To determine which chord belongs to a sequence of notes, we need to know:
With these knowledge many chords will fit a sequence of notes. Therefore, I define some rules with which unlikely candidates can be excluded.
To find the chord played in the InstrumentPanel, I first defined a set chords, along with the intervals they contain, as follows:
Name : Intervals Name : Intervals Name : Intervals Name : Intervals
1 : 1 7 : 1 3 5 b7 9 : 1 3 5 b7 9 11 : 1 3 5 b7 9 11
5 : 1 5 m7 : 1 b3 5 b7 m9 : 1 b3 5 b7 9 m11 : 1 b3 5 b7 9 11
major : 1 3 5 maj7 : 1 3 5 7 maj9 : 1 3 5 7 9 maj11 : 1 3 5 7 9 11
m : 1 b3 5 6 : 1 3 5 6 9sus4 : 1 4 5 b7 9 11+ : 1 3 5 b7 9 #11
dim : 1 b3 b5 m6 : 1 b3 5 6 6*9 : 1 3 5 6 9 m11+ : 1 b3 5 b7 9 #11
+5 : 1 3 #5 7sus2 : 1 2 5 b7 m6*9 : 1 b3 5 6 9 13 : 1 3 5 b7 9 11 13
m+5 : 1 b3 #5 7sus4 : 1 4 5 b7 7-9 : 1 3 5 b7 b9 m13 : 1 b3 5 b7 9 11 13
sus2 : 1 2 5 7-5 : 1 3 b5 b7 m7-9 : 1 b3 5 b7 b9
sus4 : 1 4 5 m7-5 : 1 b3 b5 b7 7-10 : 1 3 5 b7 b10
7+5 : 1 3 #5 b7 9+5 : 1 b7 b9
m7+5 : 1 b3 #5 b7 m9+5 : 1 b7 9
7+5-9 : 1 3 #5 b7 b9
m7+5-9 : 1 b3 #5 b7 b9
It is easier to store these intervals as semitone distances to the root note. For instance, the semitone distances of the minor chord are (1-1, b3-1, 5-1) = (0 3 7). The list of known chords is actually stored as follows:
Name : Intervals Name : Intervals Name : Intervals Name : Intervals
1 : 0 7 : 0 4 7 10 9 : 0 4 7 10 14 11 : 0 4 7 10 14 17
5 : 0 7 m7 : 0 3 7 10 m9 : 0 3 7 10 14 m11 : 0 3 7 10 14 17
major : 0 4 7 maj7 : 0 4 7 11 maj9 : 0 4 7 11 14 maj11 : 0 4 7 11 14 17
m : 0 3 7 6 : 0 4 7 9 9sus4 : 0 5 7 10 14 11+ : 0 4 7 10 14 18
dim : 0 3 6 m6 : 0 3 7 9 6*9 : 0 4 7 9 14 m11+ : 0 3 7 10 14 18
+5 : 0 4 8 7sus2 : 0 2 7 10 m6*9 : 0 3 7 9 14 13 : 0 4 7 10 14 17 21
m+5 : 0 3 8 7sus4 : 0 5 7 10 7-9 : 0 4 7 10 13 m13 : 0 3 7 10 14 17 21
sus2 : 0 2 7 7-5 : 0 4 6 10 m7-9 : 0 3 7 10 13
sus4 : 0 5 7 m7-5 : 0 3 6 10 7-10 : 0 4 7 10 15
7+5 : 0 4 8 10 9+5 : 0 10 13
m7+5 : 0 3 8 10 m9+5 : 0 10 14
7+5-9 : 0 4 8 10 13
m7+5-9 : 0 3 8 10 13
Given a set of intervals (for instance (0 5 10) = (1 4 b7)) it is now possible to compute which chord this might be. I simply compare these intervals to those in the table of known chords. If each given interval is also in the list of intervals of a known chord, it might be this chord. Given (0 5 10), the possible chords are therefore 7sus4 : 0 5 7 10 and 9sus4 : 0 5 7 10 14. Although both chords contain all the given intervals, it doesn't make much sense to choose 9sus4. The "best chord" for a given set of intervals is defined as the possible chord with the lowest index in the table above.
Suppose we have the notes (e,a,d) and want to know the intervals, given the root note e. This is done my measuring the distance, in semitones, between each note and the root note. For this example, this yields (e-e,a-e,d-e) = (0 4 10). As we saw, the best chord for this set of intervals is 7sus4. Because the root note is e, we therfore know it is a E7sus4 chord.
We now know how to match intervals to a known chord, and also how to compute intervals given a set of notes and a root note. Therefore, we can also match a set of notes and a root note to a known chord. Suppose we have the notes (a# c d f d# g) and specify that the root note is c. First we compute the intervals given the root note (a#-c c-c d-c f-c d#-c g-c) = (10 0 2 5 3 7). Unfortunately, there is no known chord with the intervals ( 0 2 3 5 7 10). The reason is that when computing the intervals of a set of notes, the interval can never become larger than 11 semitones. For example a#-c=10, b-c=11, c-c=0, c#-c=1. Therefore the intervals of the known chords should also be confined to the range [0-11]. For instance the chord m11 : 0 3 7 10 14 17 is converted to m11 : 0 3 7 10 2 5. This does correspong to the intervals (10 0 2 5 3 7). Therefore, the notes (a# c d f d# g) with root note c correspond to a Cm11 chord.
What if the root note is unknown? The procedure is to simply try each note c..b as a root note, compute the intervals, and match the best chord. For the notes (a d e).
Root note : Intervals = Known chord (intervals) CAI RNP ANP R=B
a d e
c : 9 2 4 = C6*9 (0 4 7 9! 14=2!) Y
c# : 8 1 3 = C#m7+5-9 (0 3! 8! 10 13=1!) Y
d : 7 0 2 = Dsus2 (0! 2! 7!) Y Y Y
d# : 6 11 1 = X
e : 5 10 0 = E7sus4 (0! 5! 7 10!) Y Y
f : 4 9 11 = X
f# : 3 8 10 = F#m7+5 (0 3! 8! 10!) Y
g : 2 7 9 = G6*9 (0 4 7! 9! 14=2!) Y
g# : 1 6 8 = X
a : 0 5 7 = Asus4 (0! 5! 7!) Y Y Y Y
a# : 11 4 6 = X
b : 10 3 5 = Bm11 (0 3! 7 10! 14=2 17=5!) Y
An X means that now known chord contains all given intervals. So, given the notes (a d e), the possible chords are C6*9, C#m7+5-9, Dsus2, E7sus4, F#m7+5, G6*9, Asus4, Bm11. Their intervals are also listed. An exclamation mark means this interval is actually played.
The list above shows the naivety of the procedure: no mucisian would call (a d e) a Bm11. The root note and fifth aren't even played! It is simply included, because Bm11 contains the intervals 10 3 5. Therefore, several "filters"/"rules" should be applied. These are liste below. Which chords conform to the rules is depicted in the table above.
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