Intervals

An interval between 2 notes is named firstly for the number of lines and spaces it occupies when written (including both ends). The actual number of semitones is only a secondary consideration. This is an unfortunate throw-back to the 7-note scale. A diatonic interval is one in which both notes belong to the same scale. Semitones with different note names are diatonic, while those with the same name are chromatic. Enharmonic notes are those which are written differently but actually have the same pitch. Intervals greater than an octave are called compound. For example, a twelfth is an octave plus a fifth.

The intervals usually described as perfect are the fourth and fifth. Most other intervals have a major and minor form (ie large and small). However, on this basis, the unison and the octave might also reasonably be described as perfect. Intervals can also be augmented (larger than normal) or diminished (smaller than normal). A diminished second is an enharmonic interval. There doesn't seem to be a name for a second where the top note is lower than the bottom one, eg E# & Fb.

Perfect Octaves

Perfect Unisons

Enharmonic Notes

Chromatic Intervals

Diminished Seconds

Minor Seconds

Major Seconds

Augmented Seconds

Diminished Thirds

Minor Thirds

Major Thirds

Augmented Thirds

Diminished Fourths

Perfect Fourths

Augmented Fourths

Diminished Fifths

Perfect Fifths

Augmented Fifths

Diminished Sixths

Minor Sixths

Major Sixths

Augmented Sixths

Diminished Sevenths

Minor Sevenths

Major Sevenths

Augmented Sevenths

Note that pairs of intervals are complementary (make an octave together). The formulaes are: major Nth = minor (9-N)th and aug. N = dim. (9-N). So a major 6th has a complementary minor 3rd and a diminished 5th has a complementary augmented 4th. So just as diminished 2nds are enharmonic unisons, augmented 7ths are "enharmonic" octaves.

Chords & Arpeggios

A chord is 3 or more different notes played simultaneously (2 notes aren't really enough). Chords are the basis of harmony. Those two things are probably true for all musical systems. The rest of this section is based on the European system. Originally all chords were harmonious. Then some were developed that were slight discords requiring resolution to another chord. This could be a note temporarily suspended from one chord to the next or an extra note.

An arpeggio is a chord broken into an ascending or descending sequence of notes. The word means "like a harp". This is because, in accompaniment, the harp often has chords extended as a sequence of notes rippling up or down the instrument. However, a harp player can pluck several strings simultaneously (just like a lutist or guitar player can). Any chord could also be played as an arpeggio. There is even a shorthand notation for this - a rippling vertical line to the left of the chord.

Triads

The simplest chord is the major or minor triad (12 different ones of each). They consist of the 1st, 3rd and 5th notes of major or minor scales.

Major Triads
	C	C#	D	Eb	E	F	F#	G	Ab	A	Bb	B
Minor Triads
	Cm	C#m	Dm	Ebm	Em	Fm	F#m	Gm	Abm	Am	Bbm	Bm

The most important thing about a chord is its root or key (the scale from which it is taken rather than the key of the music). However, the next most important thing is its bottom note. If the bottom note of a triad is the tonic (1st note) of its scale, then the triad is said to be in root position. If the bottom note is the mediant (3rd note), then the triad is a first inversion. If the bottom note is the dominant (5th note), then the triad is a second inversion.

Simple triads can also be numbered by the position of their root within the key of the piece of music. Roman numerals are used and any inversion is shown as a small letter: b for first inversion and c for second inversion. So in the key of C major: CEG is "I", FAD is "IIb", BEG is "IIIc", FAC is "IV", BDG is "Vb", EAC is "VIc" etc. In figured bass terms, root position is 5 3 (or just 5 or nothing), first inversion is 6 3 (or just 6) and second inversion is 6 4. The notes don't have to be in close formation as in the examples below - it is the bottom note that matters. Generally, the bottom note of a chord can be further from the top notes than they should be from each other.

Root Triads in C major
	I	II	III	IV	V	VI	VII
First Inversions in C major
	Ib	IIb	IIIb	IVb	Vb	VIb	VIIb
Second Inversions in C major
	Ic	IIc	IIIc	IVc	Vc	VIc	VIIc

Root Triads in A minor
	I	II	III	IV	V	VI	VII
First Inversions in A minor
	Ib	IIb	IIIb	IVb	Vb	VIb	VIIb
Second Inversions in A minor
	Ic	IIc	IIIc	IVc	Vc	VIc	VIIc

The second inversion sounds incomplete and is not often used. It can occur in passing to a chord in root position with the same lowest note. For example, in the sequence Ib V I the dominant is the bass note of the first two chords.

Other Chords

The 3rd in a chord can be replaced by the 4th (or 4th plus 2nd). This is usually a suspension from the previous chord that resolves to the major or minor triad. There are 12 of these (or 24 with the 2nd as well). The 5th of a major chord can be augmented so that all 3 notes are exactly 4 semitones apart. There are only 4 different augmented triads, eg: C E G#, C# F A, D F# A# and D# G B. Of course there is more than one name and order for these notes. The 5th of a minor chord can be diminished to make another minor 3rd. With the diminished 7th (or major 6th) added, all 4 notes are exactly 3 semitones apart. Ignoring names and order, there are only 3 different diminished chords, eg: C D# F# A, C# E G A# and D F G# B. However, it is common to have just 3 of the 4 notes making a diminished triad.

Augmented Triads

Diminished Chords

Starting from a major or minor triad, the 6th or 7th can be added. The 6th is normally major and the 7th minor regardless of the original triad's scale. However, a major triad plus 6th has exactly the same notes as the related minor triad plus 7th. Only the context can tell them apart. The major 6th added to a minor triad is an interesting chord that is less demanding of resolution than some. Sometimes a major 7th is added to a major triad. This is more discordant than the minor 7th because it makes an interval of only a semitone if the tonic is next to it. The major 7th note is usually resolved to the octave. A minor 6th makes an interval of only a semitone if the dominant is next to it.

	Dm6	Dm7	F6	F7	Fmaj7

The most common note to be added to a major triad is its minor 7th. This chord started off as a combination of V and VII called the dominant 7th. VII is not a normal triad because it has a diminished 5th. Since both intervals are minor thirds it could be made into a complete diminished chord but combining it with V works well. The result is clearly a V with an extra minor 7th which is actually the 4th of the original key. The dominant 7th traditionally requires resolution to another chord (usually I or VI). However, its use has spread beyond its original position of V (eg to most chords in rock and roll music).

(Dominant) 7th Chords
	Bb7	B7	C7	C#7	D7	Eb7	E7	F7	F#7	G7	Ab7	A7

Cadences

A cadence is a sequence of two different chords at the end of a melodic phrase. It is the equivalent of punctuation in writing (eg comma, semicolon or full-stop). When constructing chord sequences, consecutive octaves and fifths are not normally allowed. Although music "rules" can be broken, it is unwise to do so without a good musical reason.

The most final sounding cadences are the perfect (or full close) and the plagal. They both end on the tonic - ie in the correct key. A perfect cadence is from dominant (V) to tonic (I) chord. A plagal cadence is from subdominant (IV) to tonic (I) chord. The tonic chord should be in root position, otherwise the finality is lost.

An imperfect cadence (or half close) ends on the dominant to suggest a continuation. It may be the reverse of the perfect, from tonic (I) to dominant (V). This would be plagal if the key had moved to the dominant. Another common form is subdominant (IV) to dominant (V). A rarer version is supertonic (II) to dominant (V). This would be perfect if the key had moved to the dominant. Note that in a minor key the supertonic is diminished and sounds better in its first inversion.

An interrupted cadence is from the dominant (V) to a chord other than the tonic (I). For example, dominant (V) to submediant (VI) is quite common. It gives the impression of being a perfect cadence gone wrong and requires resolution.

Examples in C major and A minor:

Perfect Cadences V I V I

Plagal Cadences IV I IV I

Interrupted Cadences V VI V VI

Imperfect Cadences I V I V

Imperfect Cadences IV V IV V

Imperfect Cadences II V II V

The leading note normally rises to the tonic. This means it shouldn't appear twice in a chord. The triad on the leading note (VII) is diminished and sounds better in its first inversion. Note that as this chord is just a dominant 7th without its root note, it can make a perfect cadence (VIIb to I). In a major key, the interrupted cadence to the submediant is possible (VIIb to VI). In the chord on the mediant (III), the leading note is the fifth. In a major key it can fall (I-III-VI or I-III-IV). In a minor key, the chord contains an augmented fifth which is a discord. This should only appear in passing, held over from the previous chord and rising to the next (V-III-I or V-III-VI).

VIIb I VIIb I VIIb VI

I III VI I III IV

V III I V III VI