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22edo/Unque's compositional approach: Difference between revisions

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| [[Orwell]]
| [[Orwell]]
| E♭, B𝄪
| E♭, B𝄪
| Often considered more dissonant than a true 7/6.
| Often considered less concordant than a true 7/6.
|-
|-
| 6\22
| 6\22
| 327.3
| 327.3
| [[6/5]]
| [[6/5]]
| [[Orgone]] (actually in [[11edo]])
| [[Orgone]]
| F♭, C𝄪
| F♭, C𝄪
| Somewhat contentious JI interpretation (see below).
| Orgone is inherited from 11edo
|-
|-
| 7\22
| 7\22
Line 76: Line 76:
| 8\22
| 8\22
| 436.3
| 436.3
| [[9/7]], [[14/11]]
| [[14/11]], [[32/25]], [[9/7]]
| [[Sensamagic]]
| [[Sensamagic]]
| E
| E
|  
| Sensamagic is inherited from 11edo
|-
|-
| 9\22
| 9\22
Line 93: Line 93:
| [[Joan]]
| [[Joan]]
| G♭, D𝄪
| G♭, D𝄪
| Somewhat contentious JI interpretation.
| Joan is inherited from 11edo
|-
|-
| 11\22
| 11\22
Line 118: Line 118:
| 14\22
| 14\22
| 763.6
| 763.6
| [[14/9]], [[11/7]]
| [[14/9]], [[25/16]], [[11/7]]
| Sensamagic
| Sensamagic
| A♭, E𝄪
| A♭, E𝄪
Line 133: Line 133:
| 872.7
| 872.7
| [[5/3]]
| [[5/3]]
| Orgone (actually in [[11edo]])
| Orgone
| G♯, C𝄫
| G♯, C𝄫
|  
|  
Line 179: Line 179:
|  
|  
|}
|}
[[Ups and downs notation]] uses the accidentals ^ and v to modify a given note by one step of 22edo
The intervals of 22edo can be notated using the familiar Circle of Fifths; additionally, [[ups and downs notation]] uses the accidentals ^ and v to modify a given note by one step of 22edo, so that B𝄫 may for instance be more concisely notated as ^A♭ to avoid the potentially confusing double-flat.


=== The thirds of 22edo ===
=== The thirds of 22edo ===
Line 437: Line 437:
|<nowiki>4|0</nowiki>
|<nowiki>4|0</nowiki>
|LmLsLmLsLs
|LmLsLmLsLs
|C - C♯ - E♭ - E - F - F# - ^G - ^G♯ - B♭ - B - C
|C - C♯ - E♭ - E - F - F# - A♭ - A - B♭ - B - C
|Aeolian
|Aeolian
|Contains the Wolf fifth (^G)
|Contains the Wolf fifth (^G)
Line 499: Line 499:
|<nowiki>0|4</nowiki>
|<nowiki>0|4</nowiki>
|sLsLmLsLmL
|sLsLmLsLmL
|C - D♭ - D - vF♭ - vF - G♭ - G - A♭ - A - C♭ - C
|C - D♭ - D - E♭ - E - G♭ - G - A♭ - A - C♭ - C
|Mixolydian
|Mixolydian
|Contains the Wolf fourth (vF)
|Contains the Wolf fourth (vF)
Line 521: Line 521:
|<nowiki>8|0</nowiki>
|<nowiki>8|0</nowiki>
|LssssLssss
|LssssLssss
|C - C♯ - ^D - D♯ - ^E - E♯ - ^G - G♯ - ^A - A♯ - C
|C - C♯ - ^D - D♯ - F - vF♯ - ^G - G♯ - ^A - A♯ - C
|Wolf Minor
|Wolf Minor
|Contains the wolf fifth (^G)
|Contains the wolf fifth (^G)
Line 527: Line 527:
|<nowiki>6|2</nowiki>
|<nowiki>6|2</nowiki>
|sLssssLsss
|sLssssLsss
|C - vC♯ - ^D - D♯ - ^E - E♯ - G - G♯ - ^A - A♯ - C
|C - vC♯ - ^D - D♯ - F - vF♯ - G - G♯ - ^A - A♯ - C
|Dynamic Major
|Dynamic Major
|Dual thirds are subminor and major
|Dual thirds are subminor and major
Line 533: Line 533:
|<nowiki>4|4</nowiki>
|<nowiki>4|4</nowiki>
|ssLssssLss
|ssLssssLss
|C - vC♯ - D - D♯ - ^E - E♯ - G - vG♯ - ^A - A♯ - C
|C - vC♯ - D - D♯ - F - vF♯ - G - vG♯ - ^A - A♯ - C
|Static Major
|Static Major
|Laterally symmetrical (flipping the scale upside-down yields the same pitches)
|Laterally symmetrical
|-
|-
|<nowiki>2|6</nowiki>
|<nowiki>2|6</nowiki>
|sssLssssLs
|sssLssssLs
|C - vC♯ - D - vD♯ - ^E - E♯ - G - vG♯ - A - A♯ - C
|C - vC♯ - D - vD♯ - F - vF♯ - G - vG♯ - A - A♯ - C
|Static Minor
|Static Minor
|Messiaen treated this rotation as the "default"
|Messiaen treated this rotation as the "default"
Line 545: Line 545:
|<nowiki>0|8</nowiki>
|<nowiki>0|8</nowiki>
|ssssLssssL
|ssssLssssL
|C - vC♯ - D - vD♯ - E - E♯ - G - vG♯ - A - vA♯ - C
|C - vC♯ - D - vD♯ - vF - vF♯ - G - vG♯ - A - vA♯ - C
|Dynamic Minor
|Dynamic Minor
|Dual thirds are minor and supermajor
|Dual thirds are minor and supermajor
|}
|}
=== 4L 5s ===
The [[4L 5s]] scale is another quite useful scale available in 22edo <s>and a perfect example of 22edo being better than 31</s>; it is closely associated with [[Orwell]] temperament, and as such is best utilized alongside Orwell Chords for functional harmony (see below).
{| class="wikitable"
|+Modes of 4L 5s
!UDP
!Step Pattern
!Notation
!Notes
|-
|<nowiki>8|0</nowiki>
|LsLsLsLss
|C - C♯ - E♭ - ^D♯ - G♭ - ^F♯ - ^A♭ - ^A - A♯ - C
|^F♯ is actually the Wolf Fifth in this context, and F♯ is the desirable interval!
|-
|<nowiki>7|1</nowiki>
|LsLsLssLs
|C - C♯ - E♭ - ^D♯ - G♭ - ^F♯ - ^A♭ - A - A♯ - C
|
|-
|<nowiki>6|2</nowiki>
|LsLssLsLs
|C - C♯ - E♭ - ^D♯ - G♭ - F♯ - ^A♭ - A - A♯ - C
|
|-
|<nowiki>5|3</nowiki>
|LssLsLsLs
|C - C♯ - E♭ - D♯ - G♭ - F♯ - ^A♭ - A - A♯ - C
|
|-
|<nowiki>4|4</nowiki>
|sLsLsLsLs
|C - vC♯ - E♭ - D♯ - G♭ - F♯ - ^A♭ - A - A♯ - C
|
|-
|<nowiki>3|5</nowiki>
|sLsLsLssL
|C - vC♯ - E♭ - D♯ - G♭ - F♯ - ^A♭ - A - vA♯ - C
|
|-
|<nowiki>2|6</nowiki>
|sLsLssLsL
|C - vC♯ - E♭ - D♯ - G♭ - F♯ - A♭ - A - vA♯ - C
|
|-
|<nowiki>1|7</nowiki>
|sLssLsLsL
|C - vC♯ - E♭ - D♯ - vG♭ - F♯ - A♭ - A - vA♯ - C
|vG♭ is actually the Wolf Fourth in this context, and G♭ is the desirable interval!
|-
|<nowiki>0|8</nowiki>
|ssLsLsLsL
|C - vC♯ - vE♭ - D♯ - vG♭ - F♯ - A♭ - A - vA♯ - C
|Subminor third above the root is unavailable in this mode.
|}
== Chords of 22edo ==
22edo provides a number of ways to construct and notate chords; while I cannot provide a comprehensive list of every possible chord available in 22edo, I will discuss and generalize the forms that I find to be most important for 22edo's structure.
=== Tertiary Triads ===
Having four types of thirds rather than two, 22edo has a more colorful array of tertian triads than common practice music does.  The 5-limit Major and Minor triads can be approximated using 6\22 and 7\22 as the thirds, and the 7-limit triads can be approximated using 5\22 and 8\22.
Additionally, there are four distinct types of diminished triads that can be formed using the two types of minor thirds; due to the [[225/224|Marvel comma]] being tempered out in 22edo, however, there are only two distinct types of augmented triads that can be formed using the two types of major thirds, as 7 + 7 + 8 = 22.
{| class="wikitable"
|+Tertiary Chords
!Name
!Symbol
!Notation
!Formula
!Otonal
!Utonal
|-
|Major
|C maj
|C - vE - G
|7\22 + 6\22
|4:5:6
|1/(15:12:10)
|-
|Minor
|c min
|C - ^E♭ - G
|6\22 + 7\22
|10:12:15
|1/(6:5:4)
|-
|Supermajor
|C saj
|C - E - G
|8\22 + 5\22
|14:18:21
|1/(9:7:1)
|-
|Subminor
|c sin
|C - E♭ - G
|5\22 + 8\22
|6:7:9
|1/(21:18:14)
|-
|Orwell Dim.
|c<sup>ow</sup>
|C - E♭ - G♭
|5\22 + 5\22
|36:32:49
|1/(49:32:36)
|-
|Utonal Dim.
|c<sup>ut</sup>
|C - E♭ - ^G♭
|5\22 + 6\22
|30:35:42
|1/(7:6:5)
|-
|Otonal Dim.
|c<sup>ot</sup>
|C - ^E♭ - ^G♭
|6\22 + 5\22
|5:6:7
|1/(42:35:30)
|-
|Orgone Dim.
|c<sup>og</sup>
|C - ^E♭ - vG
|6\22 + 6\22
|25:30:36
|1/(36:30:25)
|-
|Marvel
|C+
|C - vE - ^G
|7\22 + 7\22
|16:20:25
|1/(25:20:16)
|-
|Marvel (1st inv.)
|vE+/C
|C - vE - vG♯
|7\22 + 8\22
|20:25:32
|1/(40:32:25)
|-
|Marvel (2nd inv.)
|vG♯+/C
|C - E - vG♯
|8\22 + 7\22
|25:32:40
|1/(32:25:20)
|-
|Sensamagic
|C×
|C - E - G♯
|8\22 + 8\22
|21:27:35
|1/(45:35:27)
|}
=== Quartal Inversions ===
In diatonic tunings with a sharp-tempered 3/2, such as the [[Superpyth]] diatonic found in 22edo, quartal harmony is often more concordant than tertian harmony; as such, musicians interested in using Superpyth should familiarize themselves with the Quartal Triad and its inversions.
{| class="wikitable"
|+Quartal Inversions
!UDP
!Symbol
!Notation
!Formula
!3-limit JI
!7-limit JI
|-
|<nowiki>2|0</nowiki>
|C<sup>4</sup>
|C - F - B♭
|9\22 + 9\22
|9:12:16
|16:21:28
|-
|<nowiki>1|1</nowiki>
|G<sup>4</sup>/C
|C - F - G
|9\22 + 4\22
|6:8:9
|16:21:24
|-
|<nowiki>0|2</nowiki>
|D<sup>4</sup>/C
|C - D - G
|4\22 + 9\22
|8:9:12
|14:16:21
|}
== Functional Harmony ==
Useful harmonic progressions may arise in a number of ways depending on the scale being used and depending on what chord the composer wishes to tonicize.  Here, I will document some examples of how functional harmonic progressions may be created in the different scales of 22edo, with concepts that can be extended to apply to any scale.
Note that I will be constructing these chord progressions from back to front; this means that we will start with the resolution, then find the dominant chord, and then find a subdominant to precede it.
=== Elements of Functional Harmony ===
Just like in common-practice music theory, chords in 22edo have a tendency to rotate about the Circle of Fifths.  This means that in Diatonic music and other scales that contain 13\22, the chord built off of that note will be useful as a dominant.
Due to 22edo's smaller step size, there are two types of leading tones.  The smaller of the two can be found in the Superpyth Diatonic scale, where it is much tenser than the leading tone found in common-practice music; and the larger of the two is the small step of the Zarlino diatonic scale, which while less tense than the smaller leading tone is more useful as a dissonant chord tone to provide some extra tension.
Finally, it is important to notice certain tense intervals that have a tendency to voice lead by contrary motion to certain other intervals.  Specifically, the semioctave has a tendency to resolve either inwards or outwards to yield a perfect fourth/fifth; additionally, the subminor third may resolve outwards to a major third, and the supermajor third may resolve inwards to a minor third.
=== Example: Chord Progression in C Mixolydian (Superpyth) ===
The C Ionian scale in Superpyth is a perfect showcase of how common-practice harmony differs from that of 22edo; specifically, quartal harmony tends to provide a more resolved sound than tertian harmony does, due to the sharp fifth exaggerating the size of the chroma.  Thus, our progression can resolve quite nicely to the tonic chord, Csus4 or G<sup>4</sup>/C.
A supermajor triad can be found over the fifth degree of the Ionian mode, which is much tenser than the major triad found in Meantone tunings due to the major seventh (the third of the V chord) being significantly sharper; this increases the strength of the pull towards the tonic, as well as destabilizes the V chord, making it clear that this is not a place of rest and thus negating the need for a dominant seventh over that chord.  This allows us to restrict our progression to three-part harmony rather than switching between three- and four-part harmony as is often done in common-practice progressions.
Note also that a leading tone occurs between the notes E and F; this allows us to achieve a false resolution onto the Fsus4 chord, or C<sup>4</sup>/F, in our progression by way of voice leading from the Orwell diminished chord on E.
Thus, our four-chord progression is Csus4 - E<sup>ow</sup> - Fsus4 - Gsaj.  This series of chords uses a combination of voice leading, tension-and-release paradigms, and circle of fifths movement to provide a dynamic and functional harmonic progression.
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