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

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

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.

When do we bifurcate?

Okay, a DJ created a mashup of Rick Astley and Nirvana (yes, you are about to be Rick-Rolled). Listen to/watch the video and tell me if Rick and Nirvana are using the same tonic, or two different tonics. Show your work.



I heard it as one tonic, with Rick creating some very extended harmonies and dissonant non-chord tones at points. Is this because I expect blues-influenced music to go beyond the triadic norm, or because my brain wants to find the simplest pattern to fit all the information? In other words, did I come to this conclusion top-down (schema driven) or bottom-up (sequential event expectancies)? Likely the two "strategies"* reinforced each other. I do know that if there were fewer common tones, or if the genre had led me to expect more melodic tonal closure, I would have heard two separate tonics, a la Milhaud.

Just Feet It!

The organist of Trinity Episcopal Church on Wall Street in New York City gave a tribute to Michael Jackson for the organ postlude:

A new view of musical space?

Mind Hacks reports on a a new study in Perception which determined that people's comfort zones for personal space was altered by listening to music on headphones. The music hampers the ability of people to aurally track other people, especially important when those people are out of visual range (behind the listener). So more personal space is needed to feel safe.

What is a musical experience?

Okay, it took a week to recover from the Mannes Institute. It was fabulous, but the intensive reading and thinking involved required many days of decompression. One thing that really struck me was a line in David Huron's plenary talk. He said he poses the following question to his graduate students every once in a while: "If you had a chance to talk to God, and could ask Her anything about music, what three questions would you ask?" David calls these the God questions, and says these are the questions that should impell a research agenda. It made me think about what things I'm really curious about, tempered by what my training and resources permit. I also had a great chat with Steve Larson on metaphor theory while jogging through Central Park, giving me a much better handle on that mode of understanding music. I'll start posting weekly about Lerdahl's Tonal Pitch Space, working out my understanding of this complex idea as I attempt to explain it to you.

But for now, I'm bothered by something I read today in Harold Fiske's Understanding Musical Understanding. On pages 25-26, Harold writes,

[Sculpture can be observed by any angle and for as long as the viewer desires.] Not so for music, where 'viewing' time for music is controlled exclusively by the performer. And once over, returing to the room in which the performance occurred would not afford continued experience with the same work. Once the piece is over, it is over. Even playing the piece again (a recording for example) does not represent a continued experience with the piece but rather a different experience, though it is the same music being listened to.

First of all, I don't believe the performer controls all aspects of the performance time. Besides the instructions of the composer, there are constraints placed on the performer(s) by cultural expectations, acoustic limitations of the venue, and general perceptual limitations. And the listener does have control of what performance aspects to which s/he pays attention. This attention is a main part of Fiske's thesis that musical time is very different from clock time, thus the listener does control his/her musical time experience.

But the second thing is more troubling, that rehearings are different experiences rather than a continued experience. It is troubling, because part of me agrees that rehearings (whether a physical rehearing or a mental replay) are indeed different experiences. The listener has different knowledge by the time of the replay, etc. But the two hearings share a commonality of the schemata of the performance. The order of the notes, the timings, the timbres are all the same, just as a sculpture keeps the same physical features. So I really want to say that the rehearing is a continuation of the experience of the piece, just as when I revisit the sculpture it continues my experiences with that artwork, even though I have different knowledge from the last time I experienced it. What say you? Is a rehearing a completely new experience, or is it a continuation of the same musical experience?

Are you a Mannes or a Mouse(s)?

Tomorrow I am heading out to New York for the Mannes Institute for Advanced Studies in Music Theory. As you can see from the preparation page, I have been rather busy doing my assigned readings for the last month. I'm in two workshops, Tonal Tension with Fred Lerdahl, and Music and Embodiment with Eric Clarke. Preparing for Fred's workshop is killing me, trying to get a grasp on his Tonal Pitch Space. But it is also very invigorating, makes me feel like I'm in grad school again. In fact, I will be with many of my former grad school compatriots, like Rich Randall and Ian Quinn. And to go really old school, I will be rooming with a friend from my undergraduate days, Peter Martens. This is more of a think tank than a workshop, as the website describes it: "The 2009 Mannes Institute on Music and The Mind will be a seminal collaborative think tank for serious music cognition and perception scholars from around the world." I'm very excited, and hope to have time to blog about it each day. Oh, plus I've been rocking the house with Rock Band 2, my Father's Day present.

Viral Dancing?

My sister sent me this Youtube video of a "happening" in Antwerp. As a promo for a Belgian reality show searching for Maria for a revival of Sound of Music, 200 dancers started dancing to "Do Re Mi" in the Antwerp Central Station. Even though it has a commercial purpose, it does capture the spirit of brightening peoples days with unexpected art. This sounds like an interesting project to get my college students involved with, not necessarily dance mind you. Especially not me dancing, except to horrify my friends and family.