User:Unque/22-TET Plagal Theory
"Plagal Theory" (Eleutheric: Χωρη Πλαγικη, Xoree Plagikee /ʃori plaʝici/) is a fictitious alternative music theory popularized across Europe in an alternate history. As such, the concepts discussed here, and the history thereof, should be understood to be fictitious, and therefore not necessarily applicable to the real world; specifically, the scales and their modes are named in accordance with the in-universe traditions, and are not legitimate proposals for conventions to use in real-world xenharmonic theory.
Overview
Plagal Theory is a system of functionality based on several scales of 22-EDO. The main defining feature of compositions in this tradition are the modulation from one mode of a scale to another by sharpening a single note; for instance, a piece written in the 5|1 mode of 5L2s would traditionally modulate to the 6|0 mode (with the same root pitch) to add a sense of grandeur and finality to the last movement of a piece. These modes are often grouped into "plagal pairs" that indicate how scales are related and how modulation tends to occur between pairs of scales.
The main scales considered in Plagal Theory are the MOS scales 5L2s (known as Monotonic or Unitonic), 3L4s (known as Bitonic), and 2L5s (known as Enarmonic).
Monotonic (5L 2s)
Modes of Monotonic
The modes of the Monotonic scale are grouped into three plagal pairs, with one additional mode that lacks a plagal partner. Shown below is the general UDP notation and step sizes for each mode, as well as the degrees of 22-EDO contained by that mode (with the root implied at 0\22).
| Mode Pair | Standard Mode (general) | Standard Mode (22-EDO) | Plagal Mode (general) | Plagal Mode (22-EDO) |
|---|---|---|---|---|
| Lydian | 5|1 (LLsLLLs) | 4 8 9 13 17 21 22 | 6|0 (LLLsLLs) | 4 8 12 13 17 21 22 |
| Dorian | 3|3 (LsLLLsL) | 4 5 9 13 17 18 22 | 4|2 (LLsLLsL) | 4 8 9 13 17 18 22 |
| Phrygian | 1|5 (sLLLsLL) | 1 5 9 13 14 18 22 | 2|4 (LsLLsLL) | 4 5 9 13 14 18 22 |
| Locrian | 0|6 (sLLsLLL) | 1 5 9 10 14 18 22 |
The Locrian mode is considered to be a "mixed" mode with no plagal form, and is seldom considered in traditional composition.
Bitonic (3L 4s)
Modes of Bitonic
The modes of Bitonic number a total of four plagal pairs, and an additional "mixed" mode.
| Mode Pair | Standard Mode (general) | Standard Mode (22-EDO) | Plagal Mode (general) | Plagal Mode (22-EDO) |
|---|---|---|---|---|
| Lydian | 4|2 LssLsLs | 6 7 8 14 15 21 22 | 5|1 LsLssLs | 6 7 13 14 15 16 22 |
| Phrygian | 0|6 ssLsLsL | 1 2 8 9 15 16 22 | 1|5 sLssLsL | 1 7 8 9 15 16 22 |
| Dorian | 3|3 sLsLsLs | 1 7 8 14 15 21 22 | 4|2 LssLsLs | 6 7 8 14 15 21 22 |
| Mixodorian | 2|4 sLsLssL | 1 7 8 14 15 16 22 | 3|3 sLsLsLs | 1 7 8 14 15 21 22 |
| Locrian | 6|0 LsLsLss | 6 7 13 14 20 21 22 |
The so-called "Mixodorian" mode is a later invention by theorists, especially experimentalists, in an attempt to "fix" the issues presented by the traditional Dorian pair: traditional Plagal Dorian was the same pattern as Standard Lydian, which left two "mixed" modes instead of just one. The Mixodorian mode does not see widespread usage, but it is not entirely unheard of either.
Just like with Monotonic, the "mixed" modes do not see widespread usage.
Enarmonic (2L5s)
Modes of Enarmonic
Like monotonic, the Enarmonic scale has three non-overlapping plagal pairs and only one "mixed" mode.
| Mode Pair | Standard Mode (general) | Standard Mode (22-EDO) | Plagal Mode (general) | Plagal Mode (22-EDO) |
|---|---|---|---|---|
| Lydian | 5|1 LsssLss | 6 8 10 12 18 20 22 | 6|0 LssLsss | 6 8 10 16 18 20 22 |
| Phrygian | 0|6 sssLssL | 2 4 6 12 14 16 22 | 1|5 ssLsssL | 2 4 10 12 14 16 22 |
| Dorian | 2|4 ssLssLs | 2 4 10 12 14 20 22 | 3|3 sLsssLs | 2 8 10 12 14 20 22 |
| Locrian | 4|2 sLssLss | 2 8 10 12 18 20 |
Once again, the "mixed" Locrian mode does not see widespread usage in composition.
Notation
The notation system used in Plagal Theory utilizes three types of clefs to describe the three primary MOS scales: the G-clef (𝄞) is used for the Monotonic Scale, the F-clef (𝄢) is used for the Bitonic scale, and the C-clef (𝄡) is used for the Enarmonic scale. Each scale has seven nominals and uses accidentals to alter those nominals, just like in common-practice Western theory in the real world.
The accidentals ♭ and ♯ respectively lower and raise the note by the chroma of the current scale; in other words, they are the difference between a large and small step. A circular accidental placed above or below a note also occurs in later iterations of musical notation, and is used to respectively raise or lower the note by a single step of 22edo, regardless of the scale being used.
Note that this notation system arbitrarily treats G as the "default" starting point, akin to how real-world descriptions of musical structure often treat C as the "default" tonic.
| EDO steps | Monotonic (G Dorian) | Bitonic (G Dorian) | Enarmonic (G Dorian) | Harmonic Category (See below) |
|---|---|---|---|---|
| 0 | G | G | G | |
| 1 | A♭ | A | Hard Dissonance | |
| 2 | B♭ | B♭ | A ~ F♯ | Hard Dissonance |
| 3 | G♯ | C♭ | Soft Dissonance | |
| 4 | A | F♯ | B ~ G♯ | Soft Consonance |
| 5 | B | G♯ | Soft Consonance | |
| 6 | C♭ | A♯ | A♯ ~ C♭ | Soft Consonance |
| 7 | A♯ | B | Hard Consonance | |
| 8 | B♯ | C | B♯ ~ D♭ | Soft Dissonance |
| 9 | C | D♭ | Hard Consonance | |
| 10 | D♭ | E♭ | C ~ E♭ | Soft Consonance |
| 11 | E♭ | Hard Dissonance | ||
| 12 | C♯ | B♯ | D | Soft Consonance |
| 13 | D | C♯ | Hard Consonance | |
| 14 | E | D | E ~ C♯ | Soft Dissonance |
| 15 | F♭ | E | Hard Consonance | |
| 16 | D♯ | F♭ | D♯ ~ F♭ | Soft Consonance |
| 17 | E♯ | G♭ | Soft Consonance | |
| 18 | F | A♭ | G♭ | Soft Consonance |
| 19 | G♭ | D♯ | Soft Dissonance | |
| 20 | E♯ | F | Hard Dissonance | |
| 21 | F♯ | F | Hard Dissonance | |
| 22 | G | G | G | Hard Consonance |
Note that, unlike in real-world musical notation, the semitones of 5L2s occur between A-B and D-E, rather than B-C and E-F. Additionally, the Enarmonic notation is only productive for notating 11edo, because the intervals of the 2L5s scale can only be used to reach every other note of 22edo.
Finally, the Monotonic and Ditonic scales each have a singular "blind spot" note that cannot be reached by a single alteration of the nominals. This note must instead be reached by use of double-accidentals or the over/under dot accidentals.
Harmony
Over time, elements of chord-based harmony begin to take shape in the Plagal system, beginning with Parallel Generator dyads and later extending to triads and tetrads. Unlike in real-world Western classical theory, chords in Plagal Theory do not have a conventional method of construction by stacking flavors of thirds, and instead are built off of the mode that they begin on.
All intervals are categorized as "soft" and "hard" consonances and dissonances, with "hard" categories being the strongest resolutions and tensions, and "soft" categories being passing intervals that are used more often, without extreme weight placed on their harmonic content. These categories arose from rules of counterpoint composition, with the most concordant intervals being favored as ending points and the least concordant intervals being used as controlled dissonances to provide tension and set up for resolutions.
In the earliest forms of contrapuntal polyphony, chords were traditionally built by combining two hard consonances either over one another or over the same root, such that the third interval introduced was a soft consonance: for instance, seven steps and thirteen steps are both hard consonances, and their difference of six steps are a soft consonance, which means that [0 7 13]\22 is a valid chord. Later expansions of this counterpoint system utilized more soft consonances and soft dissonances as a form of controlled tension; for instance, the intervals at five and eight steps are both soft consonances, which means that [0 5 13]\22 and [0 8 13]\11 are valid chords under the later expansion that were not valid prior.
The Plagal Fugue
The Plagal Fugue is a specific prescriptive style of composition that arises from the rules of modal counterpoint. A Plagal Fugue is built around a progression of underlying chords that determine where the melody and its countermelodies can and cannot move. Voicings in this Plagal Fugue can be broken down into four main parts:
- The Meelus (μηλος) is the most prominent voice, and plays the primary melodies.
- The Antees (αντης) is the second-most prominent; it usually plays a part that is homorhythmic with the Meelus, but using different chord tones.
- The Uphras (ουφρας) is typically a higher-pitched, fast-paced line that plays overtop the Meelus, with tones from the same chords.
- The Threenus (ϑρηνος) is a lower-pitched voice that drones a singular tone for the duration of a chord.
Plagal Fugues tend to be divided into three short movements: a slower Prelude that introduces the modality (which is foregone in many pieces), a Core that states the theme and provides variations upon it, and a bombastic Finale that modulates from the standard into the plagal mode and restates the melody under this new harmonic context.