View source for User:Unque/5L 3s Tonal Theory

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The 5L 3s scale, or Oneirotonic, is particularly interesting as far as scales go.  Its doesn't arise very intuitively from low-complexity harmonic relationships in the same way as the diatonic scale or the third-generated smitonic and mosh scales; however, once its structure is asserted, it can be a very fruitful system to use as a basis for many useful harmonic ideas.  On this page, I aim to document a coherent theory of tonal harmony based on Oneirotonic and related [[MODMOS scale]]<nowiki/>s.

It should be noted that the notation and interval categories used here are based on my own derivations from Oneirotonic structure, mostly because I found that other existing systems were unsatisfactory in describing it.

== Defining Terms ==
For the purposes of this page, I will be describing the degrees of the Oneiro scale using a system based on a system of seven nominals and one interordinal; this allows for the octave to land on the interval of equivalence, maintaining the octave complement relationships (e.g. m7 ~ M2, M3 ~ m6, etc.)
{| class="wikitable"
|+Degrees of Oneiro
!Degree
!Soft Range
!Hard Range
!General Notation
!Notes
|-
|1
|1/1
|1/1
|C
|
|-
|chroma
|1/13 - 1\5
|1/1 - 1\13
|C♯
|Doesn't occur directly in the scale, but very useful to note anyways
|-
|m2
|1/1 - 1\13
|1\13 - 1\8
|D♭
|
|-
|M2
|2\13 - 1\5
|1\8 - 2\13
|D
|
|-
|m3
|1\5 - 2\8
|3\13 - 1\4
|E♭
|
|-
|M3
|4\13 - 2\5
|1\4 - 4\13
|E
|
|-
|d4
|1\5 - 4\13
|4\13 - 3\8
|F♭
|Enharmonically equivalent to M3 in basic tuning ([[13edo]])
|-
|P4
|5\13 - 2\5
|3\8 - 5\13
|F
|
|-
|t
|2\5 - 6\13
|6\13 - 1\2
|G♭
|Minor Tritone
|-
|T
|7\13 - 3\5
|1\2 - 7\13
|G
|Major Tritone
|-
|P5
|3\5 - 8\13
|8\13 - 5\8
|H
|
|-
|A5
|9\13 - 4\5
|5\8 - 9\13
|H♯
|Enharmonically equivalent to m6 in basic tuning ([[13edo]])
|-
|m6
|3\5 - 9\13
|9\13 - 3\4
|A♭
|
|-
|M6
|10\13 - 4\5
|3\4 - 10\13
|A
|
|-
|m7
|4\5 - 11\13
|11\13 - 7\8
|B♭
|
|-
|M7
|12\13 - 2/1
|7\8 - 12\13
|B
|
|-
|8
|2/1
|2/1
|C
|Assuming justly-tuned octave
|}
Additionally, the eight modes of the Oneirotonic scale have been given names based on the Dreamlands from the works of H.P. Lovecraft:
{| class="wikitable"
|+Oneiro Modes
!Brightness
!Pattern
!Name
!Notation
!Notes
|-
|7
|LLsLLsLs
|Dylathian
|C - D - E - F - G - H♯ - A - B - C
|
|-
|6
|LLsLsLLs
|Ilarnekian
|C - D - E - F - G - H - A - B - C
|Natural major mode
|-
|5
|LsLLsLLs
|Celephaïsian
|C - D - E♭ - F - G - H - A - B - C
|Harmonic minor mode
|-
|4
|LsLLsLsL
|Ultharian
|C - D - E♭ - F - G - H - A - B♭ - C
|Natural minor mode
|-
|3
|LsLsLLsL
|Mnarian
|C - D - E♭ - F - G♭ - H - A - B♭ - C
|
|-
|2
|sLLsLLsL
|Kadathian
|C - D♭ - E♭ - F - G♭ - H - A - B♭ - C
|
|-
|1
|sLLsLsLL
|Hlanithian
|C - D♭ - E♭ - F - G♭ - H - A♭ - B♭ - C
|Natural diminished mode
|-
|0
|sLsLLsLL
|Sarnathian
|C - D♭ - E♭ - F♭ - G♭ - H - A♭ - B♭ - C
|
|}

== Chords of Oneirotonic ==
Unlike in the diatonic scale, the Oneirotonic scale has its generator an odd number of steps above the root, which means that it cannot be split into two intervals of the same category; because of this, the Oneirotonic scale does not have a dichotomy of tonalities.

Rather than the dichotomy, there are two primary ways to categorize useful chords in Oneirotonic: tertiary chords (which generally display a three-way distinction), and genspan chords (which have up to a five-way distinction).

=== Tertiary Triads ===
The first class of chords is built by stacking major and minor thirds; as such, I will call this class of chords "tertiary."  There are three types of tertiary triads which occur in the Oneirotonic scale, and one type which can be extrapolated:
{| class="wikitable"
|+Tertiary Triads
!Quality
!Symbol
!Formula
!Notation
!Modes
|-
|Major
|
|M3 + m3
|C - E - G
|Dylathian, Ilarnekian
|-
|Minor
|
|m3 + M3
|C - E♭ - G
|Celaphaïsian, Ultharian, Mnarian
|-
|Diminished
|
|m3 + m3
|C - E♭ - G♭
|Kadathian, Hlanithian, Sarnathian
|-
|Augmented
|
|M3 + M3
|C - E - G♯
|N/A
|}
As can be seen here, the distribution between major, minor, and diminished triads in Oneirotonic is rather even.  In hard tunings, these chords may provide a useful place of rest, since the major triad resembles [[13/10|10:13:15]] and the diminished triad resembles [[6:7:8:9|6:7:8]].  In soft tunings, however, these chords are much less stable, as the tritones approach the semioctave.

MODMOS scales of Oneirotonic can be designed to target specific combinations of triads, including the usage of augmented triads.

=== Genspan Triads ===
The other primary approach to chords in Oneirotonic is to use chords that split the perfect fifth into two unequal parts; I will hereby refer to these as "genspan" chords, as they have a span of one generator.  The fifth can be broken up into the sum of a third and fourth; because the two constituent intervals do not fall on the same degree, each mode of the scale has two genspan triads over the root, rather than just one.
{| class="wikitable"
|+
!Quality
!Symbol
!Formula
!Notation
!Modes
|-
|Fourth
|C<sup>4</sup>
|P4 + m3
|C - F - H
|Ilarnekian, Hlanithian
|-
|Third
|C<sup>3</sup>
|M3 + d4
|C - E - H
|Ilarnekian
|-
|Flat-Fourth
|C<sup>♭4</sup>
|d4 + M3
|C - F♭ - H
|Sarnathian
|-
|Flat-Third
|C<sup>♭3</sup>
|m3 + P4
|C - E♭ - H
|Celephaïsian, Sarnathian
|}
In soft tunings, these chords are useful resolutions, since C<sup>4</sup> resembles [[6:8:9]], and C<sup>3</sup> resembles [[4:5:6]].  In hard tunings, however, these chords are much less stable, as C<sup>4</sup> better resembles 13:17:20 and C<sup>3</sup> better resembles 13:16:20.

== Chord Functions ==
The tertiary chords and genspan chords occupy two distinct sects of Oneirotonic harmony, and effectively "switch places" between soft and hard tunings of the scale; in this respect, I find them comparable to the tertiary and suspended chords in Diatonic, which similarly seem to "switch places" between soft and hard tunings.

Because of this dichotomy, I will be outlining two primary sets of chord functions: those that describe soft Oneirotonic, and those that describe hard.

=== Soft Oneiro Functions ===
In soft tunings of the Oneirotonic scale, the Fourth genspan chord will be used as the primary tonic function, since that chord is the most consonant, and is included in six of the eight modes, making it an efficient choice for a general resolution.

==== Dominant ====
In order to define a dominant chord, we need three primary features:

# Creates satisfying motion around a relevant circle, especially a generator
# Contains a tension that resolves cleanly to the tonic chord
# Is regularly contained within the same mode as the tonic

While there are a number of chords that satisfy one or two of these properties, there are very few that satisfy them all; however, the Third genspan chord built on the fifth degree of the scale manages to rise above those other options.  It creates motion around the generator, which makes it relevant to many scales and allows secondary dominants to be generalized; it creates tension due to the third above the fifth degree being a leading tone into the tonic; and allowing it to be altered to a Flat-Third chord makes it recognizably appear in four of the six modes that contain the tonic Fourth chord.