Monday, February 19, 2018

T and I: Twice-Related Chords in Some Music of Schoenberg

Arnold Schoenberg started and finished the first five of his Six Little Piano Pieces, op. 19 on this day 107 years ago. Here is a score of the fourth piece in which I have selected ten groups of notes, each identified by a colored enclosure and a roman numeral.

Some of the groups of contiguous notes relate to one another by 1. disregarding both the octave in which each note appears and when it appears relative to the others in the group, 2. placing each note of each group on a circle of half steps, forming a polygon, and 3. relating two polygons as either differing by a certain rotational distance or through a certain reflection.

For example, as shown below, the groups II and V have the same triangular shape, and one shape is a reflection of the other shape around an axis that falls between the notes B and C on the circle of half steps, shown with a dashed line. Music theorists call this relation inversion (I).

As another example, as shown below, the groups III and VIII have the same pentagonal shape, and one shape is, at minimum, a 150° rotation away from the other shape (allowing for either clockwise or counterclockwise rotation). When this rotation is directed, music theorists call this transposition (T).

This second example does not surprise, as the notes in group VIII are, in chronological order, all transposed down a perfect fifth from the notes in group III. The inversion of a perfect fifth is a perfect fourth, which is five-twelfths of an octave; likewise, 150° is five-twelfths of 360°.

However, as shown below, the same 150° rotational difference also relates the seven-note groups IX and X, each of which fill a full measure and have a similar single-line-then-big-chord presentation. This suggests a covert and distinctive employment of the perfect-fifth or perfect-fourth transposition, the most important transpositions in Western classical tonal music.

It stands to reason that a certain relation would be even more distinctive if it were a composite of both a certain rotational difference (one of six possible) and a reflection around a certain axis (one of twelve possible). One such composite relation occurs at least twice in this piece. As shown below, the five-note groups I and IV are related both by a 60° rotational difference and a reflection around the axis that falls on the note C, and so are the three-note groups VII and VIII.

I will leave it to the reader to find, and comment upon, two two-contiguous-note groups -- melodic intervals -- that are related by this same composite relation of 60° rotation and reflection around C. These two intervals also have the same articulation and same melodic-contour context.

Wednesday, January 31, 2018

Schubert and the Colors of Hierarchy

Franz Schubert was born 221 years ago today.

This video uses different colors to demonstrate the hierarchical embedding of pitch centers -- roots of chords and tonics of scales -- in Schubert's song "Mein!" from Die schöne Müllerin. Relationships between such centers by perfect fifths are indicated by the near-complement relation between colors, and those relationships by step are indicated by colors very similar to one another.

The small dark blue circle toward the right of the thumbnail below is immediately within a purple oval, which is within a red oval, which is within the pastel green oval. This indicates a chord that is a F#°7 chord, which is vii°7 in G minor, a key which is vi in B-flat major, a key which is bVI in the song's key of D major: thus, the chord may be understood as vii°7 / vi / bVI.

After a introduction to the graph, the video synchronizes a scrolling presentation of the graph with a performance by Genaro Mendez, to whom I am very grateful for lending his voice to this little project.

Saturday, December 2, 2017

Time Follows Pitch In "Carol of the Bells"

Below is one notation of the four-note ostinato, derived from Ukrainian folk-song literature, that serves as the basis of what later became known as “Carol of the Bells.” I have placed the ostinato in G minor, the key of multiple popular recorded versions (Mannheim Steamroller, Wynton Marsalis, Trans-Siberian Orchestra), but the key doesn't matter for what follows.
This ostinato has three properties. The first two are more common among melodies, particularly simple melodies, and they make the third more probable. Nonetheless, even with the first two properties in place, the third property is rather unusual, even for an ostinato with only four notes.

1) The ostinato uses exactly two different kinds of melodic intervals, irrespective of direction: seconds and thirds. I will call these pitch intervals one-steps and two-steps, respectively, since thirds are twice the size of seconds, generically speaking.

2) The ostinato uses exactly two different kinds of durations: some length of time and twice this length of time. (As I did above, these timespans are usually transcribed as an eighth note and a quarter note, respectively, when this carol is notated.) I will call these temporal durations one-spans and two-spans, respectively.

3) Every note in the ostinato that finishes an X-step commences an X-span. For example, the only instance of scale degree 2 comes after scale degree 3 in the ostinato; therefore, it finishes a descending-second interval, or, more generally, a one-step. Scale degree 2 is then held for a one-span. For this note, X = 1. This holds true for all four notes in the ostinato.

The diagram to the right visualizes this property. The circle shows the four notes of the ostinato, identified by scale degree (in green) and position in the triple-time meter (in red) and arranged so that clockwise order reflects temporal order. However, these notes are evenly spaced on the circle: their pitch and durational proportions are shown with the kite figures outside and inside the circle, respectively. The fact that the red kite’s position is rotated 90° clockwise from the green kite’s position means that each temporal interval immediately follows its corresponding pitch interval.

I can think of another repeating four-note motive from the music of Robert Schumann that has all three of these properties, but the shape is a rectangle instead of a kite. I’ll reveal it next month, unless someone beats me to it by identifying it in a comment below. Also below feel free to identify melodies that achieve some variation of these properties: a different number of notes, rotations by some amount other than 90° (including 0°), a non-repeating melody, absence or alteration of the first and second properties, etc.

Sunday, November 5, 2017

Some 027s in NPR's Music

If you have listened to National Public Radio, then you have probably heard many 027 chords. A 027 chord is a group of three notes that can be arranged to form a stack of two perfect fourths or two perfect fifths. These chords can be called 027 chords, because it is possible to assign those three numbers of 0, 2, and 7 to each of the notes in some register, and any two notes will differ by the same number of semitones as their corresponding numbers differ by. For example, as shown below, the group of notes B, E, and F# make a 027 because 1) they can be stacked in perfect fourths (F#-B-E), or 2) they can be stacked in perfect fifths (E-B-F#), or 3) if some E is assigned to 0, then some F# is 2 semitones higher than this E, and some B is 7 semitones higher than this E.

The chords F#-B-E, B-E-A, and E-A-D are all 027 chords. You can find all three of these chords in this passage from around the middle of Wycliffe Gordon’s arrangement of Don Voegeli’s theme for NPR’s program All Things Considered, which I’ve transcribed below. This is the arrangement that NPR is currently using for All Things Considered. The first three loud measures are typically heard uncontested in the mix, and then the remaining soft measures (which continue beyond my transcription) underlie the spoken word. The F#-B-E chord is in the muted trumpets throughout, and the B-E-A and E-A-D chords are played by the piano.

Even though this underscore is quite repetitive, it is also quite rich. The bass unfolds more unusual augmented fourths (F#-C) instead of the more standard perfect fourths (e.g. F#-C#). The return of the eight-note theme (marked "(theme)") in A major creates a clash between C# in the theme and the bass’s C natural. These two pitches reside at the two ends of a line of perfect fourths C#-F#-B-E-A-D-G-C. However, of these eight notes, the G is never sounded, even though it is the tonic of the theme in the transcription’s first two measures. The three 027 chords mentioned earlier are exactly those that fit on this line and avoid both the C and C# ends and the missing G. Moreover, just as each 027 chord is composed of three notes set in a line by a fixed unit distance of a perfect fourth, the three 027 chords of this music are also set in a line by a fixed unit distance of a perfect fourth. The diagram below, in which the double-tipped arrow represents a perfect-fourth span, shows these relationships.

Moreover, the divisions of time are quite varied and progressive: the bass line changes notes every three beats (seen easily in the page layout of my transcription), the piano chords occur every four beats, and the beat itself is divided into five parts.

NPR’s Morning Edition turns 38 today. The final moments of the NPR Morning Edition theme music, written by BJ Leidermann and arranged by Jim Pugh, express the 027 sound in a strikingly systematic and symmetrical way. An “origin story” for this music could be told using the examples below.

1: Three voices start on the same G above middle C. One leaps up by perfect fifth to D, one leaps down by perfect fifth to C, and one stays on the G to make a 027 chord, which I will mark *3.

2: These two perfect-fifths leaps are filled in with stepwise motion, but the outer voices move at slightly different rates—the top gets moving sooner, the bottom catches up later—producing an idiomatic tonal-harmonic progression: V7/IV (G7) resolving to IV (C).

3: Or, the outer voices move at the same time as they fill in their perfect fifths, producing two different three-note harmonies marked as †1 and †2. (A 027 chord stacked in fourths then precedes the same 027 chord stacked in fifths.)

4 and 5: Or, the outer voices move at very different, but still coordinated, rates to produce different 027 chords on their way to the final 027 chord. These contrasting 027 chords are marked with *1 and *2.

6: All three different 027 chords can be included in a similar progression, but this would require one of the two outer voices to give up its beeline and “dogleg” twice before getting to its final destination. In Example 6, it is the top voice that doglegs: G up to C, then down to A, then up to D.

7: The top voice’s gap between C and A could be filled in, smoothing out the top voice.

8: A transcription of the end of the Morning Edition theme. This music adds more top-voice activity to Example 7 that enables the utterance of every one of the three-note chords labeled in Examples 1-6, but in such a way that the progressions from these earlier examples are embedded within one another, as shown with the nested and overlapping slurs.

Lastly, the three 027 chords in the Morning Edition ending have the same relationships among them as the three 027 chords in the All Things Considered music: laid out in a line of perfect fourths, with the one in the middle of the line at the end of the music:

Wednesday, October 18, 2017

A Melody That Can Accept a Canon at "Annie" Time or Pitch Interval

Annie, the character from the 1982 movie based on the 1976 musical based on the 1924 comic strip, turns 95 years old today: in the 1982 movie, her birth certificate shows a birthdate of October 18, 1922. This post is not about Annie per se. However, a certain melody with a very special property also bears a fair amount of resemblance to both some pitches and rhythms of a distinctive portion of Charles Strouse's score for the musical and movie.

Imagine that you want to write a melody that can be combined by itself at a certain transpositional level and/or at a certain time delay: what is generally known as a canon. Even with the restriction that each non-embellishing harmonic intervals is a consonance -- a third, fifth, sixth, or octave -- the composition of such a canon is easy, and can be done in countless ways. Here's the beginning of a familiar one.

Now, imagine that you want to write a more versatile melody that can be combined with itself in different ways; in particular, using different pitch transpositions and/or different time delays. J.S. Bach has written melodies like this. The chart below shows all of the different combinations he uses in the first fugue from the first book of his Well-Tempered Clavier, all transposed to the fugue's key of C major and stripped of accidentals. The labels in angle brackets indicate the transposition of the later voice (e.g. +5 is up a fifth) and how much later the later voice is.

As versatile as this melody is, it still does not allow for certain canonic time intervals. For example, it is impossible to write a canon -- or stretto, as it is called in a fugue -- with Bach's melody using an eighth-note time delay and produce only diatonic consonances for the non-embellishing harmonic intervals, regardless of what note upon which you begin.

What melody could be consonantly combined with itself at any transpositional level or at any time delay? Such a melody may sound difficult to make, but, if one does not mind a highly patterned melody, it is quite simple. For example, as suggested by the notation below, this could be accomplished by a melody that continuously ascends by step in even durations (the left half below) or by a melody that alternates back and forth between two different intervals (the right half below). Think of each half of the music below not as one eight-voice music composition, but as seven two-voice compositions, each using the bottom line as the bottom voice. Notice how each of the seven members of the C-major scale start one of the seven delayed lines.

In this context, the following eight-note melody is quite special. It has no recurring intervallic patterns—never does one interval or a series of intervals immediately repeat—with the exception of the repetition of the entire melody. And yet it can be combined consonantly with another version of itself at any time delay or at any transpositional level. Only around .05% of all eight-note melodies (ignoring octave differences for individual notes) have all of these properties, and every such melody is a transposition, inversion, and/or rotation -- that is, starting at another part of the repeating loop -- of the melody below.

If one adds to the melody above a dotted rhythm, binds into one duration immediately repeated notes, and rotates it so that its first note starts on beat 2 of the common-time meter, then the result is rather similar to the music for the words "It's the hard-knock life for us" from "It's the Hard-Knock Life" from Annie. It has the same triadic arpeggiation as "It's the hard knock life" and the same rhythmic syncopation as "life for us." The example below demonstrates time intervals from one to seven eighth notes and pitch transpositions from unison through seventh; as before, notice how each of the seven places within the diatonic scale start one of the seven delayed lines. The clef is not fixed, but should be the same for any two voices in canon: this allows for certain scale modifications to keep melodic and harmonic fourths and fifths perfect.

It has the added advantage of working as a "stacked canon" in eight voices, creating a descending-third harmonic progression. Regardless of clef, this stacked canon will have a few melodic and harmonic tritones about, but not many: it's hardly hard-knock music to sing or hear.

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.