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: Eb

Level b: Eb


Level c: Eb


Level d: Eb
G Ab
D Eb
Level e: Eb E F F# G Ab A Bb B C C# D Eb

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: 056 4 66 3 57 6 26 75 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.


Daniel Wolf said...


could you summarize Lerdahl's rationale for describing the chromatic space in terms of twelve pitches? While much tonal music is playable (and representable) in equal temperament, much of it is (and often was) playable and representable in tuning environments (meantone, for one, the predominant tuning scheme during the development of common practice tonaity) in which there is no or only limited "enharmonic equalivalence" and, indeed, pitch specification at the chromatic level is rather unambiguous.

Scott said...

Daniel, he isn't exactly consistent with this. As you can see in the calculation of pitch distances, he doesn't use enharmonic equivalence. But here is his rationale: he wants space that is algebraic, with rotations that "yield a unified treatment of pc, chord and regional proximity."(p. 48) He wants each level to be about half the pitches of the next lower level. If we had Fb and E nat as two separate pitches in chromatic space, we would jump from 7 pitches in diatonic space to at least 18, and more likely 21 to be complete. This gets the system out of control for Fred's concepts, particularly as we get to calculations of chords and regions later.

Jim Plamondon said...

Lerdahl's not wrong; his unexamined use of 12-ET is just overly specific.

Rather than using 12-ET as the basis of tonal exploration, it makes more sense to use the syntonic temperament (http://en.wikipedia.org/wiki/Syntonic_temperament). In it, each specific syntonic tuning -- e.g., 12-ET, 1/4 comma meantone, Pythagorean, etc. -- is just a point on its tuning continuum.

The syntonic temperament is most easily visualized using an isomorphic keyboard (http://en.wikipedia.org/wiki/Isomorphic_keyboard), and is most generalized by using tonic solfa names ("movable Do") instead of pitch names (www.igetitmusic.com/papers/JIMS.pdf).

Moveable Do is useful because, when changing tuning smoothly in real time (Dynamic tonality, (http://en.wikipedia.org/wiki/Dynamic_tonality), the frequencies (pitches) of all notes (except the tonic) change, so naming them as pitches seems improper. Tonic solfa captures the invariance of the relationships among the intervals, without imposing particular pitch-names.

Even more generally, tonality can be approached through other temperaments, such as Magic. By adjusting the timbre to match the temperament (in its current tuning), one can generalize tonality beyond the harmonic series and just intonation (http://www.igetitmusic.com/papers/SpectralTools.pdf) without loss of consonance.

Most of the above has been published in peer-reviewed journals. However, our papers have been intensely mathematical -- to provide the rigor necessary for scientific proof -- so they haven't been terribly accessible for most musicians or even music theorists.

Jim Plamondon said...

A side-point: there is a potential physiological basis for tonal pitch space, that being the entorhinal cortex's network of grid cells (http://en.wikipedia.org/wiki/Grid_cells).

In combination with hippocampus' place cells (http://en.wikipedia.org/wiki/Place_cell), grid cells appear to play an important role in recignizing/remembering/managing movement through physical space. The grid cells are arranged in a hexagonal lattice, strongly reminiscent of an isomorphic keyboard (http://en.wikipedia.org/wiki/Isomorphic_keyboard).

While place cells are "fixed," grid cells are "relative."

So, I suspect that, just as the human brain uses a combination of relative relationships and fixed reference points to "imagine" navigating through a physical space, it uses these same cells (or cells that are very similar) to "imagine" navigating through tonal space. The tonic, anchored in a place cell, provides a reference point, while the tonal relationships of a rank-2 temperament are captured in a hexagonal network of grid cells (or somethig very similar). This would help explain both absolute pitch (more reference points) and relative pitch (one reference point). Likewise, the tuning invariance of isomorphic keyboards (i.e., grid cells) would help explain the ability of the human brain to recognize tonal relationships even in tunings that are considerably distant from just intonation.

Not that I'm an expert in any of this, mind you...