General Theory

What is an Interval?
An interval measures the distance between two notes. To identify an interval we must know its numerical size and quality.

Numerical size of intervals
By counting the number of notes in an interval we obtain its numerical size. The first and last note must be counted. For example from C to E we have a third (C-1, D-2, E-3). In the next figure you can see the relationship between the number of notes and the numerical size of intervals:

image001

Yet, not all intervals of the same numerical classification are of the same size. That is why we need to specify the quality by finding the exact number of whole and half steps in the interval.

Whole and Half Steps
In the equal temperament tuning system the octave is divided exactly in twelve notes. The distance between each note is known as a half step. On the keyboard any key is at the distance of a half step from the next and previous key:

image003

A whole step contains two half steps. All the white keys on the keyboard separated by a black key are at a distance of a whole step. The keys that are not separated by a black key are at a distance of a half step:

image005

Notes corresponding to the white keys on the keyboard are called C, D, E, F, G, A and B. These notes are considered natural notes. They can be raised a half step with a sharp or lowered with a flat. A black key, for example the one between C and D, can be considered a C sharp or a D flat:
image007


Interval Quality
By using the keyboard to count the number of half steps between notes we can see that intervals with the same numerical classification can contain a different number of half steps. For example, the second between C and D has one whole step while the second between E and F has only one half step:

image009
The same thing happens with other intervals. For example the third D-F has 1? whole steps or 3 half steps while C-E has 2 whole steps or 4 half steps:

image011
This is the reason why we need to specify the quality of an interval. Please refer to each interval for more information.

Seconds
Seconds can be major, minor, augmented or diminished. Below you can see the number of half steps according to the quality of a second:

image013
Diminished second, 0 half step

image015

Minor second, 1 half step

image017

Major second, 2 half steps or one whole step

image019

Augmented second, 3 half steps or 1? whole steps
Identifying seconds
Seconds are probably the easiest intervals to identify. Yet, it is very important to master the identification of seconds since it will be used when identifying other intervals.
To identify the quality of a second we must know:
* the number of half steps contained in each type of seconds
* the order of musical notes (C, C#-Db, D, etc.). We must remember that between all natural notes, with the exception of E-F and B-C, there is a distance of one whole step.
With this in mind, we can count the number of half steps in a second:

image021
Another way of identifying seconds
If both notes are natural, we don't have to count the number of half steps if we remember that only the seconds E-F and B-C are half steps. If there are accidentals, we can use the following method:
* Make all notes natural and determine the quality.
* Add the accidentals and see how the interval is affected.
Example: G#-A#:

image023
* Make all notes natural. G-A is a major 2nd (only E-F and B-C are minor).
* Add a sharp to G. The interval is now smaller, it becomes a minor 2nd Add a sharp to A.
* The interval is now larger, it becomes a major 2nd.
Another example: C#-D double sharp:

image025
* Make all notes natural. C-D is a major 2nd (only E-F and B-C are minor).
* Add a sharp to C. The interval is now smaller, it becomes a minor 2nd.
* Add a sharp to D. The interval is now larger, it becomes a major 2nd.
* Add a second sharp to D. The interval is now even larger, it becomes an augmented 2nd.

Thirds
Thirds can be major, minor, augmented and diminished. Below you can see the number of half steps according to the quality of a third:
image027

Diminished third, 1 whole step or 2 half steps

image029

Minor third, 1? whole steps or 3 half step

image031

Major third, 2 whole steps or 4 half steps

image033

Augmented third, 2 1/2 whole steps or 5 half steps
Identifying thirds
A third can be identified by analyzing the seconds between the lower and higher notes and a middle note inside the third. For example, the third C-E has two seconds: C-D and D-E. Using the following table we can find out the quality of the third:

If the seconds are: then the third is:
minor - minor diminished
major - minor minor
major - major major
augmented - major augmented





Following this method we find that the third C-E is a major third because both seconds (C-D, D-E) are major seconds.
If any note has accidentals, we can determine the quality of the interval without accidentals and then analyze the effect of the accidentals:
Example: Ab-Cb:

image035
* Make all notes natural. A-B is a major second, B-C is a minor second, so A-C is a minor 3rd.
* Add a flat to A. The interval is now a major third.
* Add a flat to C, the interval is now a minor third.
Other ways of identifying thirds
* Associating thirds with scales, triads, etc. For example the third D-F# can be associated with the I and III degrees of the D Major scale or with the third of the D Major triad. If we know that the third from the I to III degrees in major scales and the third of a major chord are major, we know then that D-F# is also a major third.
* Memorizing all major and minor thirds. Start with major thirds and continue with minor thirds. Anyway, you will learn them with practice.
* Learning the number of steps for each type of third and counting the whole and half steps (not recommended).

Fourths
Fourths can be perfect, augmented or diminished. Below you can see the number of steps according to the quality of the interval:
image037

Diminished fourth, 2 whole steps or 4 half steps

image039

Perfect fourth, 2? whole steps or 5 half steps

image041

Augmented fourth, 3 whole steps or 6 half steps
Identifying fourths
When analyzing the quality of a fourth we should know that:
* the interval is a perfect fourth if all the notes are natural with the exception of the fourth F-B which is an augmented fourth.
If there are accidentals you should identify the interval without accidentals and then analyze the effect of the accidentals.
Example: G-C#:

image043
Another example: C#-F#:

image045
Identifying fourths by counting whole and half steps is slow and confusing.

Fifths
Fifths can be perfect, augmented or diminished. Below you can see the number of steps according to the quality of the interval:

image047
Diminished fifth, 3 whole steps or 6 half steps

image049

Perfect fifth, 3? whole steps or 7 half steps

image051

Augmented fifth, 4 whole steps or 8 half steps
Identifying fifths
When analyzing the quality of a fifth we should know that:
* the interval is a perfect fifth if all the notes are natural with the exception of the fifth B-F which is a diminished fifth.
If there are accidentals you should identify the interval without accidentals and then analyze the effect of the accidentals.
Example: D-A#:

image053
Another example: Gb-Db:

image055
Identifying fifths by counting whole and half steps is slow and confusing.



Sixths
Sixths can be major, minor, augmented or diminished. Below you can see the number of steps according to the quality of the interval:

image057
Diminished sixth, 3? whole steps or 7 half steps

image059

Minor sixth, 4 whole steps or 8 half steps

image061

Major sixth, 4? whole steps or 9 half steps

image063


Augmented sixth, 5 whole steps or 10 half steps
Identifying sixths
The easiest way to identify the quality of a sixth is by inverting the interval and identifying the resulting third. For example, the interval C#-A#:
* The inversion is A#-C#.
* We identify the resulting third.
* A#-C# is a minor 3rd. so C#-A# is a major 6th.

image065
A major 6th. becomes a minor 3rd. when inverted
Construct the following intervals:
1) Diminished fifth ascending (from Sol.)
2) Major sixth   ascending (from sol.)
3) Major seventh ascending (from la.)
4) Major third ascending  (From sol)
5) Perfect fifth ascending ( From Mi)
6) Diminished third ascending ( From La)
7) Minor sixth ascending ( From Do)
8) Diminished fifth ascending ( from sol)
9) Augmented forth ascending ( from do)
10) Major sixth ascending (from fa)


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