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:
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 c:||F||Bb||D |
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:
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.