Sunday, June 23, 2019

Missing the beat

Van Halen's Right Now is a great tune with a musical experience that, for me, doesn't diminish with repeated listenings. I enjoy it so much that last week I had decided to learn the first minute or two on my keyboard. While learning it I had quite the time wrestling with its ambiguous rhythm. When listening casually to the song, my ear naturally heard the left-hand bass as coming in on the 16th note before the downbeat of a bar, like so:



This made sense aurally to me because the G on the downbeat of the second measure above would then act like a suspension down to F, and suspensions typically happen on strong beats like 1 and 3. When I played it, however, I kept wanting hear the left-hand bass to be the downbeat of a measure, like so:



It wasn't until discussing this with a friend of mine that he pointed out that neither of those were correct! The piece is arranged such that the left-hand bass comes in on the second 16th note of the bar, like so:



What? This really messed with my head until I sat down and actually played it with a metronome. It sounded great and incredibly interesting to perceive the rhythm phrased that way. Later on in the song Van Halen does a recapitulation of the piano intro which confirms that rhythm. Even more daring, lead singer Sammy Hagar injects an inversion of that syncopation while the keyboard concurrently plays the intro riff - he lands a strong vocal note on the 16th note before the downbeat. This completely de-emphasizes the downbeat by sandwiching it between two accented pitches on weak beats:


I like to think that the longevity of my interest in this song is owed at least in part due to the syncopated song writing. It's fun to listen to it and a challenge to try and figure out where the real downbeat is!

Saturday, July 21, 2018

Programming with triadic transformations

I decided to make use of a 5-hour train ride from Toronto to Ottawa recently by writing a small program that performs all the fun Neo-Riemannian triadic transformations music theorists at the graduate level know and love. 

At the heart of the program is the Triad class which coordinates the triadic operators:

data class Triad(val root: PitchClass,
                 val third: PitchClass,
                 val fifth: PitchClass) {

    fun parallel() = Triad(root = root,
            third = when {
                isMinor() -> third.transpose(desiredInterval = 1)
                isMajor() -> third.transpose(desiredInterval = -1)
                else -> onNonMinorOrMajorTriad()
            },
            fifth = fifth)

    fun relative() = when {
        isMinor() -> Triad(root = third,
                third = fifth,
                fifth = root.transpose(-1, -2))
        isMajor() -> Triad(root = fifth.transpose(1, 2),
                third = root,
                fifth = third)
        else -> onNonMinorOrMajorTriad()
    }

    fun leadingTone() = when {
        isMinor() -> Triad(root = fifth.transpose(1, 1),
                third = root,
                fifth = third)
        isMajor() -> Triad(root = third,
                third = fifth,
                fifth = root.transpose(-1, -1))
        else -> onNonMinorOrMajorTriad()
    }

    fun nebenverwandt() = relative().leadingTone().parallel()

    fun slide() = leadingTone().parallel().relative()

    fun hexatonicPole() = leadingTone().parallel().leadingTone()

    private fun isMinor() = root.isMinorThird(third) && third.isMajorThird(fifth)
    private fun isMajor() = root.isMajorThird(third) && third.isMinorThird(fifth)
    private fun isDiminished() = root.isMinorThird(third) && third.isMinorThird(fifth)
    private fun isAugmented() = root.isMajorThird(third) && fifth.isMajorThird(fifth)

    private fun onNonMinorOrMajorTriad(): Nothing =
            error("Triad was neither minor nor major")
}

Transposing musical pitches programmatically always takes some creativity because of the peculiar way our system of music is organized. Specifically, some intervals between pitch letters are whole-tones (e.g. C-D) while some others are semi-tones (e.g. E-F). For this application I solved that problem by asking the developer to provide both the number of letter changes (e.g. C-D would be a value of 1) in addition to the number of semi-tone changes (e.g. C-D would be a value of 2). This makes it easy to reconcile those aforementioned peculiarities and any accidentals that ensue following a transposition. 

With that transpositional logic, if I wanted to invoke a somewhat unusual interval like an augmented 2nd (e.g. C-D#) then as a developer I'd just have to provide the value 1 for the number of letter changes, and 3 for the number of semi-tone changes. In code this could look like cNatural.transpose(1, 3). I feel like this is a pretty reasonable and readable solution. 

From a set theory perspective all I would need is the number of semi-tone changes to perform a transformation, but because music deals with letters and accidentals that can have enharmonic spellings (e.g. B# == C), two arguments are necessary to get exactly what the composer/musician/developer is asking for.

A few neat takeaways from this are that:
1) The R (relative) and L (leading tone) transformations end up just being rotations of the pitch classes the triad is composed of (with one triad member being transposed). For example, after an R transformation the root becomes either the third or fifth and the remaining triad members follow the same shift.
2) The N, S, and H transformations use chaining and as you might expect the programmatic approach to performing this is very clean and easy (see code above). 

I haven't experimented with how exhaustive one can be composing transformations like the N, S, and H transformations, but a few others I came up with in pseudocode were:

dominant = LR
tritone = PRPR
neapolitan = SP (or, more primitively, LPRP)

You can conveniently find all this code on one of my public GitHub repositories.

Thursday, December 22, 2016

Pitch Class Set Calculator

Made a pitch class set calculator. It includes a list of all set classes with cardinalities three through nine represented with a RecyclerView, one of Android's most powerful components.

You can get it right here: https://play.google.com/store/apps/details?id=com.nihk.github.pcsetcalculator

Source code!

Saturday, January 9, 2016

Composing T and I operations


Almost four years ago to this day I submitted a simple assignment to a Transformational Theory course I was attending at college. The task was to compose different combinations of two of the so-called Twelve-Tone Operators (T and I) into just one operation. The Tn operation is transposition by the value of n, whereas the In operation is inversion followed by a transposition by the value of n. Note that this is all done within a given modulo universe; in music, we operate within mod 12.

Back to the task at hand: If you apply, for example, T4 and then I7 to a set of pitch-classes, is there an alternate way to reach the same state the set is in but with just one operation? Undoubtedly there is. The previous example can be more simply done with just one stroke of an I3 operation. But how can one confidently generate that singular operation given any two operators T and I in any order? The following set of rules is exactly how:


Tm · Tn = Tm+n
Im · Tn = Im+n
Tm · In = In-m
Im · In = Tn-m

NB All resulting values from the above operations on n and m must be mod 12, e.g. I9 · I1 = T-8 mod12 = T4

My assignment was to generate all possible combinations, which results in a 24x24 sized matrix. At the time I wasn't quite the computer wizard I am now, so I painstakingly plugged in more or less each value into an Excel file. This was slow and also harboured the ominous risk of inputting an inaccuracy by mistake. It also wasn't very flexible in the event that I want to compose T and I operators in a universe of a different size (as unlikely as that may be); I'd have to rewrite a ton of it!

Looking back now on the assignment from the perspective of a programmer made me rethink how I would've done it at the time. Any programmer knows that solving repetitive tasks and programming are happily married. Below I complete the assignment with some C# code. The (admittedly crude in format, but complete) output follows it. 


namespace TransformationalTheory.Assignment1 {
    class Program {
        // Music is in modulo 12
        public const int ModUniverse = 12;
        // Two twelve-tone operators: T and I
        public const int NumTTOperators = 2;
        // Set dimensions of the matrix based on the above values
        public const int Rows = ModUniverse * NumTTOperators;
        public const int Columns = Rows;
        public const int Halfway = Rows / 2;

        public enum Operator {
            T, I
        }

        static void Main(string[] args) {
            // Create and set the matrix
            string[,] matrix = new string[Rows, Columns];

            for (int m = 0; m < Rows; m++) {
                for (int n = 0; n < Columns; n++) {
                    if (m < Halfway) {
                        if (n < Halfway) {
                            matrix[m, n] = TmTn(m, n);
                        } else {
                            matrix[m, n] = TmIn(m, n);
                        }
                    } else {
                        if (n < Halfway) {
                            matrix[m, n] = ImTn(m, n);
                        } else {
                            matrix[m, n] = ImIn(m, n);
                        }
                    }
                }
            }

            // Write to a file
            using (System.IO.StreamWriter file =
                    new System.IO.StreamWriter(@"C:\Users\Nick\Desktop\Assignment1.html")) {
                file.WriteLine("<table>");
                for (int m = 0; m < Rows; m++) {
                    file.Write("<tr>");
                    for (int n = 0; n < Columns; n++) {
                        file.Write("<td>" + matrix[m, n] + "</td>");
                    }
                    file.WriteLine("</tr>");
                }
                file.Write("</table>");
            }
        }

        public static string TmTn(int m, int n) {
            return Compose(Operator.T, AddMod(m, n));
        }

        public static string ImTn(int m, int n) {
            return Compose(Operator.I, AddMod(m, n));
        }

        public static string TmIn(int m, int n) {
            return Compose(Operator.I, SubtractMod(n, m));
        }

        public static string ImIn(int m, int n) {
            return Compose(Operator.T, SubtractMod(n, m));
        }

        public static string Compose(Operator a, int b) {
            return string.Format("{0}<sub>{1}</sub>", a, b);
        }

        public static int AddMod(int m, int n) {
            return (m + n) % ModUniverse;
        }

        public static int SubtractMod(int n, int m) {
            return ((n - m) + ModUniverse) % ModUniverse;
        }
    }
}


File output
To navigate the matrix, firstly take any one value from the leftmost column and secondly any one value from the top rowWhere they meet in the matrix is the value of their composition with respect to the aforementioned chosen order, e.g. T2 on row 3 (leftmost column) can be composed with I4 on column 17 (top row) to make I2 (which is therefore living in cell 3, 17). I've bolded these rows for clarity. NB that the matrix logic only works with these directions; if you choose to start with values from the rightmost column or bottom row, or pick the top row first then the leftmost column second, you'll likely end up with an incorrect composition.

You might also be wondering why it isn't possible to compose any operation with T0 in this matrix. T0 is the identity operation; i.e. T0 composed with x will always result in x. It's inclusion would therefore be quite trivial!

T0T1T2T3T4T5T6T7T8T9T10T11I0I1I2I3I4I5I6I7I8I9I10I11
T1T2T3T4T5T6T7T8T9T10T11T0I11I0I1I2I3I4I5I6I7I8I9I10
T2T3T4T5T6T7T8T9T10T11T0T1I10I11I0I1I2I3I4I5I6I7I8I9
T3T4T5T6T7T8T9T10T11T0T1T2I9I10I11I0I1I2I3I4I5I6I7I8
T4T5T6T7T8T9T10T11T0T1T2T3I8I9I10I11I0I1I2I3I4I5I6I7
T5T6T7T8T9T10T11T0T1T2T3T4I7I8I9I10I11I0I1I2I3I4I5I6
T6T7T8T9T10T11T0T1T2T3T4T5I6I7I8I9I10I11I0I1I2I3I4I5
T7T8T9T10T11T0T1T2T3T4T5T6I5I6I7I8I9I10I11I0I1I2I3I4
T8T9T10T11T0T1T2T3T4T5T6T7I4I5I6I7I8I9I10I11I0I1I2I3
T9T10T11T0T1T2T3T4T5T6T7T8I3I4I5I6I7I8I9I10I11I0I1I2
T10T11T0T1T2T3T4T5T6T7T8T9I2I3I4I5I6I7I8I9I10I11I0I1
T11T0T1T2T3T4T5T6T7T8T9T10I1I2I3I4I5I6I7I8I9I10I11I0
I0I1I2I3I4I5I6I7I8I9I10I11T0T1T2T3T4T5T6T7T8T9T10T11
I1I2I3I4I5I6I7I8I9I10I11I0T11T0T1T2T3T4T5T6T7T8T9T10
I2I3I4I5I6I7I8I9I10I11I0I1T10T11T0T1T2T3T4T5T6T7T8T9
I3I4I5I6I7I8I9I10I11I0I1I2T9T10T11T0T1T2T3T4T5T6T7T8
I4I5I6I7I8I9I10I11I0I1I2I3T8T9T10T11T0T1T2T3T4T5T6T7
I5I6I7I8I9I10I11I0I1I2I3I4T7T8T9T10T11T0T1T2T3T4T5T6
I6I7I8I9I10I11I0I1I2I3I4I5T6T7T8T9T10T11T0T1T2T3T4T5
I7I8I9I10I11I0I1I2I3I4I5I6T5T6T7T8T9T10T11T0T1T2T3T4
I8I9I10I11I0I1I2I3I4I5I6I7T4T5T6T7T8T9T10T11T0T1T2T3
I9I10I11I0I1I2I3I4I5I6I7I8T3T4T5T6T7T8T9T10T11T0T1T2
I10I11I0I1I2I3I4I5I6I7I8I9T2T3T4T5T6T7T8T9T10T11T0T1
I11I0I1I2I3I4I5I6I7I8I9I10T1T2T3T4T5T6T7T8T9T10T11T0

Tuesday, July 7, 2015

Music Theory Roman Numeral Analysis app

Do you find a lack of block-chords to analyze with the tools of music theory keeps you up at night? Is your one purpose in life to attribute Roman and Arabic numerals to music as rigidly and literally possible? I would venture a guess that 99.8% of my readership fall into this demographic. Guaranteed. To sate your insatiable quest to better your musicotheoretical chops I've developed a simple app for Android phones and tablets.  

You can find it and install it here; it's free and has no ads forever!: https://goo.gl/EiQnzU

Here's what it looks like screenshotted from an emulator and then photoshopped onto a fake phone and then Blogger added a gross black background:

Tuesday, December 30, 2014

Idioms of popular music (from the perspective of a classical musician)

I'm going to start writing a series of posts which examine aspects of popular music. These will be aspects which I find frequently in popular music's many genres but very rarely or not at all in common practice era music, and which can be described adequately with the tools of music theory. I don't intend this list to be a comprehensive study by any means; it's just a selection of things I've noticed from many years of listening to both popular and classical music. I'm not trying to come to any definition of what popular music is either, but I'd imagine that if one were to incorporate the list of things I plan to write about into their own tonal compositions, one would likely then be able to characterize their work as popular music.

I won't include songs in this series that are explicitly jazz or blues pieces because those genres contain a ton of unique elements to distinguish them from classical music. Instead, I focus on popular, radio-friendly, mass-consumed, mainstream works from roughly 1960 onward. I'll try to make each item interesting to read about by avoiding obvious differences from classical music like syncopation and orchestration. Below is the first part of this series.

Part 1.  Scale degrees 5-6-1 over a tonic chord

The title of this section refers to a simple musical idea -- three successive pitches in a melodic line. This idea is ubiquitous in pop music, and I've provided handful of excerpts of it occurring below. I like to understand this 5-6-1 melodic idea as an inverted cambiata figure, with the last gap-fill note omitted, moving through scale degrees 5, 6, and 1. All of this is done over a tonic chord. The main point of interest here is scale degree 6. From a classical musician's perspective, this scale degree on top of a tonic chord is considered a dissonance, since the tonic triad consists of scale degrees 1, 3, and 5. In the musical examples below it acts like a passing tone between 5 and 1, but rather than passing to the next scale degree in a diatonic scale, scale degree 7, it instead skips upward by a third to scale degree 1. This avoids any instance of the melodic line creating or alluding to semitonal movement (here, between scale degrees 7 and 1), which is the characteristic feature of a typical pentatonic scale (a scale comprising solely of major-second and minor-third intervals). My understanding of this phenomenon is that since so much popular music has a hard-on for pentatonicism, this is likely then a pentatonic-derived idea shoved into a diatonic context. Examples below show this with horizontal brackets identifying the exact location of the figure in each score.

Hall & Oates - You Make My Dreams
This song's opening line consists exclusively of scale degrees 5, 6, and 1. The pseudo-cambiata figure described above is used four times (twice in retrograde, with one instance of overlap) and scale degree 6 is freely skipped to and away from the tonic note, all over a tonic chord. 



Travis - Sing
Another instance of a song using the 5-6-1 figure.



Michael Jackson - Man In The Mirror
The first 5-6-1 figure is actually over an extended dominant chord (i.e. not a tonic chord), but I felt was still worth identifying. The second bracket shows it sounding over a first-inversion tonic chord.



Black Eyed Peas - Where Is The Love
This song uses solely a retrograde of the 5-6-1 figure: 1-6-5.




The Beatles - All You Need Is Love
This song opens with a quotation of the French national anthem, La Marseillaise, but with a few interesting edits. 



The pickup measure in the top voice moves along D-E-G, the familiar 5-6-1 figure I've been addressing in this section. This wasn't originally in the national anthem, however. In the opening of the original score, pictured below, the pickup is a simple 5-1 without any intermediary pitch E (scale degree 6). To me, this is an indication of how salient the 5-6-1 figure is in popular music. It's as if the Beatles decided a literal quotation of the anthem wouldn't have sat comfortably in the context of a popular music work, and thus edited it into the more idiomatic 5-6-1 figure.










Besides this addition, the Beatles's version also includes some fun parallel fifths, which La Marseillaise would've surely avoided.

The Four Tops - I Can't Help Myself
This last example is so saturated with the 5-6-1 figure that there really isn't a need for a score.




The 5-6-1 figure can be found in late classical impressionist works like that of Debussy's, who was no stranger to pentatonic collections. The majority of classical works, however, e.g. that of Bach's, Mozart's, and Beethoven's, are decidedly without its presence. 

Regarding copyright:
Edited mp3 files are available on this web site to illustrate particular analytical points as well as for ear training purposes. Many of these mp3s are copyrighted material, but I believe my use of them here constitutes “fair use” since the mp3s are short excerpts only and used for an academic purpose. Please do not hesitate to contact me if you disagree with my definition of “fair use” and wish to have any musical examples removed from this web site.

Tuesday, April 30, 2013

Brahmsfo(u)rmations or: Transformations within Brahms 4

You're probably familiar with the opening theme of Brahms's fourth symphony. It's the one that descends by thirds and then ascends by thirds. No, not that one. It's this one. Big deal, right? Well I had the audacity to look a little closer at it and let me tell you, the results are quite illuminating. Take a gander at them, won't you?

Let me preface my so-called analysis of the piece by saying that this is but a part of a greater study of mine on this work. It's also heavily inspired by Steven Rings's book Tonality and Transformation, which was a pretty decent read. I use a lot of its terminology, but I won't cite the pages I used because I can't be bothered. 

Somewhere in that book of his he mentions the concept of "tonal intention." This more or less describes the phenomenon of our attention being directed to the tonic when encountering some subordinate tonal element. A basic example would be the momentum created by the dominant sonority; it generates a strong pull towards the tonic within the context of a tonal work. I think that's what it means anyway. I will refer to this concept again in a minute, so don't forget it.

Let's get down to brass tacks. Take a look at the score. Here's the first page of a piano reduction for the work:

Figure 0 Johannes Brahms, Symphony No. 4 in E minor, Op. 98, i

Notice there are brackets labelled X and Y. They demarcate the aforementioned chains of descending and ascending thirds, respectively. That's all there is to it, right? Wrong. See below:

Figure 1 Network of the Main Theme in the first violins, mm. 1-9

Now to explain this. X and Y, the labels that bracketed a part of the theme within Figure 0, are modeled by Figure 1, the transformational network above. In it, nodes contain pitches accompanied by the number of their octave (e.g. middle C = C4); they are arranged approximately in pitch-space, where the highest node represents the highest registral pitch, and vice versa. Solid arrows connect the majority of nodes, and these are accompanied by an ordered pair consisting of scale-degree intervals and pitch intervals. Henceforth I'm going to refer to scale-degree intervals and pitch intervals as sdints and pints, respectively, so don't get confused. Each sdint represents an interval between sds; for example, the interval from sd 1 to sd 3 is a 3rd (or its inverse, a 6th-1). When an octave or unison is encountered, that is, when there is no change in sd, the symbol e, for identity, is used. The pints conform to pitch-space, where negative and positive integers represent intervals. The solid arrows trace the literal path of the theme's melody found in the score.

Take a deep breath and continue to persevere through this molasses-like prose.

The other two types of arrows are dotted and dashed. Dotted arrows trace the top layer of the theme; a path determined by the network's upper registral notes. This path does not happen as explicitly as the one traced by the solid arrows due to the many intermediary pitches within the score of the work. The sole dashed arrow is essentially a combination of the dotted and solid arrows; it literally appears in the score where B5 moves to E6 (in m. 4), but it also traces that upper line based on register. The rightmost node of Figure 1 has a B5 enclosed by a dotted circle. This special type of node implies that the pitch contained inside of it has not been realized within the measures pertaining to this network (i.e. mm. 1-9).

Before the significance of the dotted arrows can be explained, I'm going to talk about the significance of the arrival and emphasis of pc C on the downbeat to m. 9 until m. 13. Note that most of these measures are not included in Figure 1 (read: go look at the score to see them). Within them, C, as a pitch, alternates repeatedly between C5 and C6. Although three eighth-notes are found at the end of each measure which include pcs other than C, these deviations, don't disrupt the emphasis of pc C found on the downbeat of each of those measures. This area marks a point of contrast to the preceding measures.

So Brahms has decided to play around on this emphasis of pc C. I'd like to ask why he has decided to do so, but he's dead. Therefore, speculating on composer intention is little bit futile. Instead, I'd like to take a closer look at what the consequences of the prolonged C might be.

Looking to the score shows that before C5 arrives on the downbeat of m. 9, each of the first eight measures of the movement has two thematic notes (i.e. melodic notes within the theme) per full measure. These unfold supported by basic triadic harmonies built on diatonic scale degrees. There is a tonic pedal within mm. 1-4, but afterwards until m. 8 the theme is supported by root-position harmony. In contrast, the chordal harmonies within mm. 9-13 are far more dissonant; they include fully-diminished-seventh applied chords based on chromatic scale degrees (mm. 9, 12), as well as a second-inversion chord (m. 10). Therefore, in addition to emphasizing primarily the melodic pc C, as was discussed in the above paragraph, these measures also feature prominent harmonic instability.

As a pitch, chord, and key, C is an element that binds the symphony together. As a chord it plays an important role in the recapitulation of the first movement. As a pitch and key area it is prominent in the second movement as an inflection within the key of E-major. The third movement is fully in the key of C-major, and in the fourth movement C-major emerges as a key area despite this movement being in E-minor.

Returning to Figure 1; the dotted arrows trace the paths of the line formed by the highest registral notes. Within X, the pitches B5-C6-B5 are highlighted by these arrows. This idea, reminiscent of a neighbour-tone, sets up an expectation: a departure from B5 is followed by a return to it. In this sense, the motion away from B5 is intentionally directed back towards it. Another departure from the B5 initiates Y. This motion is a leap to E6, which then descends stepwise to D6, and C6, again demonstrated by the dotted arrows in this bracketed area. It appears that this is fulfilling a similar idea to what X worked out; a departure from B5 begins to return to its origin through stepwise motion, which is anticipated in Y because of the expectation introduced by X. By the time the C6 in Y is reached, however, the path to B5 is heavily delayed. As was mentioned before, there is an emphasis on pc C within mm. 9-13 which is supported by dissonant harmonies. Based on the intentionally directed dotted arrows of Figure 1, that emphasis on C builds up energy in its anticipation towards the B5 in the dotted circle node; a pitch that remains unrealized within the measures that this transformational network models. This B5 in the dotted node is eventually encountered in the score, but not until the anacrusis to m. 18. Here the expected B5 dramatically returns with the leaping of first violins and oboes through two of its lower octaves in m. 17. At this moment there also appears to be an allusion to the B5-C6-B5 traced by X within the top voice of mm. 17-18. This more literally outlines the neighbour-tone dotted arrow motion X first introduced. 

Figure 2 models a similar kind of reading of the theme but from a different perspective. This network uses a pitch-class space. Beginning with the upper-left node and then following the path directed by the arrows will trace the theme, not unlike to the solid/dashed arrows of Figure 1. Although the bottom row is labelled Y, it is slightly different from the Y in Figure 1: here in Figure 2, any repetitions of pitch-classes are omitted. The purpose of this network is to articulate that, unlike X, which begins and ends on pc B through a chain of descending thirds, Y begins on pc E and stops prematurely at pc C. This is one ascending third away from returning to pc E and the completion of a chain that returns to its origin. This missing pc E is articulated in the network by the dotted node, which implies that it is not realized in these measures, similar to the dotted node within Figure 1. Beyond the measures Figure 2 represents, however, are no immediately discernible candidates to satisfy this unrealized node. One might speculate that the pc E in the dotted node is in fact realized in m. 13, immediately following pc C. But because pc E here is merely a lone eighth note, supported by a dissonant harmony, and in a formal area where the theme becomes fragmented, it is quite alien to the material from mm. 1-9. Therefore, in my hearing, it does not grant a convincing relationship to the way each pc contained within the other nodes of this figure are presented in the score. The potential chain of thirds beginning and ending on pc E within Y remains incomplete.

Figure 2 Another network modeling mm. 1-9.

The horizontal dotted arrow in Figure 2 traces the unrealized path leading from pcs C to E; it is also consistent intervallically with the solid arrows preceding it on the bottom row, as was discussed above. The relevance of the dotted arrow leading vertically from pcs E to B, however, is a little more interpretive. Its addition unifies the entire network and enables it to be closed in a logical manner; but the music itself does not follow this path literally, nor does it convincingly imply an intentionally directed motion from pc E to B as strongly as the preceding pc C to E does. X and Y, however, are essentially repeated in the transition following the main theme (see Figure 0). This action restarts the network, but only after having skipped over that pc E in the dotted node. The two dotted arrows and dotted node of Figure 2 therefore represent the portion of the network that the music itself had ignored, and provide what I consider to be a more predictable and logical way for the network to return to the beginning of X and initiate the transition. In addition, this vertical dotted arrow creates a kind of symmetry with the solid arrow from pcs B to E on the right-hand side of the network, which alludes to the inversional qualities of X and Y.

The avoidance of Y's return to the originating pc E, represented by the dotted node of Figure 2, is metaphorically similar to a stepwise ascent of a major scale that begins on its tonic and halts motion once reaching the leading-tone. In this example, there tends to be an acoustic desire for a resolution to the tonic from that leading-tone pitch. In the case of Brahms's symphony, I would argue that the cessation of ascending thirds on pc C in Y impels a similar anticipation for a resolution to the tonic, pc E. Given that Brahms has already offered a complete cycle in X beginning and ending on the same pc, this only serves to add to the expectation that the first pc which initiates Y should also be its closer. Rings's concept of tonal intention is applicable here as well because it involves the process of subordinate tonal elements directing our attention to the tonic. The corner nodes of the bottom row, Y, in Figure 2, would be the tonic in this case, and the subordinate elements can be understood as the intermediary pcs between them. Although the corner nodes of X are not the tonic, in the sense that pc B is not the tonic of the symphony's key of E-minor, within this localized context of the network it does not seem to me a stretch of the imagination to conceive of our attention being intentionally directed in some way back towards pc B once the subordinate intermediary pcs of X are encountered. This skipping over of pc E in Y parallels of the avoidance of resolution to the tonic throughout the entire movement (a typical Romantic aesthetic), and its prolonged acoustic absence heightens intentional energy towards it. 

Both Figures 1 and 2 demonstrate that pc C plays a role in anticipating and delaying a resolution that is expected, represented by the dotted nodes. It is intentionally directed towards both pcs B and E, depending on which of these two networks one is engaged with. In a listening experience of the movement, however, one could claim that pc C is directed towards both of these pcs simultaneously because of its dualistic role within the first two figures.