User:Unque/22-TET Plagal Theory

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"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).

Modes of Monotonic

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.

Modes of Bitonic

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.

Modes of Enarmonic

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.

Notation

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 Monotonic 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.