User:VectorGraphics/Xenxenharmonic theory

We start with an overview of the notes as they appear in the most common paradigm:

The heptatonic scale and pentatonic subsets

There are seven tones within the octave, spaced equally in pitch, such that the step from one note to the next represents a fixed unit of pitch, which will be used to define the intervals. These are labelled [0] (the root), [1], [2], [3], [4], [5], [6]. [7] is the same pitch class as [0], and so is [-7], so the numbers [0] through [6] alone are used to refer to the 7 notes.

While all seven tones are used, only five are used at any one time; this is done by omitting two notes that are spaced three steps apart (i.e. [1] and [4], [5] and [1] (remember, [1] is equivalent to [8]), [3] and [6].) The root [0] cannot be omitted, as such there are five pentatonics on [0], which are all rotations of each other.

Basic form (Rotation A): [0], [1], [2], [4], [5] (chaotic, angry, scary)

Rotation B: [0], [1], [3], [4], [5] (dark, thoughtful, serious)

Rotation C: [0], [1], [3], [4], [6] (simple, balanced)

Rotation D: [0], [2], [3], [4], [6] (fun, innocent, bright)

Rotation E: [0], [2], [3], [5], [6] (quirky, goofy, exciting)

Here the rotations are arranged in traditional order, based on emotional associations; there is also an association to differences in standing - Rotation A embodies the extreme of something grand or superior; Rotation E something inconsequential or humorous.

The pitch labelled [0] is variable; it regularly moves around during a song and across songs. In order to specify the motion of [0] more precisely, we label a single note (0), which is set at a standard, universal pitch; notes count up and down in steps from there, without repeating at the octave. Then, the position of [0] in the middle octave is set relative to (0).

Intervals

Intervals measure the distance between two notes, in steps. There are two forms of intervals: melodic and harmonic - melodic intervals (steps or jumps up or down) appear between the upper notes in harmony (i.e. shifting from [0 2] to [0 3] is an ascending tone), and with modulations (i.e. a modulation from [0] (0) to [0] (-2) is a descending bitone); harmonic intervals (between two notes occurring at once) make up chordal harmony.

The nature of the interval is largely consistent when reflected across the octave, but usually, the wider step neutralizes the feel, making it milder. This is most seen with the tone: the tenseness of the tone above the root becomes a relatively neutral hexatone when inverted, leading to the distinction in tension between two sets of rotations that either have the tone or lack it. Additionally, the tone is a highly directional interval: an ascending tone is resolved and pointed, but a descending tone is disoriented and hesitant.

Interval	Size	Melodic	Harmonic
Tone	1 step	*	Tense
Bitone	2 steps	Twisted, confused, ambiguous	Sharp
Tritone	3 steps	Balanced, simple	Smooth

Harmony

The chordal harmony is characterized by a constant root; varying the note played on top of it varies the harmonic interval, and thus the mood. Shifts are done via direct modulation, shifting the [0] and the scale around in absolute pitch (creating melodic intervals), and parallel modulation, shifting the scale to a new rotation but keeping [0] in place, altering the set of harmonic intervals available over the root.

Melody

Melody generally "follows" the harmony, but with extra ornamentation in the form of additional notes, repeated notes, rhythm, etc. The intervals in melody share the distinct melodic interval qualities, but larger jumps are seen as important, and are often held longer. For example, you might encounter tones as a walk up to a more significant melodic note, but a hexatone would be held long, to compensate for the neutralized feel of the hexatone.

Intonationality

Up to this point, a brief overview of music has been provided involving the equal heptatonic. However, just as selecting a pentatonic can define the general feel of a song, you can also select an unequal heptatonic, allowing you an additional dimension of control over the overall feel of the song. The use of unequal heptatonics, and the practices derived from such usage, are called intonationality.

There are six simple unequal heptatonics, each of which have seven rotations; all six have their own corresponding number of substeps per octave in order to create them - these six heptatonics are constructed such that each single interval in the equal heptatonic becomes at most two in each of the unequal heptatonics. Each pair of unequal heptatonics has a pair of maximally consistent intervals, which are reflections of one another.

Heptatonic	Pattern	Substeps	Maximally consistent interval
Supratonic-A	LLLLLLs	20	Tone, hexatone
Supratonic-B	LLsLLLs	19	Tritone, tetratone
Supratonic-C	LLsLsLs	18	Bitone, pentatone
Subtonic-C	ssLsLsL	17	Bitone, pentatone
Subtonic-B	ssLsssL	16	Tritone, tetratone
Subtonic-A	ssssssL	15	Tone, hexatone

To explore the varieties of interval found in each category, we will first have to discuss just intonation. Because pitch is logarithmic, intervals correspond to pitch ratios. When the ratio is close to a simple integer ratio, the interval sounds more stable than otherwise. In the equal heptatonic, this is true for the tritone and tetratone (with ratios of 4:3 and 3:2 respectively). In contrast, the ditone sits right in between two simple ratios.