Tuesday TPS: Chords

Last week I explained the basic space of melodic pitches in Fred Lerdahl's Tonal Pitch Space. Today I will look at the chordal level of basic space. Inspired by Riemann's Klangs, Fred arranges chords by proximity in the circle of fifths and by the number of common tones. The circle of fifths is pervasive in tonal theory, Fred briefly mentions the fifth motion prevalent throughout tonal progressions, the psychoacoustic strength of the perfect fifth within the overtone series, and the ability to generate the diatonic scale through fifth motion as reasons for using the circle of fifths to measure chord distance. Each step along the diatonic circle of fifths, either clockwise or counterclockwise, is counted as one step removed in distance. So from our tonic chord of Eb in last week's example, we can move one step up to Bb major (V), two steps up to F minor (ii), or three steps up to C minor (vi). And we can move one step down from Eb to Ab major (IV), two steps down to D diminished (viio), or three steps down to G minor (iii). Note that this is the diatonic circle of fifths, so Ab descends a diminished fifth to D so we stay in the major scale of Eb.

Distance by fifths are only one half of the chord distance measurement. The other half is the number of common tones between the two chords being compared, with the idea that the fewer common tones, the more distant the chords. This isn't simply calculated by saying the two triads have one or two notes in common, but by looking at the number of common tones throughout the five levels of pitch space. Here is the tonic chord from last week:

Level a:

Eb

Eb

Level b:

Eb

Bb

Eb

Level c:

Eb

G

Bb

Eb

Level d:

Eb

F

G

Ab

Bb

C

D

Eb

Level e:

Eb

E

F

F#

G

Ab

A

Bb

B

C

C#

D

Eb

Now here is the Dominant chord, Bb major, with all the distinctive pitch classes bolded. Note that levels D and E are the same, but level C now shows the Bb triad, with the root and fifth of that chord at level B, and just the root at level A.

Level a:

Bb

Level b:

F

Bb

Level c:

F

Bb

D

Level d:

Eb

F

G

Ab

Bb

C

D

Eb

Level e:

Eb

E

F

F#

G

Ab

A

Bb

B

C

C#

D

Eb

So the total distance between the tonic Eb chord and the dominant Bb chord is 1 (step along the circle of fifths) + 4 (distinctive pitches) = 5. This will be the same distance for all tonic-dominant pairings. The summary of distances from the tonic to each of the other diatonic chords is:
I - ii: 8
I - III: 7
I - IV: 5
I - V: 5
I - vi: 7
I - viio: 8

These distances remain the same regardless of which chord comes first (it is a symmetric function). Distances between other chord pairs can also be calculated, but thankfully the relationships are transpositionally invariant. This means that if I - ii has a distance of 8, ii - iii will also have a distance of 8, as will every pair of sequential chords in the major tonality. So Fred can generalize that moving root motion by a diatonic step is a perceptual distance of 8, moving root motion by a diatonic third is a perceptual distance of 7, and moving root motion by a diatonic fourth is a perceptual distance of 5. Fred realizes this geometrically as a Chordal Space:

V	viio	ii	IV	vi
I	iii	V	viio	ii
IV	vi	I	iii	V
viio	ii	IV	vi	I
iii	V	viio	ii	IV

The horizontal axis is root motion by thirds, the vertical axis is root motion by fifth. Two dimensionally like this it keeps repeating over and over. It can be wrapped around to make a torus, a three-dimensional doughnut. One of Fred's main rules is to follow the shortest path when connecting two chords, and the length of this path demonstrates the perceptual distance in moving from the first chord to the second.

Thus far we have remained within a single key. Next week we will look at the regional level of TPS, to deal with secondary dominants and modulations.