July 2021
We are finally in a position to explain the names of intervals that we've been using. (This gets complicated quickly, so the explanation will be somewhat simplified.)
An interval's numerical value comes from the number of diatonic notes spanned by the interval. For example, the interval from C to G is a “fifth” because the diatonic sequence from C to G always contains five notes:
And so on for other diatonic scales containing C and G.
We can stack intervals but have to be careful about the way addition works. For example, CE is a third, and EG is a third, but CG is a fifth (rather than a sixth).
The most consonant intervals within an octave are said to be “perfect” intervals: unison, fourth, fifth, and octave. Consider the difference between a major and minor scale that both begin on the same note (i.e. “parallel” scales). For example, C major/minor:
#  C major  C minor 

1  C  C 
2  D  D 
3  E  E♭ 
4  F  F 
5  G  G 
6  A  A♭ 
7  B  B♭ 
Relative to the tonic, the fourth (CF) and fifth (CG) intervals are the same in both scales (as are the unison and octave, of course). The other intervals have major and minor versions.
Half steps  Interval  Example  Play 

0  Perfect unison  C  C  
1  Minor second^{1}  C  C♯  
2  Major second  C  D  
3  Minor third  C  E♭  
4  Major third  C  E  
5  Perfect fourth  C  F  
6  Augmented fourth Diminished fifth 
C  F♯ C  G♭ 

7  Perfect fifth  C  G  
8  Minor sixth  C  A♭  
9  Major sixth  C  A  
10  Minor seventh  C  B♭  
11  Major seventh  C  B  
12  Perfect octave  C  C 
The minor version of an interval is always a half step smaller than the major version.
Intervals within an octave are called “simple”, and intervals beyond an octave are called “compound”. For example, a ninth is an octave plus a second, and a (perfect) eleventh is an octave plus a fourth.
Any interval can be augmented by a half step, or diminished by a half step. An augmented perfect or major interval is called an augmented interval (e.g. an augmented fourth is a half step larger than a perfect fourth). A diminished perfect or minor interval is called a diminished interval (e.g. a diminished third is a half step smaller than a minor third).
A simple interval can be inverted by raising the lower note an octave, so it becomes the higher note. For example, a minor sixth inverts to a major third .
The following patterns apply to all such inversions:
The second is a dissonant interval with major and minor versions, even though the second notes of parallel scales are always the same (because both scales start with a whole step).↩