Canon on White Christmas

Below is a recording I made of a canon -- at the sixth above -- that I discovered and arranged using the melody for Irving Berlin's song "White Christmas." It works rather well: I made one modification in the second half of measure 30. The bass line is a nod to that of the Air from Bach's Orchestral Suite No. 3.

This combination of Bach's and Berlin's music can also be heard in Peter Breiner's mashup on Naxos's Christmas Goes Baroque II from 1993.

The Pitches of the Music for NPR's All Things Considered Are (Just About) Right on Time

This post combines the music I considered in my post of two years ago with the methodology of my post here about some of Carl Vine's music and here about some of Sergei Prokofiev's music that show correlations between the pitches and tempos of a musical work.

Below is a partial transcription of the middle of the theme music for NPR's All Things Considered. During this middle, the music transitions from 4/4 to 5/8 with the eighth note as a common pulse. At this point of transition, there is also something of an authentic cadence in B major: a top-voice A#5 harmonized by a F-sharp-major dominant triad resolves to a top-voice B5 harmonized by what will probably heard as a B rooted chord, even though an E replaces the expected D# -- making a 027 -- and the F# is in the bass. In the 5/8 section, the change of bass from this F# to C and back again occurs every three measures.

Into the 5/8 section, there is a slight increase in a particular hypermetric frequency: the two-4/4-measures pulse (16 eighth notes long) speeds up just a tad to the bass's three-5/8-measures pulse (15 eighth notes long). When spanning two pitches, this 16:15 ratio is the just diatonic semitone. Indeed, if one transposes this particular 16:15 tempo interval of .4375/.4666... Hz up a few octaves so that it sounds as the pitch interval of a semitone, and moves the two notes up to the next available equal-tempered semitone -- if A4 is 440 Hz -- then this pitch interval is A# to B, the same top-voice melodic motion at this same point of metric transition. Furthermore, the 5/8 downbeats, which come three times as frequently as the .4375 Hz change-of-bass frequency, would correspondingly map onto an F#, which accompanies the B in the 027 harmony at the metric transition. These tempo-pitch relationships are summarized below, using a color coding from above (16:15:5). The two outside three-note groups are pitches in equal temperament, and the middle three-note group expresses the three aforementioned tempos as pitches, using a ten-octave transposition.

A Maximally Varied Stretto Fugue

Two years ago, I introduced an isochronous melody that 1) has no internally recurring patterns and 2) can be consonantly combined with itself at any transpositional level or at any time delay. This is the only such melody of eight notes -- allowing for individual octave transfers, or the melody's wholesale transposition, inversion, or rotation (moving some of the notes from one end to the other) -- that has these two properties.



I doubled the durations, moved some notes up an octave, and embellished this eight-note melody to make the following subject.

I then wrote a stretto fugue based upon this subject that demonstrates its special properties. After a standard exposition, and before a final entry in the subdominant, the subject overlaps with itself at seven different time intervals to create seven different strettos, each time interval one unit smaller than the last. Since each time interval is linked to a certain pitch interval, the choice to use an incremental acceleration of the frequency of subject entries thus stipulates the fugue's succession of key areas. However, the linear pattern of the acceleration translates into a convenient arch pattern of transpositions. After a presentation in the main key (A), this arch pattern takes the fugue along a relatively standard tour through the relative major (C), then its dominant (G) and then its subdominant (F, of sorts) before reversing course through these keys, ending with two subject entries in A minor only a single unit apart. The close proximity of the acceleration's final entries yields three more secondary strettos and requires the use of four voices, up from the three with which I decided to begin.

All of this is shown in the graphic below, and can be watched and heard in the video below.

This fugue has 10 different stretto intervals among 11 subject entries, which earns a ratio of .91 different stretto intervals per entry, and all of the stretto intervals are different. This can be compared to two stretto fugues of J.S. Bach. Contrapunctus VII of The Art of Fugue has 22 different stretto intervals that are all different, as I have shown here. However, with 26 subject entries, this fugue earns a ratio of .85 different stretto intervals per entry, a little lower than mine. The C-major fugue from Book 1 of The Well-Tempered Clavier has 8 different stretto intervals among 24 subject entries (ratio of .33), and duplicates some stretto intervals.

More Perfectly Contrarian Counterpoint in Berlioz's Symphonie fantastique

Last month, I pointed out a spot in the fifth and last movement Berlioz's Symphonie fantastique that uses successive parallel octaves. As shown in Liszt's solo piano transcription below, there is another spot in the previous movement that dwells upon perfect harmony between the outer voices, but in contrary motion. It's during the moment when the march to the scaffold first bursts forth with the entire orchestra playing fortissimo:

I hear this moment as pointedly transgressive of classical norms, which seems appropriate for music meant to accompany the opium-drugged artist witnessing his own execution, as Berlioz's own program notes describe the scene. There are innumerable examples of a diatonic progression comprising at least three chords in Western classical music in which 1) one outer voice moves by step in one direction, 2) the other outer voices alternates skipping by thirds and fourths in the opposite direction, and 3) the two voices always form imperfect harmonies. Below show 56 (7 x 2 x 2 x 2) possible versions with three chords, categorized by 1) the scale degree the stepwise line starts on (7 options), 2) whether the stepwise line goes down (d) or up (u) (2 options), 3) whether the stepwise line is on the bottom (b) or top (t) (2 options), and 3) whether the first melodic skip is a third (3rd) or fourth (4th) (2 options). I like to call these imperfect wedges.

Left out of the categorization is what mode (major or minor) the music is in, how the outer voices are harmonized, any transposition of one or both voices away from the other by one or more octaves, and so forth.

At least one of these progressions has been named: what the labeling system above designates as a 3ut3rd (for "stepwise line starts on ^3, goes up, and is on top; skipwise line starts with a 3rd"; it is enclosed in blue above) has been called the "champagne progression" by music theorist Gene Biringer and promoted at Open Music Theory. There it is recommended to "[o]nly use it with mi–fa–sol (or me–fa–sol) in the melody." Below each progression I have listed the number of instances of the progression I have found in a broad survey of Western classical music. While 3ut3rd (that is, the stepwise line starts on mi or me) is by far the most common of all of the ut3rd progressions, other ut3rd progressions are also used, particularly 1ut3rd. Moreover, the "champagne progression" is not the only imperfect wedge I would recommend as a schema: in my survey, 1db3rd (99 instances; enclosed in red above) is even more common than 3ut3rd (73 instances).

Berlioz, however, uses a perfect wedge: the same design but the outer voices are a third farther part. (I suppose you could call it a 4ub3rd perfect wedge.) Perfect wedges are much rarer in Western classical music: while I found 250 imperfect wedges in my survey, I only found 13 perfect wedges, including Berlioz's. This situates it as both atypical and perhaps also impertinent, since its design so closely resembles that of an imperfect wedge. Furthermore, in the Symphonie fantastique passage cited above, Berlioz's displacement of a third interval from more normative schematic counterpoint occurs at the same time as his displacement of the onsets from the strong beats of the meter.

More Successive Octaves in Some Music of Berlioz

As I post this, Quatuor Aeolina is performing a four-accordion transcription of Berlioz's Symphonie fantastique of 1830 at the Berlioz Festival in La Côte-Saint-André in southeastern France.

Here's more audacity: toward the end of the last movement of this work, Berlioz's combination of his original Witches' Sabbath theme and the preexisting Dies irae chant contains successive octaves. Like the successive octaves in Jerry Goldsmith's score for Patton I blogged about a year ago, these octaves take place from one compound-duple (e.g. 6/8) downbeat to the next, toward the beginning of a combination of two themes -- one sacred, one not.

A small rewrite of the Sabbath Round would have avoided these successive octaves. Here are a couple of possibilities:

I am not arguing that these successive octaves are inherently good or bad—plenty of ink has been spilled judging Berlioz's counterpoint in this manner—but they are unquestionably a deviation from the manner in which melodies were combined in classical Western music from the previous century.

Some Music of Brahms Sounds Like Some Music of Alice Mary Smith

Alice Mary Smith (1839-1884) was an English composer of choral, instrumental, and chamber music. In his 2003 edition of her two symphonies, Ian Graham-Jones states that she was the first woman in Britain "to have written and to have had performed a symphony, the Symphony in C Minor of 1863," her first. The first movement of this symphony bears some resemblance to the first movement of the first symphony of Johannes Brahms (1833-1897), also in C minor. Although the symphony was not completed until 1876, Brahms sent a draft of the first movement, without the slow introduction of the final version, to Clara Schumann in 1862.

The two movements realize their sonata forms in similar ways. Both movements use a primary theme (P) in C minor and a secondary theme (S) in E-flat major. This is no surprise, as most C-minor sonata movements do this. However, there are other less common resemblances. The P theme and the transition (Tr) sections are almost exactly the same length in terms of numbers of measures, and both insert a four-measure thematic introduction between the slow introduction and the start of the P theme, as indicated with the formal diagram below.

Lastly, each movement has a four-measure portion around the middle of its E-flat-major S theme that sounds very similar to the other, as shown below. (This portion's place in the form of each movement is indicated by the enclosure in the graphic above.) Both portions are soft, and both use alternating and echoing one-measure motives that involve the wind instruments of the orchestra. These motives essentially embellish upon a Bb-F-Bb-F treble succession. The harmony and bass line below this treble succession is the same in each portion, prolonging a first-inversion tonic triad, a standard "beginning-of-the-end" chord for a secondary theme. However, in characteristic fashion, Brahms displaces the harmonic rhythm from the barline.

Some Thoughts About the Chaconne from Holst's First Suite in E-Flat

Around this time next year will be the centennial anniversary of the first performance (June 23, 1920) of Gustav Holst's First Suite in E-Flat, one of the most well-known works written for wind ensemble. The first movement is a chaconne. The repeating chaconne melody is on the first line below: octave position may vary for all the melodies in this example. After nine statements of this melody, the chaconne is diatonically inverted to start also on E flat—shown as the second melody below—and presented twice as such. Next the third melody below—a diatonic transposition of the chaconne to start on G—is presented once, followed by a restoration of the original chaconne tune. (I am ignoring the last statement, which deviates from the three-flat collection.)

Octave aside, there are six other diatonic transpositions and seven diatonic inversions of the initial chaconne melody. The second and third melodies above are only one of each set. Why choose them, of all possible? There are many ways to answer this question. Here's one. The F-Bb succession in the original chaconne melody, highlighted with blue brackets, clearly expresses dominant function at the end of its two halves. An alternate way to express dominant function using the same diatonic interval class (fourth or fifth) is with the tritone. In the key of E-flat, the tritone is between D and Ab, highlighted with green brackets. The second melody above is the only inversion that replaces F and Bb with D and Ab. (Holst deviates from the second melody's inversion by ending on G instead of Ab—hence the dashed green bracket—but nonetheless delivers ^2 and ^5 just like the original melody, but in C minor.) Likewise, the third melody above is the only transposition that replaces F and Bb with Ab and D.

The second and third melodies are related by inversion around F. (There are many ways to recognize this: F is equidistant between the starting notes of Eb and G, and it is equidistant between the notes D and Ab, a dyad that inversion around F preserves.) The three-flat collection, such as E-flat major or C minor, inverts into itself around F, as shown above. This means that a melody in a three-flat collection that is inverted around F will maintain the same major and minor qualities of intervals. The figure above places a letter that shows the quality of each melodic interval above it: m = minor, M = major, P = perfect, A = augmented, d = diminished. The purple enclosure surrounds intervals in corresponding spots in melodies that express the same major or minor quality of seconds and thirds. This L-shaped enclosure demonstrates not only that, as aforementioned, the second and third melodies use the same quality of seconds and thirds in corresponding positions, but also that the first melody inverts these qualities exclusively in its first part, and matches them exclusively in its second.

63 Tripled Units in a 64 Span, Tune by 65daysofstatic

65daysofstatic is a twenty-first-century English experimental band. Their techno-infused track "The Distant and Mechanised Glow of Eastern European Dance Parties" appears on their third album, The Destruction of Small Ideas, which was released in the United States on the first of this month twelve years ago. Below is a YouTube recording of the track, and below that is an annotated transcription of the snare drum, kick drum, and synthesized bass from 2:09-2:55.

Before the drums recuperate at 2:16 the 4/4 time signature that has governed the track since its beginning, the synth lays down a repeating pattern of sixteenth-then-eighth, creating a three-sixteenth pulse that cuts first against the implied continuation of 4/4 and then against an explicit 4/4 when the drums re-enter. At first, it seems as if the synth bass's triple pulse will stubbornly continue its transversality. But at 2:27 it resets as it drops the octave, starting again with the sixteenth-then-eighth rhythm on the downbeat as it did when it first entered. It then resets like this every four measures, simultaneous with a change of register. Therefore, during of these four-measure spans from 2:27 to 2:55, the synth bass delivers 21 of the sixteenth-then-eighth successions, shown with the blue brackets. These total to 63 sixteenths (21 successions times 3 sixteenths each), which is one sixteenth shy of 64, the number of sixteenths in four measures of 4/4. Each of the two yellow brackets indicate this single-sixteenth difference. 

In an article where he investigates the general phenomenon of a string of threes unfolding over, but then giving way to, pure duple meter, Richard Cohn shares an awareness of examples that do so over a 64-unit pure-duple span: Bill Withers's "Ain't No Sunshine" and some music of Brazilian jazz guitarist Baden Powell de Aquino. However, in none of these examples does the string immediately repeat. At the least, this passage from 65daysofstatic provides an example that does.

But, moreover, this passage provides a compromise between the two extremes I put forth in my earlier blog post expanding upon Cohn's article. In that post, I offered two abstract examples in which, in the first, an onset in the triple pattern never falls on the start of a duple span, and, in the second — the complement of the first — an onset in the triple pattern always falls on the start of a duple span.

This passage from 65daysofstatic falls in between. Sometimes a synth-bass onset does fall on the beginning of a duple span, as shown with the 4 and 16 in green in my annotations. Sometimes a synth-bass onset does not fall on the beginning of a duple span, as shown with the 2, 8, and 32 in red in my annotations. One could hear this as a cycle, undulating between working with and working against a meter that is duple on all levels. In his discussion of a similar phenomenon in Duke Ellington's "It Don't Mean a Thing," Cohn refer to this cycle as "a wave of release and relock."

However, as cycles go, an oscillating cycle is arguably less interesting than a cycle with more than two members. To get a cycle of four members, one can use a string of five units. To restate the challenge at the end of my February 2019 post, there is a 23-second passage in a well-known song by a progressive rock band that does exactly this. I plan to blog about this music at some point, but I would much prefer it if, before then, someone else found it, revealed it, and maybe even analyzed it in a comment below. Here's a hint: the band is Yes.

Approximating π Using Lower Pi-artials

Last month I offered a post for Pi Day. The idea of intoning the number π as a melody that matches the opening of its infinite decimal (or septimal, duodecimal, etc.) representation remains dependent upon this choice of base. An intonation less dependent on such is simply π as the frequency ratio between two numbers: it sounds like a slightly flat minor thirteenth.

Three years ago my April post demonstrated how to approximate the natural logarithm (e) using musical ratios. This time around I suggest a method to do so for π, using the Wallis product.

π Sounds Classically Evil From the Start

Happy Pi Day! It can be fairly straightforward to turn π into a melody, and there are many ways to do so. For example, in π's decimal representation (3.14159...), one could assign 0 to middle C (C4) and the other nine digits to the nine white notes above C4: 1 is D4, 2 is E4, and so on, as shown below.

Or one could assign the ten digits to some other group of ten notes. Or, since the common scales of pentatonic, diatonic, and chromatic have five, seven, and twelve notes respectively, one could represent π in base 5, 7, or 12, so that each digit would correspond to a unique note in the corresponding scale, octave differences aside. The example below shows a diatonic rendition of π in base 7 (3.0663651...) in which register is freely chosen. (In base 7, .066 is quite close to .1, which is 1/7 in base 10. This is another way to see that π is very close to 3 1/7, a well-known rational approximation.)

You can find multiple examples of such representations around the internet, such as here. The resulting music sounds as one might expect: even though the digits of π are not random, the melody sounds more or less as if it were randomly generated.

However, as with so many random or apparently random phenomena, the appearance of randomness in this sequence does not preclude identifications of design. For example, one could find multiple digits in a row, such as the series of six 9's in a row that starts with the 762nd digit in the decimal representation of π. Or one could find an incremental series: 0123456789 occurs first at the 17387594880th digit. Or one could find one's birthday (mm/dd/yy, or dd/mm/yy, or otherwise) within the sequence. Any such series would be even more remarkable if π began with them, which, in the last case above, it would for someone born on this day four years ago (if one allows 3 to substitute for 03).

Aspects of design could also be identified through musical conventions. 999999 would create a distinctive sound if played as a melody -- repeated notes -- as would 0123456789 -- straight through the scale -- if adjacency of digit corresponded to adjacency within the scale. However, other aspects of design are more particular to music. For example, one distinctive design of Western classical music -- found especially in keyboard accompaniment patterns -- is a succession of evenly spaced notes whereby pairs of notes separated by a fixed time length (labeled as n notes below) are no more than a step apart in the prevailing scale, simulating smooth voice leading in multiple virtual parts. Below are some diatonic examples. The numbers below each note show the number of diatonic steps the note is away from the note that occurred n notes before it. The series of +2, +3, +3 below the excerpt from Schumann's music shows exceptions to the stepwise relations.

As shown with the first example above, π base 10, when realized as diatonic steps on and above middle C, starts with such a design with two notes in between: the next-adjacent notes F-G, D-D, and G-A are no more than a second apart from one another. This design is more infrequent with more notes in between: the first such design with three notes apart starts at digit 24 and the first with five notes apart starts at digit 28 and overlaps with the previous, as shown below. The first such design with four notes apart does not happen until digit 502, assuming the notes are in fixed registers.

Another distinctive musical design is a progression of triads, one of Western classical music's most privileged harmonies. In the second example above, which is in base 7, the first triad between successive notes is a B triad representing the digits 3, 6, 1, and 3, which begin at order position 13. However, there is no different triad immediately before or after this B triad, so there is no triadic progression. The first such triadic progression in π base 7, shown below, starts at the 696th note.

The preceding exposition provides the context for what makes the following so remarkable. Here are the opening 17 notes of π base 12, realized as notes in the chromatic scale in which 0 = C, 1 = C# or Db, and so forth. I use the digit B for 11 in base 12; it so happens that this number also stands for the note B when C is assigned to 0.

If one considers the whole-number portion of this number -- the 3 -- as a "before-the-beginning" pickup note, then π base 12, represented by chromatic notes, begins with both a triadic progression and an arpeggiated design that simulates smooth voice leading. The next such series of notes derived from the π base 12 sequence that has both of this properties does not start until the 5763rd note.

Moreover, this triadic progression is between two minor triads -- C-sharp minor and A minor -- whose roots are a major third apart and in which the "higher" triad in the minor-third relation is more like tonic -- in this case, because it comes first. As I discuss here with regard to its use in motion pictures, such a progression has been associated with villainy and the shadowy in a lot of Western music.

The four-sharp diatony of what follows, and how standard the implied harmonic progression is among the first seventeen notes, is also quite remarkable. For a Western classical musician, the opening triadic progression and its immediate continuation might as well be the equivalent of starting the fractional portion of π with 999999.

In Common Time, Ain't No Onset on a Strong Beat When Threes Unfold, Unless There Is

A couple years ago, Richard Cohn wrote an article exploring about music in "pure duple" meter — that is, music that divides time into units of powers of 2 — interacts with a rhythmic pattern that evenly divides time into a succession of 3s. There are no integers x and y such that 2^x = 3y. This means that, if a pulse with successive onsets separated by 3 unit durations -- say, sixteenth notes -- starts on the first downbeat of 4/4 music, no onset will fall on metrically relatively important moments such as beat 2 of m. 1 (4 sixteenth notes later), beat 3 of m. 1 (8 sixteenth notes later) downbeat of m. 2 (16 sixteenth notes later), downbeat of m. 3 (32 sixteenth notes later), downbeat of m. 5 (64 sixteenth notes later), and so forth. Relatively metrically important can mean, among other things, that a change of some musical aspect, such as harmony or form, is more likely to occur at these moments. The notation below demonstrates this initial stages of this pervasive non-coincidence: never is an onset from the bottom part synchronized with an onset from the top part.

The relationship between these two parts can be inverted: if in the top part, sixteenth rests and sixteenth notes are converted into one another -- producing the complement, or negative image, of the original rhythm -- then always is an onset from the bottom part synchronized with an onset from the top part, as shown below.

In his article, Cohn spends some time with Bill Withers's song "Ain't No Sunshine." The first two verses of this song each unfold over an eight-bar span, which I hear as a shortened form of the twelve-bar blues structure, with each bar in 4/4. Instead of a third verse, Withers chains together twenty-six instances of "I know" in a single breath, each sung to what would be notated as an sixteenth-eighth rhythm to match my proposed eight-bar-verse notation. The notation below shows two possible metrical readings of this music.

Cohn puts forward Reading #1. This works out quite well for many reasons:

• the "I know" chain begins a quarter of the way during the eighth measure of the second verse's span, exactly as each of the two previous verses begin a quarter way during the measure that precedes each verse's span
• the strings fade out at this reading's beginning of the third-verse substitute
• the last "know" falls on a power of 2
• the meter falls right in line with the fourth verse to come

However, I can also hear Reading #2, which corresponds to the second example above. In fact, I find that sometimes I have to work against Reading #2 in order to hear Reading #1. Something about the new musical idea prompts a resetting of the meter for me. Or perhaps Reading #2 compels, especially in retrospect, because it continuously reinforces the pure duple's junctures with onsets. In Reading #2, the word that falls on a power of 2 toggles back and forth between "I" and "know." This means one cannot stop the pattern on any power of 2, unlike Reading #1, where the word that falls on a power of 2 -- if there is a coincident onset at all -- is always "know."

One can generalize this phenomenon beyond 2s and 3s. There are no integers x,y, and z such that 2^x = zy, and z is not a power of 2. Cohn's article, and the discussion above, concern the situation when z = 3. The next largest z would be 5. I have in mind a 23-second passage in a well-known song by a progressive rock band for which z = 5 would be appropriate. However, it neither continuously avoids pure-duple moments (like my first example) nor continuously articulates them (like my second example). Rather, it inhabits a happy medium between these two extremes, creating both a pulling away from stability and a push toward resolution, all within a single perpetual process. I will blog about this music next February.

Another Protuberant 3 in Some Mozart

In January posts on this blog from two and three years ago, I recognized two late eighteenth-century sonata movements in which the recapitulation of the second theme was altered to give scale degree 3 more salience that it had during its exposition.

Here's another example, from very well-known music of Mozart: the first movement of his piano sonata in C major, K. 545. These examples show the end of the second theme in the exposition and recapitulation, respectively. The descending-fifth transposition in the recapitulation avails more room for the treble melody to stretch out in the highest register, of which Mozart takes advantage in m. 65, which is an octave higher than what one would expect given its correspondence with m. 20. It so happens that the highest note during this stretch is E6, which is scale degree 3 in C major. Likewise, Mozart stretches out in m. 69 from what would be down a fifth from m. 24, up via a new scalar arch to the high E6 (which is connected to the beam in red, and followed by another scale degree 2 beamed in red, albeit back down in the treble-clef staff).

I will leave it there. If one wonders how this might intersect with a Schenkerian reading of this entire movement (of which my reading is neither necessarily of the entire movement, nor entirely Schenkerian), I recommend consulting John Synder's intriguing approach to this movement (cited here) at some point.