Wicked cool

I was listening to "I'm Not That Girl" from Wicked today and was reminded how much I like the final melodic note, ending on low scale degree five (Sol). There are two conclusive endings in the piece. The first one makes sense, being at the end of the A section – the song is in a typical ABA form, the B section is in a different meter and harmonically rather unstable – making the A section a parallel period (or double period if you count some inner contrapuntal cadences) of one large phrase (a1) ending on the dominant, and the second phrase (a2) ending on the tonic, both melodically and harmonically. When the A section comes back (a3 and a4), I expect it to follow the same pattern, especially when it is identical up through the dominant chord that ended a1. But this time that dominant isn't the end of a3, it leads to a closing tonic, with the melodic Re descending down to Do. The 'a' phrase repeats again, and again leads to a dominant chord that could be expected to be a half cadence like a1, or to resolve to the perfect authentic cadence of a3. The former would be very unusual, a switch of the weak and strong cadences from a typical period. The latter would be more typical, and therefore show less craft. Schwartz does neither of these things, instead leading a4's dominant to a tonic, but now with that final melodic note on Sol. We aren't done yet, though. The song begins with an introduction of Csus4 - C - Csus2 - C, Csus4 - C - Csus2. Besides forming the basis for the beginning of each 'a' phrase, the instrumental progression comes back at the end of a3 and a4, easily understood as a tag, or more theoretically as a tonic expansion. Except the final tag, after a4, never resolves back to a C chord, or to even end on the Csus2. Instead, this chord is replaced with a dominant chord, in second inversion! Beethoven finished the second movement of Symphony no. 7 on an A minor chord in second inversion, but at least it was the tonic chord. Schwartz leaves everything up in the air, and quite intentionally:

"There's something about ending on the five that I think, first of all it's gorgeous, but secondly, I always remembered that because it was unusual when I first encountered it on the Weavers album. The sense of non-resolution is significant for the content of the song, of somebody whose experiencing ambivalence or isn't sure about what's going to happen. And I thought 'I'm Not That Girl' is exactly that. It's a statement that trails off. You know that she's saying she isn't, but she hopes she is. It's an ambivalent song, obviously, in a lot of ways, and therefore I thought it was good if it ended ambivalently, (which is nice for art but not so nice when you're trying to get a hand from the audience)."

Besides the ambiguous ending, the song has many descending minor sevenths that add to the emotional angst. I'll look up the theory of emotion and melodic intervals to see how this fits.

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.

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.

Hierarchic asymmetry

Back when I was teaching my first music theory class as a graduate student, I was caught completely off guard by a student's complaint. We were talking about the difference between simple meter and compound meter, which I regarded as a very basic topic. In one the beat is divided into two equal parts, and in the other the beat is divided into three equal parts. The exemplars for each of these would be 4/4 and 6/8. The quarter note beat in 4/4 is divided into two eighth notes, and the dotted-quarter beat in 6/8 is divided into three eighth notes. Easy, right? Well, this student, who was quite bright, said that 6/8 wasn't really a compound meter, because the eighth notes themselves didn't divide into three equal parts. I thought he was making a common mistake of assuming that eighth notes were the beat in 6/8, but he wasn't . He expected that if the beat is divided into three equal parts, then each subsequent lower level in the metric hierarchy would also divide by threes. This is actually a very rare phenomenon, not even covered by normal metric notation. There is no time signature that indicates a beat that will be divided into 9 equal parts, much less 27 or 81 parts. The basic assumption in Western notated music is for beats to be grouped by 2's or 3's (or some combination thereof) and to be divided by 2 or 3, but that all lower levels of subdivision are by 2.

I never really thought about this lack of symmetry, until I was recalling an argument about chord and key relationships. Petr Janata and his colleagues have done some interesting work on brain imaging and tonal sensitivity. But at one conference he presented a torus mapping of key relations, playing along a chord sequence that modulated from one key to the next. This mapping was meant to represent brain activity, showing how the brain interprets different modulations as close or far. As expected, the modulations followed a circle of fifths pattern, but what I found quite disturbing was that mode was not taken into account. At one point the progression went from C minor to G major. I pointed out to him that this was a very distant key relationship in music theory, which he found surprising. After all, C minor chords go to G major chords all the time. We then started a debate about the difference between chord relationships and key relationships, with Carol Krumhansl coming up to keep us from punching each other. (Not really, but it was an interesting and sprited debate, including the aforementioned Dr. Krumhansl). Carol did the seminal work on tonal hierarchies, using probe tones to determine the closest cognitive notes to a given tonic, and helped develop a key-finding algorithm that might mimic how our brains determine what key we are in. What I realized that I knew to be false, but that these scientists had assumed to be true, was that the hierarchy of relationships between notes and chords would extend to keys without change.

Why is it that Western music* does not encourage complete symmetry among all levels of a given feature? Could it be like the asymmetric design of the diatonic scale, working as a signpost to help us identify where we are? The assymmetry between keys and chords could help distinguish the scope of relations we are perceiving, so we don't get the two mixed up. Likewise with meter, to help identify the beat more clearly.

*If anyone knows about any other music traditions that do include non-binary metric symmetry or note/harmony/tonality symmetry, let me know.

Schenker's tonality, Part I

Yesterday I realized that it has been eight years since I thoroughly reviewed the various theories of tonality, when I was studying for my qualifying exams. So I thought I'd create a regular series of posts going through the details of some theories, though the posting schedule will definitely be aperiodic. First I will look at Heinrich Schenker's theory of tonality, as explained in his final monograph, Free Composition.

First, there was the tone. This tone creates overtones, the images of which create the triadic chord, called the Chord of Nature. Most contemporary theorists ignore the chord of nature part of Schenker's theory, for reasons that I listed here. Instead, we accept that Schenker's tonality is based upon the triad, and upon major or minor modes. These postulates are accepted because the music literature of 1600-1880(ish) exhibit this behavior and the analyses based upon Schenker's theory work with this starting postulate. Recent theories suggest that jazz tonality is based upon the seventh chord rather than the triad.

This triadic chord is unfolded through time by stepwise descending motion in the upper voice, from either the third or fifth* of the chord to the root tonic, which Schenker calls the fundamental tone**. Schenker equates this melodic motion, the fundamental line(Urlinie) with our own life-impulses, striving towards a goal. The melodic motion is accompanied by an arpeggiation in the lower voice up a fifth and back down again, as a foundational counterpoint. Why the fifth? From the overtone series, but also from contrapuntal practice: in three-voice strict counterpoint, if the closing scale degrees 2 and 7 are in upper voices, only scale degree 5 is allowable to complete the triad, and was the only leaping bass line found at the close (same as a cadence) in modal counterpoint.

Schenker has a few terms for this counterpoint: background, fundamental structure (Ursatz), and diatony. This fundamental structure is further expanded through middleground and foreground levels by transformations, prolongations, and elaborations, until the actual musical composition is realized (the surface level). The fundamental structure provides the unity, the primary identity of the tonal piece, by marking the goal and the direct path to that goal. Schenker defines "tonality" as the sum of the fundamental structure with all the elaborations and prolongations.

I'll stop here for the first part, quoting from p. 5 of Free Composition:

As the image of our life-motion, music can approach a state of objectivity, never, of course, to the extent that it need abandon its own specific nature as an art. Thus, it may almost evoke pictures or seem to be endowed with speech; it may pursue its course by means of associations, references, and connectives; it may use repetitions of the same tonal succession to express different meanings; it may simulate expectation, preparation, surprise, disappointment, patience, impatience, and humor. Because these comparisons are of a biological nature, and are generated organically, music is never comparable to mathematics or to architecture, but only to language, a kind of tonal language.

Next time I will describe dissonance as it affects the fundamental structure, and begin to explain the transformations that elaborate this fundamental structure.

* Schenker does allow for the traversal of an octave, from root down to root, but later analyses show that he doesn't think it is very common.
**Or at least that is how Ernst Oster translates it from the german.