The distance between any two points is an interval:
"Harmony, melody: it's all about intervals and how you use them"
A series of theory tutorials for the theory challenged or curious. This time:
Intervals
After much consideration, I have decided that the next instalment should be on intervals.
Several reasons occur to me for doing this, but here are two good ones:
everything we build in the pitch domain (chords, melodies, counterpoint) is based on intervals
a working knowledge of intervals is important for the next instalment, key independent chord construction
Audio examples note
All audio examples in this tutorial are presented so that we hear the two notes of an interval:
together
low to high (ascending)
high to low (descending)
Basic Ideas
An interval is the distance between two notes. This distance is described by two parts:
the size of the interval, and
it's quality.
Size
Is the number of lines and spaces between the notes; we also count the lines and spaces the notes are on/in
Quality
Is the number of semitones the upper note is above, but not including, the lower note. (A semitone is the distance between successive notes on the piano (moving to the black or white key directly above the current note) or successive frets on the guitar)
For each of the above intervals here is a keyboard diagram showing the number of semitones (halftones) found in the interval:
For the intervals above the semitone distances are:
Big and little
As we explore the coming examples, we should see that there are two different semitone sizes for 2nds, 3rds, 6ths, and 7ths. Each of these has a big and a little version: we derive the names for these different sizes from the Latin words maior (meaning elder or bigger) and minor (meaning younger or smaller); our words are major and minor.
Compare the interval qualities found when we build intervals above each note of the C major scale:
What's so perfect?
We use the word Perfect to describe intervals of the Unison, 4th, 5th and Octave. This word has been passed down to us by Pythagoras, who was the first person to identify the ratio of frequencies of the two notes making up an interval. The Unison is in the ratio of 1:1 (both notes have the same frequency), while the Octave is in the ratio of 2:1 (the octave note is at twice the frequency). The 5th is in the ratio of 3:2 and the 4th is in the ratio 5:4. We define these small whole integer ratios as being perfect.
Larger and smaller
There are also some different sizes for 4ths and 5ths; the example above shows a 4th that is larger than the Perfect 4th and a 5th that is smaller than the Perfect 5th. We use the word Augmented to indicate an interval that has been made even larger and the word diminished to indicate an interval that has been made even smaller. These two qualities can also be applied to the other intervals found (2nds, 3rds, 6ths, 7ths).
Summary of Interval Sizes in Semitones
Single Octave intervals
The following table illustrates the relationship between size and quality in terms of the number of semitones found in the interval:
U
2nd
3rd
4th
5th
6th
7th
8ve
Augmented
1
3
5
6
8
10
12
13
Major

2
4


9
11

Perfect
0


5
7


12
Minor

1
3


8
10

Diminished
1
0
2
4
6
7
9
11
Compound intervals
Intervals larger than the octave can be worked out by starting at the octave and counting up further  we call these intervals compound intervals, because they are built from the basic intervals with one or more octaves added to them
8ve
9th
10th
11th
12th
13th
14th
15ma (2 8ves)
Augmented
13
15
17
18
20
22
24
25
Major

14
16


21
23

Perfect
12


17
19


24
Minor

13
15


20
22

Diminished
11
12
14
16
18
19
21
23
The 10th, 12th and 14th are usually reduced to their first octave names of 3rd, 5th and 7th.
Still larger intervals are possible, but are usually reduced to either the 1st octave or the 2nd octave compound interval name.
Chromatic notation, Enharmonic notes and Musical Fictions
Chromatic notation means using sharps and flats. It is required as soon as we delve into the larger world that is interval notation.
We might use sharps (), flats (), double sharps (), double flats (), triple sharps () and triple flats() to allow us to correctly notate any or all of these intervals, depending on what the first note is. It is important to remember that sharps and flats, double sharps and flats and certainly triple sharps and flats represent different views of the same note, e.g.:
G is A,
G is A,
G is A is B
and so on.
The triple sharp and triple flat are considered to be musical fictions but are used in describing intervals so that notes stay relative to the starting note and the required interval is shown correctly in notation.
Intervals built above each note in the first octave
The following are examples of the intervals in the first octave that can be built above the name key note:
Intervals above A
Intervals above A#
Intervals above Bb
Intervals above B
Intervals above C
Intervals above C#
Intervals above Db
Intervals above D
Intervals above D#
Intervals above Eb
Intervals above E
Intervals above F
Intervals above F#
Intervals above Gb
Intervals above G
Intervals above G#
Intervals above Ab
What is the true value of all this?
The overarching impact of learning about intervals is that it opens the door to many possibilities.
Add ear training (aural training) to this mix and you begin down the path to being able to write down your melodies simply by thinking about them  if you can hear it in your mind, notating is simply taking the sound and writing the intervals. Of course you might also need some training in rhythmic notation, but that can be easily done as well. Some training in chord identification may also be useful. Following is a list of ear training sets found at teoria.com, a site in Puerto rico, which I've been using in my teaching since 1998:
You can also work out music that you like but for which there is no sheet music readily available, by applying the skills acquired with ear training.
This is the beginning of the path to being able to play anything that is notated, on your instrument of choice, once you begin making the connection between the written note and the played note.