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::|CONTENTS

1. Scales, Major and Minor Scales, Intervals
2. Chord Theory
3. Chord Progressions and Cadences
4. Seventh and Extended Chords
5. Other Chords of Note
6. Other Modes
7. Non-Diatonic Chords
8. Modulation
9. More Scale Types
10. Non-Conventional Harmony
11. Helpful Links and References

**THIS ARTICLE IS UNDER CONSTANT CONSTRUCTION AND EDITING!**

Harmony theory is split from the other page because there is so much content. For other aspects of music theory see here.

TODO: someone else add a more elaborate intro here EDIT

Scales, Major and Minor Scales, Intervals

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A scale is a set of notes arranged in order by their pitch. A mode assigns a tonic to one of the notes. A key further assigns an absolute pitch to this tonic.

The major scale consists of a root note, that note's major second, its major third, its perfect fourth, its perfect fifth, its major sixth, its major seventh, and its octave. It is also known as the Ionian mode.
So, we have a pattern of whole, whole, half, whole, whole, whole, half.
The C major scale is C, D, E, F, G, A, B, C.

The (natural) minor scale consists of a note, its major second, its minor third, its perfect fourth, its perfect fifth, its minor sixth, its minor seventh, and its octave. It is also known as the Aeolian mode.
We have a pattern of whole, half, whole, whole, half, whole, whole.
The C minor scale is C, D, E♭, F, G, A♭, B♭, C.

The scale degree of a note is simply its position relative to the first note of the scale, e.g. the scale degrees of E and G in the C major scale are 3 and 5 respectively, so are E♭ and G in the C minor scale. Sharps or flats can be applied to them as well.

Now that we have our 12 notes, we have to figure out how they work relative to each other. An interval is the relative difference between two pitches, consisting of a quality, and a distance in terms of scale degrees in the Major key.

A way of looking at relative pitch is examining consonance and dissonance, which are respectively defined as when notes, when played simultaneously, either work 'well' together, or 'clash' with each other.
The intervals relative to a base (or tonic) are displayed below. Those italicized are relatively important!

1: 0 semitone difference = unison (the same note)
♭2, m2: 1 semitone difference = minor second
2, M2: 2 semitone difference = major second
♭3, m3: 3 semitone difference = minor third
3, M3: 4 semitone difference = major third
4, P4: 5 semitone difference = perfect fourth
♯4/♭5, A4/d5: 6 semitone difference = augmented fourth/diminished fifth/tritone
5, P5: 7 semitone difference = perfect fifth
♭6, m6: 8 semitone difference = minor sixth
6, M6: 9 semitone difference = major sixth
♭7, m7: 10 semitone difference = minor seventh
7, M7: 11 semitone difference = major seventh
8, P8: 12 semitone difference = perfect octave

This is rather confusing at first glance. However, there seem to be recognizable patterns. For example, a minor <x> is always closer to the tonic than a major <x>.
The perfect intervals are the most consonant intervals, other than the octave!

Notice that only the 2nd, 3rd, 6th and 7th intervals can use "major" or "minor" as the quality, whereas "perfect" applies to 4th, 5th, and 8th. "Augmented" means to raise one semitone from a major/perfect interval, "diminished" lowers a semitone from a minor/perfect interval.

It is possible to rank these intervals by their consonance:
- perfect octave
- perfect fifth
- perfect fourth (can also be more dissonant than 3rd/6th in some cases)
- third, sixth
- second, seventh
- all augmented / diminished intervals are dissonant
All this has to do with frequency intervals and ratios (see here).

Intervals are not the same as scale degrees; to avoid ambiguity, a caret (^) is usually put on top of scale degrees, and accidentals superscripted to the right of the number. Hence 6̂^♭ always refers to A♭ in C Major, but ♭6 simply means a distance of 8 semitones that corresponds to a sixth.

Chord Theory

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Now that we've learned the two most widely-used scales, we can assign chords to them. A chord is three or more notes played either simultaneously, or sequentially. A chord played sequentially is called a broken chord or an arpeggio.
A chord is formed by stacking thirds on top of a lowest scale note, called the root note, along the scale on which the chord is built. The simplest case consists of the root, the third, and the fifth; this is called a triad. Examples:
D triad on C major scale: D-F-A
A♭ triad on C minor scale: A♭-C-E♭
Later we will see more advanced chords constructed from other means.

The placement of chord tones gives rise to a voicing. Each chord tone may be transposed to any octave, repeated across multiple octaves, or even omitted in certain cases, as long as the root note remains the bass (but see below for "inverted chord"). In the default construction, the stacked thirds are as close to each other as possible; this is called close position, anything else is in open position.

In Roman relative chord notation, all chords are represented by a Roman numeral. The value of the numeral determines the scale degree of the chord's root. The major chords (which use a major third) are represented by capitalized numerals, while minor chords (which use the minor third) and diminished chords (which uses a minor third and flats the fifth to a tritone) are represented by a lowercase, with the diminished chords differentiated with a small circle next to the numeral.
In a major scale, the tonic chord is represented by a Roman numeral I.
The chords are named as follows:

I = tonic
II = supertonic
III = mediant
IV = subdominant
V = dominant
VI = submediant
VII = leading-tone

We will get into how each chord is used shortly.

The major scale has the following chords:

I ii iii IV V vi vii°

The natural minor scale has the following chords:

i ii° III iv v VI VII

As you can see, both of these (and all other modes of the diatonic scale) have three major chords, three minor chords, and a single diminished chord. The order is even the same! The only difference is which chords take prevalence in the mode. This leads us to one of my favorite revelations in all of music theory.
A relative key is a key that has the same notes and chords as another key, but the parallel key is just shifted to become major or minor! Case example:
C major consists of C, D, E, F, G, A, and B. Therefore, its chords are:

I - C = C-E-G
ii - Dm = D-F-A
iii - Em = E-G-B
IV - F = F-A-C
V - G = G-B-D
vi - Am = A-C-E
vii° - B° = B-D-F

Now, shifted forward so that vi becomes i, we get A minor!

i = Am = A-C-E
ii° = B° = B-D-F
III = C = C-E-G
iv = Dm = D-F-A
v = Em = E-G-B
VI = F = F-A-C
VII = G = G-B-D

It's almost magic to me! And you can do this with any mode, even into modes that are not major or minor! (See here)

A parallel key is a key that has the same root note as another key, but has different notes, and therefore, different chords! C major and C minor are parallel keys!

Chord Progressions and Cadences

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A chord progression is a sequence of chords played one after another, usually looping several times.

A cadence is a portion of a chord progression that 'leads' to another section in a piece, whether it be a new progression, or the same progression.

An authentic cadence has only one form: V -> I. This just sounds right to our ears, and it's used in the majority of music, ever since the Baroque period.

A plagal cadence is IV -> I. In contemporary pop and rock, this is used even more than the authentic cadence, by a significant amount.

An inauthentic cadence is literally any other cadence which is not V -> I. These tend to fake us out, or give a feeling of dissonance. They can be well-applied, however.

TODO: T / S / D properties of diatonic chords, their implications to chord progression tendencies

Common chord progressions are:
I - V - vi - IV, the famous 'Four Chords' used in many pop and rock hits.
I - V - vi - iii - IV - I - IV - V, the progression made famous by Pachelbel's Canon in D.
i - VII - VI - V, the Andalusian cadence.
I - ii - V
I - V - IV
I - IV - V
I - V
I - IV

TODO: chord progressions in minor. harmonic minor scale. difference between subtonic and leading-tone

Seventh and Extended Chords

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A seventh chord is a triad that has a seventh tacked on to it. There are five main types of seventh chords:

A dominant seventh chord adds the minor seventh of the key to a major triad. For example a C dominant 7th chord (C7) is C-E-G-B♭.
A minor seventh chord adds the minor seventh to a minor triad. For example a C minor 7th chord (Cm7) is C-E♭-G-B♭.
A major seventh chord adds the major seventh to a major triad. For example a C major 7th chord (CM7 or Cmaj7) is C-E-G-B.
A half diminished seventh chord adds the minor seventh to a diminished triad. For example a C half diminished chord (C^ø7 or Cm7♭5) is C-E♭-G♭-B♭.
A fully diminished seventh chord adds the major sixth to a diminished triad. For example a C full diminished seventh chord (C^o7 or Cdim7) is C-E♭-G♭-A (B♭♭).

There are plenty of other seventh chords which are less common and you can explore those yourself.

An extended chord is a chord with notes beyond the octave of a root note. To learn about these, we need to define more intervals.

♭9: 13 semitone difference = minor ninth (octave+minor second)
9: 14 semitone difference = major ninth (octave+major second)
♭10: 15 semitone difference = minor tenth (octave+minor third)
10: 16 semitone difference = major tenth (octave+major third)
11: 17 semitone difference = perfect eleventh (octave+perfect fourth)
♯11/♭12: 18 semitone difference = augmented eleventh / diminished twelfth (octave+tritone)
12: 19 semitone difference = perfect twelfth (octave+perfect fifth)
♭13: 20 semitone difference = minor thirteenth (octave+minor sixth)
13: 21 semitone difference = major thirteenth (octave+major sixth)
♭14: 22 semitone difference = minor fourteenth (octave+minor seventh)
14: 23 semitone difference = major fourteenth (octave+major seventh)
15: 24 semitone difference = double octave (octave+octave)

A ninth chord adds some sort of ninth to a seventh chord, usually a major ninth to a dominant seventh to for the dominant ninth chord. For example, a C9 chord is C-E-G-B♭-D. This pattern of tacking on notes can be repeated, especially with elevenths and thirteenths. These additional notes are called tensions. Changing the perfect fifth into ♭5 or ♯5 is also called a tension.

Tensions are spelt with odd numbers instead of even numbers, regardless of the chord's voicing. With this, all 12 chord factors can be spelled using the intervals as follows:
1, ♭9, 9, ♭3/♯9, 3, 11, ♭5/♯11, 5, ♭13/♯5, 6/13, 7, maj7.
The underlined intervals are called altered tensions or simply alterations. Any seventh or extended chord consisting of at least one altered tension is called an altered chord.

The spelling of extended chords follows a similar pattern as triads and seventh chords. The root is first spelled, followed by the quality of the third / seventh, and then the highest unaltered tension, and finally a bracketed list of all alterations. Example:
C-E-G-B-D♭-F♯ = Cmaj7(♭9,♯11)
E-G♯-B♯-D-F♯♯(G) = E7(♯5,♯9)
A-B♭-C-D♭-E♭-F-G = Am7(♭5,♭11,♭13) = A7(♭5,♯5,♭9,♯9)
As seen in the last one, tensions may be replaced with their enharmonic equivalents, and both alterations of the same interval may occur at the same time. In particular, this is sometimes called as the altered chord, and notated using "A7alt". Furthermore, this is an example of representing a mode as a single extended chord and, unsurprisingly, this is called the altered scale.

Adding ninths, elevenths, and thirteenths gives the chord more color and more of a jazzy feel, but they can be tricky to use at first. It is often undesirable to play all notes of an extended chord, since this results in a muddy sound. The following intervals are retained:
- major/minor third
- major/minor seventh
- the highest tension in the chord
- all altered tensions
The following are often omitted:
- root (the bass note handles this, whether it is the same as the chord's root)
- perfect fifth
- perfect eleventh
- major ninth

Other Chords of Note

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A slash chord is a chord with the same notes, but the root of the chord is not the bass note, in other words, it doesn't have the lowest pitch. An inverted chord is a special case where the root reappears in the high notes of the chord.
For every chord, there are at least two inversions.

For example: The C major chord is composed, in order, of the notes C, E, and G.
Its first inversion would therefore be E, G, C (because the C has to be moved up an octave)
Its second inversion would be G, C, E.
A first inversion chord can be notated either as C/E (in the case of C Major) or as I⁶ in relativistic notation.
A second inversion would be notated as C/G or as I⁶₄.
As a note, seventh chords can have three inversions, due to the extra note. For a G dominant seventh chord, they would be notated as: G7 (V⁷), G7/B (V⁶₅), G7/D (V⁴₃), G7/F (V⁴₂).

A fifth chord or power chord, often used in rock music or punk, consists of only a root note, its perfect fifth, and its octave. It is neither major nor minor, due to its lack of a third.
Another thirdless chord is the suspended chord, which suspends the third, and replaces it with either a major second (a suspended second chord) or a perfect fourth (a suspended fourth chord). In C, these can be notated as Csus2, and Csus4, respectively.

The augmented chord is a major triad, with the fifth sharpened to an augmented fifth. It can be represented with a superscripted plus sign, for example, a C augmented chord can be represented as C⁺.

A sixth chord is a thirteenth chord with the seventh omitted. It behaves as if the thirteenth replaces the seventh. If the chord has no tensions, it will be equivalent to the first inversion of another seventh chord. For example:
C6 = Am7/C = C-E-G-A
Cm6 = Am7♭5/C = C-E♭-G-A
A sixth chord may also have a ninth and/or eleventh note. A "seventh-sixth chord" is really just a special voicing of a thirteenth chord with a 9th but no 11th.

Other Modes

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We know by now two scales! You may have been wondering that if we can shift the major scale a certain amount to become the minor scale, can't we shift it different amounts to become different scales? And indeed, we can!
These other modes are listed below, by the chord from the major scale that becomes the tonic.

ii - Dorian
iii - Phrygian
IV - Lydian
V - Mixolydian
vii° - Locrian

These modes are not as commonly used as their major/minor counterparts, but are more common than one may think, and are often confused by novices for major/minor in various pieces. Each of these modes have their own unique feels different from minor and major.

Here are the scale modes we have looked at so far in C going from "most sharp" to "most flat":

C Lydian: C, D, E, F♯, G, A, B, C = Cmaj13(♯11)
C Major: C, D, E, F, G, A, B, C = Cmaj13
C Mixolydian: C, D, E, F, G, A, B♭, C = C13
C Dorian: C, D, E♭, F, G, A, B♭, C = Cm13
C Minor: C, D, E♭, F, G, A♭, B♭, C = Cm11(♭13)
C Phrygian: C, D♭, E♭, F, G, A♭, B♭, C = Cm11(♭9,♭13)
C Locrian: C, D♭, E♭, F, G♭, A♭, B♭, C = Cm11(♭5,♭9,♭13)

Non-Diatonic Chords

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You don't just have to use the triads provided to you by a key! A borrowed chord is a chord that is not in the key, but is instead borrowed from another, usually closely-related key. In contrast, a diatonic chord belongs exactly to the key. A related key is a key that shares many similar chords to another key. For example, in A major, C and G are not chords in that key, but they are present in the parallel minor key! The use of parallel major/minor keys is called major-minor mixture. There are at least two families of borrowed chords: (actually the latter is pretty rare)

Parallel minor in major key: i - ii^o - ♭III - iv - v - ♭VI - ♭VII
Parallel major in minor key: I - ii - ♯iii - IV - V - ♯vi - ♯vii^o

If this related key is neither a major nor minor key, but still shares the same tonic, the generalized version is known as modal mixture. Many extended chords can be derived as chords borrowed from these modes. Example:

Mixolydian in minor key: I - ii - ♯iii^o - IV - (v) - ♯vi - (VII)

There is much discrepancy among how to represent extensions in these Roman numerals.

Another way of obtaining chords outside the current key is to assume that one of the diatonic chords becomes the tonic locally. This process is known as tonicization. It may not seem to make much sense at first, but this actually happens all the time.

The classes of chords that appear frequently in tonicized passages are called applied chords, as follows where "?" is the chord that follows:
- Applied dominant, notated "V/?", always a dominant (seventh) chord;
- Applied leading-tone, notated "vii^o/?", can be half or fully diminished if the seventh is present;
- Applied supertonic, notated "ii/?", can be minor or (half-) diminished;
- Applied subdominant, notated "IV/?", often major.

Applied chords themselves can be tonicized! A circle-of-fifth progression from E to C can be notated as "V/V/V/V - V/V/V - V/V - V - I". When the target chord is diatonic, the applied one is called a secondary chord.

V/ and vii^o/ have dominant properties, and must lead towards their target chords. IV/ and ii/ have subdominant properties so they can only appear before V/ or vii^o/ leading to the same chord.

Finally, a number of special subdominant chords appear frequently in music theory: (they are identical whether in a major or minor key)
- The Neapolitan sixth, notated N⁶, is equivalent to ♭II⁶, e.g. D♭/F in the key of C;
- The French sixth, notated Fr⁺⁶, is equivalent to the second inversion of the 7(♭5) chord on the supertonic, e.g. D7(♭5)/A♭ in the key of C;
- The German sixth, notated Gr⁺⁶, is like the French sixth but with an extra ♭9, also enharmonic to a dominant seventh chord, e.g. A♭7 in the key of C; (A♭7 = (D)-F♯-A♭-C-E♭)
- The Italian sixth, notated It⁺⁶, omits the root from the French sixth, e.g. A♭-C-F♯ in the key of C.

Modulation

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Modulation is the act of changing keys in a single piece. It is usually achieved through a cadence involving a chord in the new progression, or through just using the next key or two up (colloquially known as a truck driver's gear change)

Examples:
C major -> C♯ major
C major -> G major (G is the V chord in C major)
C major -> C minor

TODO: common-tone, tonicization, chromatic mediant, constant structure, tritone substitution

More Scale Types

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A pentatonic scale is essentially a stripped-down version of a regular scale, that is very useful for composition and improvisation. The major pentatonic scale has only the root, major second, major third, perfect fifth, major sixth, and octave. The minor pentatonic scale uses the root, minor third, perfect fourth, perfect fifth, minor seventh, and octave. Pentatonic scales can be shifted to other modes as well.
The C major pentatonic scale is C, D, E, G, A, C.
The C minor pentatonic scale is C, E♭, F, G, B♭, C.

Another scale we can derive from these pentatonic scales are the blues scales. These scales are often used in jazz.
The C minor blues scale (usually simply known as "blues"): C, E♭, F, F♯, G, B♭, C.
The C major blues scale (less common): C, D, D♯, E, F, G, A, C.

TODO: move this to above
In other styles of music such as jazz, modern classical, and traditional music from other cultures, other scales may be used in addition to or in place of the diatonic scale. One common example is the harmonic minor scale, equivalent to a minor scale with a major seventh instead of a minor seventh. By changing the root of this scale, we can create new modes such as Phrygian dominant (equivalent to the diatonic Phrygian mode with a major third).
The C harmonic minor scale is C, D, E♭, F, G, A♭, B, C.

The melodic minor scale is a special case, as it has different notes, depending on whether you are ascending or descending it. Ascending, it is the minor scale, or natural minor scale, with the minor sixth and minor seventh sharpened to their major counterparts. Descending, the scale is identical to the natural minor scale.
The C melodic minor scale ascending is C, D, E♭, F, G, A, B, C.
The C melodic minor scale descending is C, B♭, A♭, G, F, E♭, D, C.

The Egyptian scale uses a root, a minor third, a major third, an augmented fourth, a perfect fifth, a minor seventh, a major seventh, and octave.
The C Egyptian scale is C, E♭, E, F♯, G, B♭, B, C.

Whole-Tone Scales
As the name implies, whole-tone scale is made up entirely of tones (or major seconds). Unlike a conventional scale like the major scale, the pattern goes "tone, tone, tone, tone, tone, tone."
There are only two different transpositions of the whole-tone scale available:
C, D, E, F♯, G♯, A♯, C
and
C♯, D♯, F, G, A, B, C♯

Whole-tone scales are a simple yet effective way of disguising tonality. As there are no semitones or leading notes in this scale, there's no sense of resolution. People often say that this scale gives a "mysterious" feel to a piece of music.
Debussy's "Voiles" is a good example of a piece of music that is almost entirely composed of whole-tone scales.

Octatonic Scales
In the same way that a pentatonic scale is a scale with 5 different tones, an octatonic scale is one that uses 8 different tones. It is also sometimes referred to as a "diminished scale" in jazz theory, because it can be obtained through a combination of two diminished 7th chords. The intervals in this scale alternate between a semitone gap and a tone gap (so the pattern goes; semitone, tone, semitone, tone, semitone, tone, semitone, tone).

An octatonic scale starting on C would go: C, D♭, D♯, E, F♯, G, A, B♭, C.
This version of the octatonic scale can be transposed twice. If we transpose up by a semitone each time, we can get:

C♯, D, E, F, G, A♭, A♯, B, C♯.
D, E♭, F, G♭, G♯, A, B, C, D.

If we transpose up by a semitone again, we get D♯ E F♯ G A A♯ C C♯, which if we compare it to our original octatonic scale on C, we notice that we end up with the same notes, but starting on D♯.

There are two different versions of the octatonic scale. The first one we've just seen started off with a semitone interval. The other type starts off with a tone interval, and then alternates between tone and semitone. So, if we start on C, we get:

C, D, E♭, F, G♭, G♯, A, B, C.

Just like the other type of octatonic scale, this one can be transposed twice. Again, using this scale can be effective when combined with other scale types. Octatonic scales are more difficult to listen out for due to their highly chromatic nature.

Non-Conventional Harmony

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Microtonality
Microtonality is the use of pitches in intervals that are smaller than a semitone. Microtonality is relatively uncommon, due to its jarring nature to those introduced to it, and it difficulty to perform on most instruments due to their being designed for music in standard semitone intervals.

Atonality and Chromaticism
The chromatic scale uses all twelve notes, and is usually associated with chromaticism, which is the synthesis of different scales, or atonality, which is the lack of a musical center (aka, a root note).
Atonality is usually coupled with an artistic statement which is similar to visual art styles such as abstract-expressionism or cubism: that music (or art) does not have to sound (or be) a certain way, and therefore does not need a tonal center, in other words, no key. Atonality also frequently relies on the equal treatment of all 12 notes of the chromatic scale. This doesn't mean it has to sound bad, or even strange, many pieces such as Schoenberg's 2nd String Quartet (4th movement) pioneered modern atonality, and are still pleasant by normal standards. Atonality is often coupled with widely varying time signatures and tempo.

Bitonality and Polytonality (or polymodality)
Bitonality is the use of two key signatures, or two "tonalities" across different parts at the same time (also referred to as vertical harmony). This can be achieved by having one part or set of instruments in one key signature (for example, C major), and other part or set in a different, often contrasting key signature (for example, F♯ Major). Combining different chords or key signatures can produce a range of different effects in a piece. Two consonantly related keys/chord (for example, tonic and dominant) can produce lush or more pleasant sounding chords, whereas less closely related chord combinations (tonic and supertonic) can be more clash-y and very dissonant.

Igor Stravinsky's Rite of Spring is a good piece to study for this sort of technique. The first dance combines both an E major and E♭7 chord in the lower strings to produce a brilliantly dissonant sound.

Bi/polymodality is based on the same principle, except you're combining "modes" rather than scales or key signatures. One example of this would be combining a pentatonic scale starting on E with a whole tone scale in another instrument.

Helpful Links and References

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Arps
List of Scales
Music Theory - Case Examples and Analysis
Music Theory - History of Modern Theory and Tonality
Hooktheory

- fantastic for testing melodies and chord progressions!