Sunday, September 17, 2017

Sherlock Holmes is Derek Flint's Musical Sidekick

The sidekick: that secondary character that flanks a main character, paralleling most of the primary hero's moves. From Frank Miller's 1986 graphic novel The Dark Knight Returns, here's a visual example of sidekick trailing, mid-leap. 

Twenty years earlier saw the release of Our Man Flint, the American spoof of James Bond with James Coburn as the impossibly talented Derek Flint. Composer Jerry Goldsmith provided the score. In his main title music, the main theme is first presented in E minor by the guitar, which is also the principal key and instrument of Bond's main theme. The orchestra then picks up the main theme in A minor. The transcription below shows both this A-minor presentation of the melody and an inner voice that mostly follows the main theme's melody down a sixth.

This particular brand of chromaticism for cat-and-mouse games between masterminds has resurfaced in various places in the last fifty years, like in Ennio Morricone's theme for Al Capone from The Untouchables (1987) and Michael Giacchino's end title music for the TV show Alias (2001-2006). Here's a more recent version that returns to the east side of the Atlantic: the inner voice of Goldsmith's melody strongly resembles the A-minor melody for a central theme from BBC's Sherlock (2010-) with Benedict Cumberbatch in the title role and Michael Price and David Arnold as the show's composers.

Wednesday, August 30, 2017

A Pedagogical Piece Proceeding from Paganini (Part IV)

I have metaphorically described the opening of my Paganini variation as flitting between the southern part of the United States and the northern part of South America. (Last month’s post will hopefully clarify this statement if it makes no sense.) For example, the first progression in my variation from an A-minor triad to an E-major triad could be compared to a flight from Caracas to New Orleans. (In Paganini’s variation, the first eight measures travel back and forth like a dual homeowner or commercial pilot between these two harmonic coastal cities, whereas I (discontentedly?) journey to different but nearby ports of call.) Musically, this means that these initial chords are just a semitone (or two, with some of my chords) away from the augmented triad C-E-G#, which is represented in my geographic analogy by the Caribbean Sea plus the Gulf of Mexico. For example, both the A-minor triad and the E-major triad require only a single semitone of change to transform into this augmented triad.

In both the first eight measures of both Paganini’s original caprice and my variation on these measures, the augmented triad does not appear as a harmony, even as a waystation from consonant triad to consonant triad. (Augmented triads assume this function to some extent later in my variation.) Likewise, a typical flight from Caracas to New Orleans goes over the Caribbean Sea and the Gulf of Mexico without stopping there: most planes aren’t equipped to land on water, and most people don’t live on water. And yet it is reasonable to represent such a flight as a Caribbean flight. Likewise, Richard Cohn encourages his readers to think of the C- E-G# augmented triad as an absentee emblem for progressions like the first eight measures of either Paganini’s caprice or my variation.

Perhaps it is not completely absent, if a broader category expands our notion of what “it” is. The photo below shows how the C-E-G# augmented triad tilts the four-pan scale. All three notes are in the same pan, so the scale unsurprisingly tips straight toward that single pan.

There are exactly two other ways to put three notes on the four-pan scale so that it tilts in the same direction as C-E-G#, although these ways may not tilt the scale with the same magnitude. (By design, my scale does not measure magnitude: either it tips one way, touching a side to the table, or it does not.) These two ways are shown below.


Each of these first two motives in Paganini’s caprice—consisting of three different notes each, octaves and repetitions aside—tilts in each of these two ways, that is, in the same direction as the C- E-G# augmented triad. My chromatic variants of these two motives (in my mm. 5-6) do the same, but the two ways are swapped between the motives. All four tilts are shown below. Thus, on the four-pan scale, Paganini’s motives and my variations of these motives are in the same watery region of the musical globe as the patch of less hospitable harmony that both Paganini and I repeatedly fly over during our first eight measures.

Thursday, July 6, 2017

A Pedagogical Piece Proceeding from Paganini (Part III)

While the first nine measures of my variation mostly stayed in one sector of the round four-pan "scale," the rest of my variation features two circumnavigations around its perimeter, each using this progression: AM-Bbm-DM-Ebm-GM-G#m-CM-C#m. The first circumnavigation maintains more of an allegiance to Paganini's original tune than the second, while the second is a clearer triadic version of the first. As with our spherical home Earth, there are many ways to chart out such an Verne-like journey, and some of the routes are more popular than others.

The graphic below shows, among other things, the relative direction that every major (M), minor (m), and augmented (+) triad leans on the four-pan "scale." For example, in last month's post, I showed how the F-major triad tilts the "scale" 30° farther clockwise than the A-minor triad. On the graphic below, the F-major triad (FM) is located 30° farther clockwise than the A-minor triad (Am).

In this image, I have also used parts of Earth's geography as an analogy for this musical space. Each trio of major or minor triads that tilt in the same direction are associated with three port cities relatively near to one another on our globe. Two trios whose major or minor triads are no more than 30° apart match trios of cities on the same continent. Two trios whose major or minor triads are 60° apart are on opposite sides of a substantial body of water, which corresponds to an augmented triad. The triads in the first nine measures of my variation mostly hang out in South America, with the occasional jaunt over the Caribbean Sea to visit US cities on the Gulf of Mexico. But, after m. 9, a more potent wanderlust takes over.

When circumnavigating this space as smoothly as possible using exclusively major and minor triads, one alternates between a leg of the journey on land (a 30° difference) and a leg of the journey at sea (a 60° difference). Eight legs -- four each on land and sea -- can bring you back to where you started: 30° + 60° + 30° + 60° + 30° + 60° + 30° + 60° = 360°. (Like last month, there is that number 8 again, which is conveniently quite a frequent phrase-rhythm length.)

This geographic metaphor also provides a way to remember how voice leading interacts with triadic location, if you know your American mythology. The post last month introduced the general principle that, if you divide the number of degrees two major or minor triads differ by 30, you get the minimal number of semitones needed to change the notes of one triad to the other. So, in this case, one (semitone) if by land, and two (semitones) if by sea. Almost. (More on that later.)

Every leg of the circumnavigation can end in one of three triadic ports. That means that, starting from a specific part of the globe -- say, Salvador, Brazil -- and choosing either east or west (my variation's progression, which is highlighted in red and given a start/finish arrow in green in the graphic above, goes east, i.e. clockwise), there are 6,561 (3^8) different ways of smoothly circumnavigating the four-pan "scale" exclusively with major and minor triads such that you return to the eastern coast of South America.

There is a smaller set of globe-trotting itineraries that are more regular. The roots of two triads spanning a 60° seafaring leg can be apart by either a minor second, minor third, or perfect fifth. For example, as shown with last month's video, the first progression in my variation -- Am to EM -- moves 60° counterclockwise on the image above, west (and north) across the Caribbean Sea and Gulf of Mexico. These two roots -- A and E -- span (at its smallest) a perfect fourth. Two other possible moves 60° counterclockwise -- Am to AbM, and Am to CM -- would create root motions by minor second and a minor third, respectively. Therefore, every 60° leg can be further described by one of these three root motions; for example, Am to EM could be "60° P4." Direction of circumnavigation (clockwise (east) or counterclockwise (west)) and root interval (ascending or descending) need not be specified!

The roots of two triads spanning a 30° leg across land can either be the same or apart a major third. For example, as shown with last month's video, the fifth progression in my variation -- Fm to DbM -- moves 30° clockwise, east (and south) across the continent of South America. These two roots -- F and Db -- span a major third. Two other possible moves 30° clockwise -- Fm to AM, and Fm to FM -- would create a root motion by major third and no root motion at all, respectively. But now it appears that there are two different motions for "30° M3": Fm to DbM and Fm to AM. Again, 1 (semitone) if by land, and 2 (semitones) if by sea -- normally. For example, the cross-country Fm to DbM is fine: only 1 semitone of voice leading is needed (C-->Db). However, the cross-country Fm to AM is not: 3 semitones of voice leading are needed (F-->E, Ab-->A, C-->C#), even though Fm and AM differ by only 30°. This discrepancy is what the word "almost" in last month's and this month's posts both referred to.

However, this discrepancy is useful here, because it helps us distinguish the two types of "30° M3" from one another: one uses 1 semitone of voice leading ("30° M3") and the other uses 3 semitones of voice leading ("30° M3 3").

These six types of smallest legs have synonyms in neo-Riemannian theory:

Degrees apart
Root distance
Semitones of voice leading
(if not degrees/30°)
Neo-Riemannian label
30° (by land)


60° (by sea)




The aforementioned "more regular globe-trotting itineraries" refer to those that alternate between the same kind of leg as they progress: each land leg is always the same one of the three possible, and each sea leg is always the same one of the three possible. This leads to nine (3 times 3) possible regular itineraries.

In his book, Cohn identified examples of seven of these nine. In my review, I identified an example of one of the missing two. YouTube audio of some of these examples are linked in the table below. (Some of these aren't complete circumnavigations, but all have enough to make the regularity of the trajectory clear.) The nine thick enclosures are the nine combinations of the three possible land legs with the three possible sea legs. Eight of the nine enclosures have at least one link. The one that does not -- the alternation between 60° m2 (S) and 30° M3 3 (H) -- is the one that my Paganini variation uses, because the roots of the triads that span 60° (e.g. AM and Bbm) differ by a minor second, and the roots of the triads that span 30° (e.g. Bbm and DM) differ by a major third, and require a minimum of three semitones of voice leading.

By land
30° P1 (P)
30° M3 (L)
30° M3 3 (H)
By sea
60° m2 (S)
60° m3 (R)
60° P4 (N)

* This one takes place over the entire slow introduction, in which each triad in the journey is potentially tonicized with other harmonies.

The subdivision within each thick enclosure indicate the directed transposition between every other triad. Clockwise or eastward journeys match the + symbol; counterclockwise or westward journeys match the - symbol. This is why I was drawn to Paganini's caprice in writing music that used the S-H combination, because it required a perfect-fourth root relationship between every other triad. The second half of Paganini's caprice features ascending-fourth root motion: AM, Dm, GM, CM. Creating a S-H progression simply required converting the Dm to DM -- which is not ideal, but other composers have distorted this music far more -- and interpolating the proper minor triads.

Cohn's book goes into much greater detail into these and other topics, and introduces a number of graphs and terms that I have mostly sidestepped. In fact, someone familiar with Cohn's ideas may be wondering why I appear to be reinventing wheels here. In part, it is because Quinn's four-pan "scale" allows us to make another connection between Paganini's original caprice and my variation that Cohn's concentration on consonant triads does not. I'll bring this up next month.

P.S. The S-L combination -- specifically, with the +m3 direction -- has become quite popular recently in mainstream Hollywood film music: I can think of three movies last year that made clear use of it, and here's an example from a movie this year that, like my progression, uses exactly 8 triads in its world tour, not quite ending up where it started but providing a fun ride nonetheless.

Wednesday, June 28, 2017

A Pedagogical Piece Proceeding from Paganini (Part II)

Next month's post will divulge the rare chord progression that was adumbrated in last month's post and takes place in the last two-thirds of my variation. This month's post takes a closer look at the first third of the variation, which also exemplifies another notion in Cohn's book. However, my discussion of this notion follows from an approach by music theorist Ian Quinn, one of Cohn's colleagues at Yale University.

The picture below is of a musical balance "scale" that has four "pans" represented by colored thumbtacks and distributed symmetrically around a circle, instead of the two pans of a typical balance scale. Assuming octave and enharmonic equivalence, there are only four augmented triads, which are each assigned to one of these four "pans," such that augmented triads a whole step apart are across from one another on the circle. Any group of notes can be "weighed" on this "scale." For example, the picture below evaluates an A-minor triad, the tonic triad of Paganini’s famous caprice. The single orange ring on the blue thumbtack represents the root A, and the two orange rings on the white thumbtack represents the third C and fifth E. The circle thus leans directly toward the camera, touching the table at a point along the circle’s circumference a third of the way from the C-E-G# augmented triad to the C#-F-A augmented triad, or 30° away from the C-E-G# pan.

Here's another example, using the F-major chord. Now the circle touches the table at a point 30° farther clockwise away from where the A-minor chord touched the table. Therefore, we'll say that the A-minor chord and the F-major chord differ by 30°; more precisely, we'll say that the F-major chord is 30° clockwise away from the A-minor chord.

Quinn and Cohn use the structures that "scales" like this make for different purposes:

Quinn is more interested in the magnitude by which different types of chords tip the scale, and less interested in the direction of the tip. Any augmented triad would tip this scale the most, while a chord of the fully-diminished type or four-note cluster type (among others) are perfectly balanced and do not tip this scale at all. The magnitude of the tilt can give an idea of how "major-thirdy" a certain type of chord is.

Cohn is more interested in the direction that a chord of a certain type tips the scale, and less interested in the magnitude of the tip. He has noticed that this scale can give a relatively good idea how far the notes in one chord would need to travel to move to notes in another chord. In particular, when you divide the number of degrees two three-note chords are apart by 30, this sometimes, or perhaps even often, tells you how many semitones in total the notes in one chord minimally need to move to produce the notes in the other chord. For instance, in the example above, the A-minor triad and the F-major triad differ by 30°. Divide this by 30 and you get 1, which is the sum of all of the semitones minimally needed to change the notes of an A-minor triad to the notes of an F-major triad: A-->A (0 semitones), C-->C (0 semitones), E-->F (1 semitone) (0+0+1=1). Furthermore, if the change is clockwise, the minimal voice leading goes up; if the change is counterclockwise, the minimal voice leading does down. In this case, the change is 30° clockwise; therefore, the single semitonal motion is up (E-->F).

I write "relatively good idea" and "sometimes, or perhaps often" because this method works more frequently for some chord types than others. Between augmented triads, it always works. Between major and minor triads, it almost always works. The closer the notes can be put together, irrespective of octave, the less frequently it works. Cohn focuses his attention on major and minor triads, so this approach is pretty reliable.

My Paganini variation begins, as expected, with an A-minor chord, and then follows it with eight other major or minor triads, one for each measure: EM, FM, CM, Fm, DbM, AbM, C#m, and AM. (The number 8 is useful in how it matches the number of measures in the first section of Paganini's caprice.) These triads are exactly the eight major or minor triads that are no more than 60° away from the A-minor tonic triad. Three are exactly 60° away: EM, CM, and AbM. Three are exactly 30° away: FM, DbM, and AM. Two are exactly 0° away; that is, they tilt the circular "scale" in the same direction as Am: Fm, and C#m. The order of these triads both accommodates the general shape of Paganini's tune (while, at the same time, semitonal shifts of parts of Paganini's tune in mm. 5-7 accommodate the triads) and moves consistently down by (sometimes enharmonic) major third among these three trios: Am.........Fm......C#m, ...EM...CM......AbM, and ......FM......DbM......AM.

Moreover, as one triad progresses to the next in this variation, the number of degrees the circular "scale" shifts divided by 30 matches the minimal number and direction of semitonal changes needed for the triadic progression, as the video below demonstrates. The few progressions that contradict this correlation are avoided here. (One of those progressions will be examined more closely next month.) Overall, the voice leading from one triad to the next is relatively small, because no triad in this group of nine is more than 90° away from any other.

Sunday, May 28, 2017

A Pedagogical Piece Proceeding from Paganini (Part I)

Five years ago, the music theorist Richard Cohn published a book called Audacious Euphony. At one point in the book, he expressed no knowledge of the existence of a certain kind of triadic succession in nineteenth-century classical music. When I wrote a review of his book, I expressed the same thing. So I decided to write some music that not only used this succession, but also demonstrated other aspects from his book. It may be twenty-first-century music, but it is based upon a well-known and oft-varied piece of nineteenth-century music: the last of the 24 Caprices of Niccolò Paganini for solo violin, whose composition was completed no later than 200 years ago this November (as shown on this manuscript). Here is my variation, with a score and my approximating performance.

I'll reveal this progression, and discuss other aspects of this piece—including why I chose Paganini's caprice to illustrate parts of Cohn's approach—in the coming months.

Here is also a PDF of the score.

Sunday, April 30, 2017

A Subject's Temporal Invitation to a Countersubject's Chromaticism: Bach's Fugue in A-Flat Minor from WTC II

In this blog, I have proposed pitch-time matches in the music of Carl Vine, George Gershwin, Steve Reich, Richard Wagner, Phish, and Sergei Prokofiev. Here is an example from an earlier composer.

Below is a piano-roll depiction of the beginning of the Fugue in A-flat Major from J.S. Bach's Book II of the Well-Tempered Clavier, aligned with the usual two-dimensional orientation of Western music notation. In this grid, the units of the x-axis are notated sixteenth notes, and the units of the y-axis are semitones. Therefore, the first note, shown on the left in gray, is an eighth note long, and descends three semitones to the next note.

This jaunty subject is built upon a frame of ascending perfect fourths -- 5-semitone intervals -- arranged in stepwise descent, as shown with the notes in red and the blue lines. The second and third perfect fourths occupy twice as much time as the first, which partition this part of the subject's timespan into a quarter-half-half division, or, more generally, a 1-2-2 division, as shown with the brackets above the grid. The filling-in of the second and third fourths -- the gray notes in between the red notes -- sustains the eighth-note pulse of the first three notes, compensating for the written-out rallentando of the ascending fourths.

The 1-2-2 division can also be found in semitones, as the intervallic division of this subject's third perfect fourth -- from top to bottom -- as shown with the brackets to the right of the grid.

This match may seem rather trivial; in fact, it should seem rather trivial. The duration succession is fairly short and indistinctive, and perfect fourths can also be divided diatonically into 2-2-1 and 2-1-2 as well; the second fourth in Bach's subject uses the latter division. But this pitch-time match has important implications for the countersubject, the line that appears in counterpoint with the subject.

Part of the enjoyment of listening to a work like a fugue is made available by the textural incrementalism of its opening: it begins with a monophonic subject that immediately appears again in tandem with a countersubject. Imagining what that countersubject might be given a subject can be a exciting, high-speed challenge for a seasoned listener hearing a fugue for the first time. One goes in with certain assumptions about this countersubject's likely features: diatonic, stepwise motion (especially if the subject is dominated by leaps from beat to beat), imperfect harmonic intervals with the subject, metrical conformity, steady pacing. The only three-note beginning of a countersubject that can completely meet the first four of these five assumptions for the subject's three melodic fourths is the following melody shown in green. The numbers in the green cells indicate the intervals between the notes in the countersubject and simultaneous red notes in the subject: they are all either thirds or sixths.

A deficiency of the beginning of this countersubject is that it lacks steady pacing: it slows down according to the same 1:2 proportions outlined earlier. The subject made up for this rallentando by filling in the fourths with passing motion and octave changes. Any stepwise melody can appear to quicken its pace by filling in its major seconds with chromatic passing tones. But in this case, the countersubject is perfectly suited to do so: by virtue of the subject's 1-2-2 temporal proportions and the placement of its fourths within the scale, a complete chromatic descent from scale degree 8 to scale degree 5 is the perfect complement for the subject, as shown below. The first of the five desiderata listed earlier (diatonicism) is traded for the fifth (steady pacing). The 1-2-2 temporal rate in which the melodic fourths of the subject unfold is matched perfectly to the 1-2-2 relationship between diatonic (Ab-G-F-Eb) and chromatic (Ab-G-Gb-F-Fb-Eb) steps in the top fourth of the major scale.

And, indeed, this is what happens, albeit first in the dominant key, which is the convention for this two-voice moment in a fugue's beginning.

Scholars rightfully highlight the disparity between the exuberance of the diatonic subject and the severity of the chromatic countersubject. As part of his wonderful commentary on the music of the Well-Tempered Clavier, the pianist David Korevaar recognizes that "the descending chromatic line from scale degree 1 to scale degree 5 is the standard signal for a lament – not at all the atmosphere promised in the subject!" It seems to me that this countersubject -- however dour -- is one of the few satisfying lines, and perhaps the most satisfying of them, that the subject promised.