; Subfourth
| Naiadic; Subfourth
Line 96: Line 96:
| ^4 <br> vv5
| ^4 <br> vv5
| up 4th, <br> double-down 5th
| up 4th, <br> double-down 5th
| ^F <br> vvG
| F^ <br> Gvv
| J
| J
| Narrow Tritone
| Narrow Tritone
Line 104: Line 104:
| ^^4 <br> v5
| ^^4 <br> v5
| double-up 4th, <br> down 5th
| double-up 4th, <br> down 5th
| ^^F <br> vG
| F^^ <br> Gv
| J#
| J#
| Wide Tritone
| Wide Tritone
Line 120: Line 120:
| ^5 <br> vv6
| ^5 <br> vv6
| up 5th, <br> double-down 6th
| up 5th, <br> double-down 6th
| ^G <br> vvA
| G^ <br> Avv
| K
| K
| Superfifth
| Superfifth
Line 128: Line 128:
| ^^5 <br> v6
| ^^5 <br> v6
| double-up 5th, <br> down 6th
| double-up 5th, <br> down 6th
| ^^G <br> vA
| G^^ <br> Av
| K#/Lb
| K#/Lb
| Minor 6th
| Minor 6th
Line 138: Line 138:
| A
| A
| L
| L
| Neutral 6th
| Neutral 6th; Submajor 6th
|-
|-
| 16
| 16
Line 144: Line 144:
| ^6 <br> vv7
| ^6 <br> vv7
| up 6th, <br> double-down 7th
| up 6th, <br> double-down 7th
| ^A <br> vvB
| A^ <br> Bvv
| L#
| L#
| Major 6th
| Major 6th
Line 152: Line 152:
| ^^6 <br> v7
| ^^6 <br> v7
| double-up 6th, <br> down 7th
| double-up 6th, <br> down 7th
| ^^A <br> vB
| A^^ <br> Bv
| Mb
| Mb
| Subminor 7th
| Subminor 7th
Line 168: Line 168:
| ^7 <br> vv8
| ^7 <br> vv8
| up 7th, <br> double-down 8ve
| up 7th, <br> double-down 8ve
| ^B <br> vvC
| B^ <br> Cvv
| M#
| M#
| Major 7th
| Major 7th
Line 176: Line 176:
| ^^7 <br> v8
| ^^7 <br> v8
| double-up 7th, <br> down 8ve
| double-up 7th, <br> down 8ve
| ^^B <br> vC
| B^^ <br> Cv
| Nb
| Nb
| Supermajor 7th
| Supermajor 7th
Line 188: Line 188:
| Octave
| 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]]

Latest revision as of 01:13, 20 August 2025

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, 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

Score

LH Diasem Dorian

RH Diasem Dorian

Score

LH Diasem Phrygian

LH Diasem Locrian

Score

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.

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