**THIS ARTICLE IS UNDER SOME SERIOUS CONSTRUCTION!
A lot of the time, you'll hear people that claim music theory is not important, or maybe even that it stifles creativity. Often cited are case examples such as Paul McCartney's inability to read sheet music. While it is absolutely true that music theory is not required to write great music, for some people, it can help greatly, and introduce new ways to express oneself through music.

Music theory is a set of concepts and a language you can use to communicate ideas about music with other people. The main benefits of learning about music theory are that you will be able to express musical ideas in a concise and accurate manner and be able to think about music in a structured way.

'[t[4]I. The Notes and Keeping Time'[/t]

A '[b]pitch'[/b] is a specific frequency of sound. This frequency may be expressed in '[b]Hertz'[/b] (Hz), a unit that measures cycles per second--in this case, oscillations per second of a pressure wave. Because of way the human ear and mind perceive sound, a pitch with a frequency that is an integer multiple of another pitch's frequency will sound very similar to it--for example, 440 Hz and 880 Hz. The pitches are clearly distinguishable, but seem to share a certain quality. Because of this effect, music has the concept of a '[b]pitch class'[/b]: a set of pitches with the same name that are considered more or less equivalent for the purposes of harmony (for example, "C"). In music, the most common way to refer to a pitch is by a combination of its pitch class and octave, such as "A4", a pitch of 440 Hz in modern tuning.

The pitch classes are as follows:
A, A♯, B, C, C♯, D, D♯, E, F, F♯, G, G♯

'[t[2]Note Value'[/t]

The '[b]value'[/b] of a note is an expression of how much time it occupies.

A '[b]quarter note'[/b] is a quarter of a common time measure (see below). As an example, a 7/4 measure can hold seven quarter notes, while a 2/4 only two.
An '[b]eighth note'[/b] simply takes up half the time of a quarter note.
A '[b]half note'[/b] is twice the length of a quarter note.
A '[b]whole note'[/b] takes up an entire common time measure.

In sheet music, adding a single dot after a note increases its duration by half; thus, a '[b]dotted quarter note'[/b] means a note with a length or '[b]value'[/b] of a quarter note plus an eighth note.

'[t[2]Time Signatures'[/t]

A '[b]measure'[/b] is a unit in music which defines the length of an arbitrary phrase, sort of like a coherent musical statement. Measures are subdivided into equal time intervals called '[b]beats'[/b]. A '[b]time signature'[/b] is the way to express the number of beats in a measure. The time signature most often used in Western popular music is 4/4, also called '[b]common time'[/b].

The top number in a time signature tells how many beats are in a measure, and the bottom tells what note value makes up a single beat (ordinary a quarter, eighth, or sixteenth note). A time signature is '[o]NOT'[/o] a fraction--3/4 are 6/8 denote two different meters (by convention, 3/4 denotes a measure split into 3 quarter-note pulses, and 6/8 denotes a measure split into 2 dotted-quarter-note pulses).

'[t[2]Tempo'[/t]
'[b]Tempo'[/b] is the speed a piece is played at, and it is usually measured in '[b]beats per minute'[/b]. For an example, a piece at 100 beats for minute (BPM) has a beat take place 100 times a minute. Pretty simple.

'[t[4]II. Intervals and The Ionian (Major) and Aeolian (Minor) Modes'[/t]
Now that we have our 12 notes, we have to figure out how they work relative to each other. A way of looking at relative pitch is examining '[b]consonance'[/b] and '[b]dissonance'[/b], which are respectively defined as when notes, when played simultaneously, either work 'well' together, or 'clash' with each other.
As a rule of thumb, the closer notes are together, the more they clash, until the difference becomes null to our brain. Notes that are one or two semitones apart are generally said to be dissonant. This has to do with frequency intervals and ratios (see Section VI).
The intervals relative to a base (or '[b]tonic'[/b]) are displayed below. Those '[o]italicized'[/o] are relatively important!

0 semitone difference = unison (the same note)
1 semitone difference = '[o]minor second'[/o]
2 semitone difference = '[o]major second'[/o]
3 semitone difference = '[o]minor third'[/o]
4 semitone difference = '[o]major third'[/o]
5 semitone difference = '[o]perfect fourth'[/o]
6 semitone difference = augmented fourth
7 semitone difference = '[o]perfect fifth'[/o]
8 semitone difference = minor sixth
9 semitone difference = major sixth
10 semitone difference = minor seventh
11 semitone difference = major seventh
12 semitone difference = '[o]octave'[/o]

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

The '[b]major scale'[/b] or '[b]Ionian mode'[/b] 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.
So, we have a pattern of whole, whole, half, whole, whole, whole, half.

The '[b]minor scale'[/b] or '[b]Aeolian mode'[/b] 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.
We have a pattern of whole, half, whole, whole, half, whole, whole.

'[t[4]III. Chord Theory'[/t]
Now that we've learned the two most widely-used scales, we can assign '[b]chords'[/b] to them. A chord is three or more notes played either simultaneously, or sequentially. A chord played sequentially is called a '[b]broken chord'[/b] or an '[b]appergio'[/b]. A chord consisting of a scale's note, that note's minor/major third, and its perfect fifth (usually, at least, the fifth can be flatted or sharpened) is known as a '[b]triad.'[/b]
In '[b]Roman relative chord notation'[/b], all chords are represented by a Roman numeral. The '[b]major chords'[/b] (which use a major third) are represented by capitalized numerals, while '[b]minor chords'[/b] (which use the minor third) and '[/b]diminished chords'[/b] (which uses a minor third and flats the fifth to an augmented fourth) are represented by a lowercase, with the diminished chords differentiated with a small circle next to the numeral.
In a major scale, the '[b]tonic chord'[/b] 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

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 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 '[b]relative key'[/b] 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
ii - Dm
iii - Em
IV - F
V - G
vi - Am
vii° - B°

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

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

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

A '[b]parallel key'[/b] 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!

'[t[2]Chord Progressions and Cadences'[/t]

'[t[2]Modulation'[/t]

'[t[2]Seventh and Extended Chords'[/t]
A '[b]seventh chord'[/b] is a triad that has had a seventh tacked on to it. There are three main types of seventh chords:

A '[b]dominant seventh chord'[/b] adds the minor seventh of the key to a major triad.
A '[b]minor seventh chord'[/b] adds the minor seventh to a minor triad.
A '[b]major seventh chord'[/b] adds the major seventh to a major triad.

There are also lesser-used seventh chords.

'[t[2]Other Chords of Note'[/t]
An '[b]inverted chord'[/b], also known as a '[b]slash chord'[/b], is a chord with the same notes, but the root of the chord is not the '[b]bass note'[/b], in other words, it doesn't have the lowest pitch.
For every chord, there are two '[b]inversions'[/b].
For example: C Major is comprised, in order, of the notes C, E, and G.
Its '[b]first inversion'[/b] would therefore be E, G, C (because the C has to be moved up an octave)
Its '[b]second inversion'[/b] would be G, E, C.
A first inversion chord can be notated either as C/E (in the case of C Major) or as I6 (most times the 6 is superscript) in relativistic notation.
A second inversion would be notated as C/G or as I6/4 (usually the 4 is in subscript).
As a note, seventh chords can have three inversions, due to the extra note.
A '[b]fifth chord'[/b] or '[b]power chord'[/b], 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 '[b]suspended chord'[/b], which suspends the third, and replaces it with either a major second (a '[b]suspended second chord'[/b]) or a perfect fourth (a '[b]suspended fourth chord'[/b]). In C, these can be notated as Csus2, and Csus4, respectively.
'[t[2]Borrowed Chords and Harmonic Dissonance'[/t]
You don't just have to use the triads provided to you by a key! A '[b]borrowed chord'[/b] is a chord that is not in the key, but is instead borrowed from another, usually closely-related key. A '[b]related key'[/b] 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!

Borrowed chords are represented one of two ways:

With a flatted Roman numeral (such as ♭III, which means it is a semitone lower than the III chord)

Or with a slash, representing which key the chord was taken from! (such as IV/V, which means you have taken the IV chord from the key in which the V chord is the I chord)
Dissonance can be used in a number of ways in harmony, such as creating emotions of suspense, dread, or horror, or simply as a transition to a more consonant chord.

'[t[4]IV. Melody'[/t]
Take a note. I will take C, and name it Do in '[b]solfege notation'[/b], which is relative, any note can be Do. Let's also give it the number it corresponds with in its major mode. With this information, we can conclude we are working in C Major.

Do = 1 = C
Re = 2 = D
Mi = 3 = E
Fa = 4 = F
Sol = 5 = G
La = 6 = A
Ti = 7 = B

Melody and harmony have been intertwined ever since music was a thing. But a good rule of thumb for beginners is:
When a certain chord is playing, or is implied, it's a good idea for a note in the melody to be a note in the chord.
Of course, this does not apply in all circumstances, or even most of them.
An '[b]accidental'[/b] is a note in the melody that is somewhat like a borrowed chord, in that it is not in the key the rest of the piece is written in. If I had a melody that went C, G, D, D♯, C, if that piece was in C Major, the D♯ is an accidental. Accidentals are useful, but a lot of composers (usually novices) avoid them like the plague. My advice is to experiment a lot, and see what works, and what doesn't!

'[t[2]Making Melody Behave With Chords'[/t]

'[t[4]V. Uncommon Modes'[/t]
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 I.

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.

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 '[b]harmonic minor scale'[/b], 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 '[b]Phrygian dominant'[/b] (equivalent to the diatonic Phrygian mode with a major third). 

'[t[4]VI. The Science of Sound'[/t]
So how does this all work? I mean, on the most fundamental level, how does it really work?
A '[b]sine wave'[/b] is the most basic of sounds, and the fundamental building block of all other sounds!

'[t[4]VII. Timbre and Sound Design'[/t]
'[b]Timbre'[/b], with pitch and tempo, are the three needed components to make music. White noise is not music, because as it contains timbre, it has no discernible pitch or tempo. Timbre is defined as the synthesis of a pitch and its overtones to create a different sound. A guitar, and a piano may play the same pitch at the same tempo, but they have different timbres.

'[b]Velocity'[/b] is the volume at which a note is played.
'[b]Attack'[/b] is the rate at which the beginning of the note reaches its full velocity.
'[b]Sustain'[/b] is how long a note stays at its peak velocity.

'[t[4]VIII. Miscellaneous Tips and Tricks'[/t]

'[t[4]IX. Helpful Links and References'[/t]
Analysis of BotBr Music (coming soon)
Analysis of famous Classical Music (coming soon)
http://www.youtube.com/watch?v=i_0DXxNeaQ0 - A great video explaining the fundamentals of timbre and sound design!
http://hooktheory.com - fantastic for testing melodies and chord progressions!
http://musictheory.net - interactive, graphical music theory lessons!