Most jazz pieces are
essentially song structures in which musicians play sequences of chords called chord
progressions. Certain chord progressions
are popular because the transition from chord to chord is musically pleasing,
and because the progression permits moving from one musical key to another,
permitting flexibility and providing musical variety.
One particularly
important chord progression is the II-V-I., in the key of C major this
progression starts by first playing the D minor seventh chord (Dm7), then by
playing the G dominant seventh chord (G7), and ends by playing the C major
seventh chord (Cmaj7). In the key of C
this is a II-V-I progression because D, the root note of Dm7, is the second
note of the C major scale; G, the root note of G7, is the fifth note of the C
major scale; and C, the root note of Cmaj7, is the first note of the C major
scale.
The II-V-I has
evolved into more complex chord progressions.
For instance, John Coltrane introduced the chord progression now known
as the Coltrane changes on his seminal 1960 album Giant Steps, where it is central to two pieces, “Giant Steps” and
“Countdown”. The Coltrane changes are an
elaboration of the II-V-I; it includes the three chords of this older
progression, but adds four more chords.
Two of these are lead-in chords to the V, and the other two are lead-in
chords to the I. The table below
provides the Coltrane changes for the key of C major.
Last summer we were interested in training artificial neural
networks to generate chord progressions: when presented a chord, a network would
generate the next chord in the progression.
To do this for the Coltrane changes, we had to determine the chord
progression for any major key. However,
this is not easy: accounts of the Coltrane changes on the web are opaque.
In
a previous blog entry, I described some ‘strange circles’ – arrangements of
notes in a circle, so that adjacent notes in the circle are a constant musical
interval apart – that we had extracted from other musical networks that we
trained. For our Coltrane project, we
found that combining these circles into more complex diagrams provided a map
that let us build our training set.
In particular, all of the tonic notes of the Coltrane
changes are represented in a ‘rose diagram’ that attaches a ‘strange circle’ of
major thirds to every note around the more traditional circle of fifths. Here is the complete ‘rose diagram’:
The inner circle provides the tonic notes for the II-V-I
skeleton of this progression, and the outer circles provide the tonic notes for
the lead in chords. To use this map, you
start with a chord from the inner circle, you then play four chords related to the
outer circle, and you end with two chords from the inner circle. The figure below shows how the Coltrane
changes falls out of the ‘rose diagram’ for the key of C major. In this figure I extended the lines ending in
G# and B so that the chords are mapped in the proper order. The dotted arrows take you chord by chord
through the changes; compare their order with the table above. If you follow this pattern starting at any
other inside note, then you will generate the Coltrane changes for some other
key.
The connection
weights revealed that the network had learned that a key aspect of the Coltrane
changes involved the relationships between the different base notes used in the
seven (inverted) chords for the changes.
These relationships only involve four different musical intervals:
unison, the major second, the minor 7th, and the major 7th. A unique connection weight defined each of
these intervals. These relationships
could be plotted as a map of movements about the vertices of a triangle, where
each vertex is the lowest note of an inverted chord.
Interestingly, to
start the same sequence in the next key the first note is the Eb; the last
arrow in this map is a pointer to the first chord in the next key. If you fill out the remaining triangles then
another elegant map of the Coltrane changes is produced:
The interesting thing about this final map is that its outer
wheel of notes, and its inner wheel of notes, are two other ‘strange circles’:
the two circles of major seconds. In
short, it seems that whenever we train networks on tasks that involve musical
chords, we find the networks represent chord regularities with these strange
circles.
Nice post, very useful :)
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