Musical Perceptions: Lerdahl

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Tuesday, July 28, 2009

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:

Level b:

Level c:

Level d:

Level e:

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:

Level b:

Level c:

Level d:

Level e:

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.

Tuesday, July 21, 2009

Tuesday TPS

As promised, I'm starting an explanation of Fred Lerdahl's Tonal Pitch Space. It certainly won't be as comprehensive as Fred's own book, but may be an introduction to the subject. First up, the different levels of Tonal Pitch Space. TPS is not a real space, but rather meant as a metaphor describing how Fred believes we understand tonal music. Later I'll go through some of the empirical research that may support TPS. Fred describes the "basic space" in five levels. The first level is root space, consisting of only the tonic pitch and its octave. So in the key of Eb major, root space is Eb. The next level is fifth space, reflecting the stability of the perfect fifth in tonality. So in our same key, fifth space consists of Eb and Bb. One rule of TPS is that a pitch found at a given level of pitch space must also be found at the higher levels of pitch space. So Eb will be found at all five levels of pitch space, and Bb will be found at four levels. The third level of basic pitch space is the triadic level, completing the tonic triad, in this case Eb G and Bb. Fred posits triads as the basic harmonic unit for Classical music, but does suggest the possibility of tonal spaces with sevenths as the harmonic basis for other genres of music, such as jazz. The fourth level is diatonic space, the complete diatonic scale for the given tonality. It could be the major scale, the natural minor scale, and Fred does show spaces with octatonic scales or whole tone scales as the basis. Our Eb tonality has the pitches Eb F G Ab Bb C D and Eb in diatonic space. The final level is chromatic space, all twelve pitches between the octave tonic pitches: Eb E F F#/Gb G Ab A Bb B C C#/Db D and Eb. Enharmonic equivalence is somewhat allowed here, so it doesn't matter if the given pitch class is spelled as F# or Gb, it has the same position in chromatic pitch space. But it does matter how the pitch is spelled in determining the distance between pitches, discussed below.

Level a:

Level b:

Level c:

Level d:

Level e:

Within each level of space, a step is considered perceptually close. Chromatic space makes sense, each half-step is very close in frequency. Likewise at the diatonic level, our concept of scale steps fits well here. Less obvious is triadic space – a step up from Eb is G – and fifth space has one step between Eb and Bb. The distance between two pitch classes is calculated by the horizontal and vertical steps needed to traverse from one to another, by the shortest path possible. It takes five steps to the right from Eb to C in diatonic space, but only two steps to the left, so the shortest horizontal path is two steps. Vertical steps are measured to get to the lowest level of the most salient pitch. A is found only at Level e, G is found at Level c. So the horizontal distance between these two pitches is 2. A to Eb is four vertical steps. Fred combines the horizontal and vertical distances to measure the distance from each pitch to the tonic pitch. Fb goes to the left to Eb and then up the four vertical levels (1+4 = 5), E goes to the right to F, then up a level, then to the left to Eb, then up three levels to root space (1+1+1+3=6).

Combined distance:	0	5	6	4	6	6	3	5	7	6	2	6	7	5	7	6	4	0
Pitch:	Eb	Fb/	E	F	Gb/	F#	G	Ab	Bbb/	A	Bb	Cb/	B	C	Db/	C#	D	Eb

So the closest pitch to the root is Bb, the fifth. The furthest pitches from the root Eb are Bbb, the flat fifth scale degree; B, the sharp fifth scale degree; and Db, the flat seventh scale degree. This last pitch relationship is the odd one to me. I totally agree that chromatic alterations of all-important Sol is moving us far away from the tonic, but Te is much closer. Fred has a solution later by shifting from major diatonic space to minor diatonic space, but even as a simple chromatic alteration Te seems more like a 5 than a 7 (to pull a number out of my keister).

Thus far we have been dealing with pitches in melodic relationships. Next week we will move to chordal relationships.