In music, an interval is the relative distance between two notes that can be measured in tones or steps and, in the equal temperament system that uses twelve notes, the smallest measurable unit is a half-step or a half-tone. The two notes can be played either melodically (one note after the other), or harmonically (two notes at the same time), and the intervals can be considered to be consonant or dissonant.
In western music culture, it is generally accepted that consonant intervals include the perfect octave, fifth and fourth, thirds and sixths; while the remainder intervals – seconds, augmented fourth and sevenths; are considered to be dissonances. If you look at the harmonic series, and keep in mind the various periods in western music history until present day, the exploration, the use and cultural acceptance of the interval tensions and consonance has been expanding. In other words, over time, we have started making music by using intervals from the lower region of the harmonic series and have been progressing to the upper regions.
To identify the interval between two notes you just have to count them, inclusively! For example, if you have an F and an A (ascending), you just count F, G and A. This is a third interval. If it was the case of a descending interval, you would count F, E, D, C, B and A. So, it would be a 6th interval.
To identify the quality of the interval, we must count how many steps or semitones are involved. Here is a table of intervals up to the first octave, considering C as our relative note:
SIMPLE AND COMPOUND INTERVALS
The previous table showed simple intervals and this only means that we are considering the intervals in the span of an octave. A compound interval is when we consider intervals that range more than one octave. To figure out a compound interval, you simply follow the same procedure to identify the number of the interval but instead of counting the interval between F and G as being a 2nd interval, you count the interval considering F as the octave. That said, a compound interval between these two notes will be a 9th interval – F and G as being the octave and the ninth, respectively.
The quality of the interval remains the same as the notes do not actually change. In the previous example we have a major 2nd so, its compound interval is a major 9th. Mind you that a major 2nd and a major 9th sound different despite the same notes are involved:
How interval relationships sound to every composer is a personal and aesthetic construction based on musical experience or acquired tastes and both of those can change with time. This means that any tone or group of tones can succeed to any other tone(s), just like any degree of tension can be extended at will, given a proper musical context. That said, we should understand the melodic and harmonic intervals of sound, in order to better understand the harmonic process.
Another related concept that will be useful in the long run is interval complementation. This only means that you just have to invert one of the involved intervals so that it completes the span of an octave. For example, if you consider the ascending interval of a 7th between C and a B, this interval is complemented when you invert C and place it one octave higher. Now you´ll have to count from B to C and this makes an ascending 2nd. But you could do the same with B. Previously you had an ascending 7th but if you complement the interval by adding a B an octave lower, now you count a descending 2nd from C to B.
The quality of the interval is also usually reversed, with exception made for perfect intervals. So, if you have an ascending minor 7th (with the notes C and B♭), then the descending interval will be a descending major 2nd (descending from C to B♭). This is called an interval inversion.
Notice how the sum of the complemented intervals equals nine. You can take this as a simple way to figure out the interval inversions really quick. For instance, if you have an ascending perfect 4th, to sum nine you will have to add a descending perfect 5th. And do not forget that perfect intervals are the exception and are complemented by perfect intervals as well; major intervals complement minor intervals; and augmented intervals are complemented by diminished intervals – always aiming to sum nine.
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