The reference system for this research is equal temperament, based on dividing the octave into intervals that are equal in size. Musical intervals are the main subject of this study. So what is an interval?
Quite simply, a musical interval is the distance between two notes: a very basic, simple element of musical construction.Intervals are usually ordered by size:
1 semitone = minor second (2m)
2 semitones = major second (2M)
3 semitones = minor third (3m)
4 semitones = major third (3M)
5 semitones = perfect fourth (p4)
6 semitones = augmented fourth (a5)
7 semitones = perfect fifth (p5)
8 semitones = minor sixth (m6)
9 semitones = major sixth (M6)
10 semitones = minor seventh (m7)
11 semitones = major seventh (M7)
12 semitones = octave
In Western music theory, an interval's name depends on the spelling and function of its notes. For example, C-Eb is a minor third, while C-D# is an augmented second. This depends on the harmonic relationships established within the reference system, whether tonal, modal or otherwise.
In this research I will refer to intervals only by the number of semitones they contain, without considering their degree of dissonance or their possible harmonic meaning.
For me, an interval is a basic element from which to try to create a musical language: a language for experimentation in composition, but above all in improvisation.
I will use intervals as a tool for creating melodies and stimulating improvisers, a kind of generator producing sequences of notes and sounds.
We will treat both the unison and the octave as interval 0.
Intervals also have a direction: ascending (+) or descending (-).
Interval 6, the augmented fourth or diminished fifth, divides the octave in half and acts as a kind of axis around which the intervals are mirrored.
An ascending minor second (1) is therefore "equivalent" to a descending major seventh: they reach the same pitch class.Starting from C and moving up a minor second (1), I reach Db. Starting again from C and moving down a major seventh (11), I also reach Db. If we adopt this rule, the intervals have a particular relationship that I will call "reciprocity".
In this sense, the reciprocal pairs are:
the minor second (1) and major seventh (11)
the major second (2) and minor seventh (10)
the minor third (3) and major sixth (9)
the major third (4) and minor sixth (8)
the perfect fourth (5) and perfect fifth (7)
the augmented fourth (or diminished fifth) has no distinct reciprocal
Following this idea, intervals can be grouped as follows:
unison/octave (0)
minor second/major seventh (1)
major second/minor seventh (2)
minor third/major sixth (3)
major third/minor sixth (4)
perfect fourth/perfect fifth (5)
augmented fourth/diminished fifth (6)
These intervals can have an ascending or descending direction, indicated by + or -.
Since I will distinguish intervals only by their size and direction, rather than by names such as second or third, a question arises about the choice of note names. To explain:
if I start from C and apply an ascending minor third (+3), should the resulting note be called Eb or D#?
Without a tonal or modal reference system, it seems clear that there should be twelve note names, no more: each note should have one name, rather than two possible names depending on the reference system.
I have therefore chosen twelve names from those offered by musical tradition:
C C# D Eb E F F# G Ab A Bb B. I have chosen two sharp names and three flat names, and will use these spellings throughout the study.