Wednesday, August 10, 2016

Modes of Limited Transposition

When a beginner is first taught an instrument, they are almost always taught major and minor scales since most of Western music (classical and otherwise) is based on these. Unfortunately, many other kinds of scales are often neglected. In theory, a musical scale is any set of notes spanning an octave. Within the standard chromatic 12-note scale, there is an enormous number of scales which exist.

Particularly interesting are the set of scales derived by French composer Olivier Messiaen called modes of limited transposition. These sets of scales bridge the gap between the chromatic scale and the diatonic scale (a fancy word for major/minor scales). The theoretical basis of the modes of limited transposition is actually very simple to understand, with the defining property of each mode being how many times it can be transposed up 1/2-step before you end up with the same set of notes with which you started.

Although they are in fact scales, Messiaen nearly always thought of his modes in terms of harmony. Going even further, he claimed to associate each mode (and each transposition of each mode) with an array of colors -- a condition known as synesthesia. He provided some very elaborate (borderline excessive) descriptions of the colors he associated with each transposition of each mode. I was hesitant to include these here since even he said that these colors are subjective (and that different people with sound-color synesthesia would have different color associations), but I ultimately decided to since they seemed so important to him.

  MODE 0

Messiaen never mentioned a "mode 0", but it is a good way to begin understanding the modes of limited transposition. It is simply the chromatic scale.

Mode 0, more commonly know as the chromatic scale. If we attempt to transpose each note +1/2-step, we have the exact same set of notes. Click on image to enlarge.
A set of diagrams illustrating the idea of limited transposition. Green indicates that the pitch is in the mode. The shaded keys on the piano diagram as well as the bold circles indicate which note the scale begins on. Click on image to enlarge.
The only transposition of mode 0.


The 1st mode is a little bit more interesting because it can be transposed. It is commonly known as the whole-tone scale (i.e. a scale consisting of only whole steps and containing no semitones) and was used very often by the French composer Claude Debussy. As a result, Messiaen rarely used it, claiming that Debussy did everything that could be done with it.

Mode 1 has two transpositions. The first line shows mode 1.1 starting on a C. If each note is transposed +1/2-step, we obtain a new set of pitches, called mode 1.2 (the 2nd transposition of mode 1). If we try and transpose mode 1.2 +1/2-step again, we end up with mode 1.1 again (starting on a D instead of a C). Click on image to enlarge.

Modes 1.1 and 1.2, respectively.


The 2nd mode was frequently used by Messiaen and is used to create some of his most distinct harmonies. It is a scale which alternates between half-steps and whole-steps.

Note: The color descriptions given for each mode refer not to the individual scale, but to the harmonies that can be created using each scale (see further down for an example of some harmonies).
  • Mode 2.1: Blue-violet rocks with flecks of gray, cobalt blue, Prussian blue with rays of purplish-blue, gold, ruby red, and mauve stars (dominant color is blue-violet)
  • Mode 2.2: Spirals of gold and silver on a background of vertical stripes of brown and ruby red (dominant colors are gold and brown)
  • Mode 2.3: Leaves of light green and prairie green with blemishes of blue, gold, and red-orange (dominant color is green)
Mode 2 has three possible transpositions. The first line shows mode 2.1 starting on a C. If each note is transposed +1/2-step, we obtain a new set of pitches, called mode 2.2. Transposing mode 2.2 +1/2-step yields mode 2.3. If we try and transpose mode 2.3 +1/2-step again, we end up with mode 2.1 again (starting on a D-sharp instead of a C). Click on image to enlarge.

Modes 2.1, 2.2, and 2.3, respectively.


The 3rd mode was another one that was used very frequently by Messiaen. In particular, he said that he thought mode 3.2 was the best of all of his modes.

  • Mode 3.1: Orange, gold, milky-white
  • Mode 3.2: Horizontal stripes of colors (listed from bottom to top) dark gray, mauve, light gray, and white with hints of mauve and yellow -- with flamboyant gold letters forming unknown writing, and a number of small, thin red and blue arcs
  • Mode 3.3: Large vertical stripes alternating between cobalt-blue and dark blue-green with red-orange lilies (dominant colors are blue and green)
  • Mode 3.4: Orange, red, a bit of blue
All four transpositions of mode 3. If we try and transpose mode 3.4 +1/2-step, we return to mode 3.1. Click on image to enlarge.

Modes 3.1, 3.2, 3.3, and 3.4, respectively.



Mode 4 has six transpositions and sounds significantly different than the previous modes because it is contains two minor 3rds (one between the 3rd and 4th pitches and another between the 7th and 8th pitches). It was used by Messiaen, but not as often as modes 2 and 3.

  • Mode 4.1: Gray, gold, a bit of blue
  • Mode 4.2: Streaks of iron-gray, pink-mauve and coppery-yellow; black and clear Prussian blue; green and purple-violet
  • Mode 4.3: Yellow and violet
  • Mode 4.4: Violet with white veins
  • Mode 4.5: Deep violet
  • Mode 4.6: Carmine red, violacious purple, mauve, gray, pink
All six transpositions of mode 4. If we try and transpose mode 4.6 +1/2-step, we return to mode 4.1. Click on image to enlarge.

Modes 4.1 through 4.6, respectively.


Modes 5, 6, and 7 were rarely used by Messiaen. Mode 6.1 was used in the first movement of his Trois Petites Liturgies de la Presence Divine. The first transposition of the sixth mode was used by the Hungarian composer Gyorgy Ligeti (1923-2006) in his Horn Trio. It also serves as the ostinato figure in his 4th piano etude Fanfares.

Mode 6.1 occurs as an ostinato passed between the right and left hands in Gyorgy Ligeti's 4th piano etude Fanfares. Click on image to enlarge.

One of the most interesting aspects of the modes is that they bridge the gap between the chromatic scale and the diatonic (major/minor) scale. If we perform the same analysis on the diatonic scales, we find that they both have twelve transpositions! Thus, going through all of the modes brings us from the chromatic scale to the set of diatonic scales.

The same diagram for diatonic scales. After 12 transpositions (only the 1st two are shown) we arrive back where we started. Click on image to enlarge.


Most of the time, Messiaen used parallel motion within the modes of limited transposition to obtain colorful, exotic harmonies.

Parallel motion chord chains in modes 2.1, 3.1, and 4.1. The light green keys are keys within the given mode and the dark green keys are the ones that are "pressed down" to play the chord shown. Click on image to enlarge.

The first four chords of modes 2.1, 3.1, and 4.1, respectively (shown in the above figure).

Note: A very good explanation of these modes (and other methods of harmonization used by Messiaen) can be found here.

Sunday, July 17, 2016

Messiaen's Turangalila-Symphonie: An Informal Analysis (Mvt. VIII -- Developpement de l'Amour)

The eighth movement of the Turangalila-Symphonie is the summit of the work and it contains one of its biggest climaxes (and perhaps one of the biggest climaxes in all of music). Like the seventh movement, it is highly fragmented and consists of many striking horizontal juxtapositions. Messiaen seemingly has no problem going from Penderecki's Threnody to the Victims of Hiroshima to The Wheels on the Bus in a matter of seconds. Click here for a map of the eighth movement (the width of each block is very approximate).


The eighth movement opens up in a mood very similar to that of the seventh movement. At this point, it is difficult to see why Messiaen calls this movement "The Development of Love" -- it sounds much more like the gates of hell opening up!

Some highlights of this section follow.

Diminishing Non-Retrogradable Rhythm

We have a repeated non-retrogradable rhythm of sixteenth note values {5-7-10-7-5} which gets diminished by one sixteenth note on each repetition. It is played by bass clarinet, 3rd bassoon, cymbals, gong, tubular bells, and basses (pizzicato). While it isn't exactly the "Statue Theme", the contours of the notes are the same as the "Statue Theme".
One cycle of the non-retrogradable rhythm. It gets diminished by one sixteenth note on each iteration. Click on image to enlarge.

The "Chord Theme"

There are two simultaneous appearances of the "Chord Theme". First, we hear the piano play the four chords repeatedly in a rhythm {4-4-3-3}, thrown about the instrument's registers and accompanied by maracas and vibraphone resonances.
The "Chord Theme" as played by the solo piano. Click on image to enlarge.

Shortly after, a very dense and spread-out orchestration of the "Chord Theme" enters in sixteenth notes played by 2 oboes, English horn, 2 clarinets, 2 bassoons, 2 horns, cornet, temple blocks, celesta, glockenspiel, violins, violas, and cellos.

Heh, it actually kind of sounds like the theme from Twilight Zone when it is heard by itself.


The transition from part 1 to part 2 is about as abrupt as it gets in terms of musical mood swings. Immediately following the terrifying last note of part 1, we have a new theme that seems to be a sort-of fast variant of the "Love Theme" -- we will refer to it as the "Joyous Theme" (this is not the same as the "Joyous Theme" in the fifth movement).
The "Joyous Theme" occurs throughout the rest of the movement in various transpositions. Click on image to enlarge.

Superimposed with the "Joyous Theme" is the "Chord Theme", hammered out in eighth notes by solo piano. We also have metallic percussion playing the {4-4-3-3} rhythm that was played by the piano in part 1.
The "Chord Theme" played by the piano. Parts like this make me wonder how anyone can possibly play the piano part (quarter note is 160 bpm). Click on image to enlarge.

The "Joyous Theme" is interrupted by a fragment of the "Love Theme", followed by the "Flower Theme" (this can be seen in the map of the eighth movement). Interestingly, the durations (in sixteenth notes) of the theme are {2-3-5-8}, which is a fragment of the Fibonacci Sequence.
This lushly-orchestrated fragment of the "Love Theme" interrupts the "Joyous Theme" throughout the eighth movement. Click on image to enlarge.

This series of events repeats itself once with everything transposed up 1/2-step (to G major) except for the "Chord Theme" and the "Flower Theme", which remain at the same pitches. Following this, we have the first of the three "Love Theme" explosions. These consist of massive crescendos in mode 2 (always with a trill in the maracas -- an instrument characteristic to Messiaen's build-ups) leading into a colorful display of the "Love Theme" (3 trumpets, ondes-Martinot, and strings) accompanied by piquant chords in the woodwinds, celesta, glockenspiel, and piano.

The "Love Theme", played by 3 trumpets, ondes-Martinot, and full strings. Harmonies are formed by using parallel motion in mode 2.3. Click on image to enlarge.

Chords in mode 2.3 played by the solo piano during the 1st explosion of the "Love Theme". Notice how one hand lags behind the other by one sixteenth note. Click on image to enlarge.


Part 3 abruptly ends the preceding climax in part 2 with a series of agitated, dramatic gestures. Messiaen also introduces a new section which uses opposing scales of chromatic durations. Group A (arguably the most prominent) plays a rising scale in mode 3.1 with durations increasing from 1 sixteenth note to 9 sixteenth notes. Group B plays a descending scale in mode 4.4 with durations decreasing from 9 sixteenth notes to 1 sixteenth note. Group C plays an ascending whole-tone scale (mode 1) in eighth notes. All of this is accompanied by figurations in the piano, piccolo, and celesta.

Opposing scales of chromatic durations in different modes are used to create tension. Click on image to enlarge.
Group A:

Group B:

Group C:

Groups A, B, and C:

Actual recording:

What follows is basically a repeat of part 2 (juxtaposition of the "Joyous Theme", the "Love Theme" fragment, and the "Flower Theme") in different keys. We come to the second explosion of the "Love Theme" (second climax), but this time it is in D major and it lasts longer.


Part 4 immediately begins with the "Joyous Theme", but this time it sounds far more desperate and passionate. No longer do we have the gentle, lush interludes comprising of the "Love Theme" fragment and the "Flower Theme" which were in the two preceding parts. Instead, the "Joyous Theme" is interrupted by dramatic gestures involving swooping glissandos played by the ondes-Martinot. The opposing scales of chromatic durations appear twice, exactly as they did in part 3 (although the 2nd appearance is transposed up 1/2-step). The whole section seems to be a race to something unbelievably and overbearingly massive -- and it is! The third explosion of the "Love Theme" (the third climax of the movement, referred to by Messiaen as the "summit" of the Turangalila-Symphonie) must be one of the biggest climaxes in all of classical music (1:24 in the audio clip below). It occurs in F# major, which is arguably the symphony's home key.

The full "Love Theme" is played, with sections repeated several times. It gradually dies down from a burning fire to just a warm glow as it becomes quieter and lower (finally played by cellos and basses).


Another strikingly abrupt transition! The coda of the eighth movement consists of the same material that the introduction consists of, although it is a little bit more complex. We have the diminishing non-retrogradable rhythm, the chord theme in the solo piano, and the chord theme in the orchestra, but now Messiaen adds two layers of the "Statue Theme" in the brass. Click here for a diagram of part 5. Referring to the diagram, this "layered" section of part 5 ends when group 1 cannot be diminished anymore (since the minimum duration reaches 1 sixteenth note).

The movement closes with another return of the "Statue Theme" followed by a horrific blast from the tam-tam, bass drum, piano (its three lowest notes), and basses.

Note: In this movement, all audio recordings either ones that I generated using Finale or from Hannu Lintu's recent (and outstanding) recording of the piece with the Finnish Radio Symphony Orchestra.

Monday, July 4, 2016

Messiaen's Turangalila-Symphonie: An Informal Analysis (Mvt. VII -- Turangalila 2)

The seventh movement, entitled Turangalila 2, is the shortest of the piece. It is also the most intense and terrifying of the 10 movements. Structurally, it is a series of contrasting events and is highly fragmented. For convenience, we will divide the movement into nine parts.

A map of the seventh movement with rehearsal numbers for reference. Click on image to enlarge.

PART 1 (Through Rehearsal No. 1)

The movement opens up with a solo piano playing a more aggressive rendition of the gentle birdsong heard in the previous movement.

PART 2 (Rehearsal Nos. 1-2)

Messiaen says that this section was directly inspired by Edgar Allen Poe's story The Pit and the Pendulum. The ondes-Martinot plays a slowly descending scale in its upper register and trombones/tuba play a slowly ascending scale in their lower registers, giving the feeling of the walls slowly closing in on the prisoner.

During this, the rest of the orchestra plays a series sixteenth notes in a Klangfarbenmelodie setting. Klangfarbenmelodie (tr. sound color melody) is a term devised by the 20th-century Austrian composer Arnold Schoenberg. It refers to a melody constructed with different timbres instead of just different pitches.

PART 3 (Rehearsal Nos. 2-3)

During part 3, only six percussionists play in three pairs. Woodblock is paired with bass drum (this is the most prominent pair), triangle is paired with maracas, and Turkish cymbal is paired with Chinese cymbal. Each of the three pairs plays a different tala, but one player in the pair reads it from left-to-right and the other in the pair reads it from right-to-left (retrograde). Click here to see a map of part 3.

PART 4 (Rehearsal Nos. 3-6)

Part 4 consists of a cello solo accompanied by birdsong in the piano and woodwinds. Vibraphone and cymbal have a repeated rhythm with 8 sixteenth notes added on each iteration -- an example of rhythmic augmentation.

PART 5 (Rehearsal Nos. 6-7)

This part is an exact retrograde of part 2 (i.e. it is part 2 read backwards).

PART 6 (Rehearsal Nos. 7-9)

This part is among the most complex in the symphony and is a compilation of several of Messiaen's rhythmic compositional techniques. Similarly to part 2 of the first movement, Messiaen layers several processes, resulting in a very dense and chaotic sound whose aesthetics are much closer to Skrillex than to, say, Beethoven. Click here to see a map of part 6.

Cell 1: Flute, Piccolo, & Celesta

Cell 1 is a five-measure long sequence of sixteenth notes. This was actually heard previously in the third movement (entitled Turangalila 1) and a variation will appear in the ninth movement (entitled Turangalila 3).
A combination of flute, piccolo, and celesta. Click on image to enlarge.

Cell 2: Solo Piano

The solo piano plays a series of three chords in a right/left hand rhythmic canon consisting of the Ragavardhana-Candrakala-Lakskmica juxtaposition. The left hand lags behind the right hand by four sixteenth notes.
A right/left hand rhythmic canon with the Ragavardhana-Candrakala-Lakskmica juxtaposition. Click on image to enlarge.

Cell 3: The "Chord Theme"

This is probably the most interesting of all cells in this section. Messiaen assigns each of the four chords from the "Chord Theme" to different groups of instruments.

The four seven-note chords which make up the "Chord Theme".
  • Chord 1: 1st trumpet, violins, violas, cellos, small cymbal (up 1 octave)
  • Chord 2: 2 clarinets, 3 bassoons, 3 horns, Chinese cymbal (down 1 octave)
  • Chord 3: 2 oboes, violins, violas, cellos, suspended cymbal
  • Chord 4: Bass clarinet, 3 bassoons, 4th horn, 3 trombones, tuba, ondes-Martinot, gong (down 2 octaves)
Referring to the graphical analysis we can see that chords 3 and 4 remain in their locations. However, chords 1 and 2 "approach" chord 3 by one 16th-note on each iteration until the first three chords sound simultaneously. Click here for a visual representation of this cell.

Cell 4: Vibraphone & Glockenspiel

Vibraphone plays the same thing that they had in part 4. Glockenspiel plays it in rhythmic canon, lagging behind the vibraphone by an eighth note.

Cell 5: Temple Blocks & Maracas

This cell consists of three sixteenth-notes played by the temple blocks followed by a trill played by the maracas. The initial value of the maracas trill is 16 sixteenth-notes and it decreases by 1 sixteenth-note on each iteration (rhythmic diminution). Click here for a diagram.

Cell 6: Side Drum

The side drum repeats a tala three times. Each cycle has a duration of 46 sixteenth notes.

Cell 7: Bass, Bass Drum, & Triangle

Here we have two scales of chromatic durations playing against each other. The triangle expands from a value of 1 sixteenth note to a value of 16 sixteenth notes. Meanwhile, the bass and bass drum contract from a value of 16 sixteenth notes to a value of 1 sixteenth note. In fact, part 6 ends when the bass and bass drum have reached the end of their scale.

PART 7 (Rehearsal Nos. 9-10)

Part 6 is immediately cut off by solo piano in a return of the birdsong heard at the beginning.

PART 8 (Rehearsal Nos. 10-12)

A series of dramatic gestures thrown about the orchestra. The "Statue Theme" makes an appearance. A fast, descending scale of cluster chords in the piano followed by an enormous gong crescendo leads into part 9.

Piano cluster chords as heard at 22 seconds in the audio clip above.

PART 9 (Rehearsal No. 12 to End)

Part 9 is a combination of parts 2 and 3. Messiaen has altered part 2 by adding violins to accompany the ondes-Martinot's descending scale. He begins the "Klangfarbenmelodie" as he does in part 2, but halfway through the part (four measures in) he reverses it so that this layer is a palindrome about the barline separating the third and fourth measures of the part. At the same time, he also has the six percussionists play what they had in part 3. The movement ends with a "lightning and thunder" effect -- a flicker of grace notes played in the upper register of the piano followed by a booming strike of the bass drum.

The palindrome in the "Klangfarbenmelodie" layer of part 9. Click on image to enlarge.

Note: In this movement, all audio recordings either ones that I generated using Finale or from Riccardo Chailly's recording of the piece with the Royal Concertgebouw Orchestra. I've been using Hannu Lintu's recording for most audio clips, but I think that Chailly does a slightly better job with the 7th movement.

Saturday, July 2, 2016

Messiaen's Turangalila-Symphonie: An Informal Analysis (Glossary of Compositional Techniques)

Throughout his career as a composer, Messiaen used a variety of compositional techniques. Many of these were developed impressively early and he was so confident in them that he even published a short book "Techniques of My Musical Language" in 1944 -- when he was only 36.

This section is by no means complete and will be gradually expanded upon.


Often times in Messiaen's music we hear strange chords which sound like major chords with "extra notes". Or sometimes we hear melodies that aren't in major or minor keys, but they also can't be justifiably called atonal. This is because Messiaen derived his own set of modes (these are not the standard modes which are often taught in music theory -- i.e. dorian, lydian, etc).

Rather, Messiaen's modes are sets of pitches which can be transposed only a certain number of times before ending up with the original set. For instance, we could call the chromatic scale the zeroth mode (although I haven't come across any reference to this elsewhere). Transposing this set up one-half step gives us the exact same set of notes. Similarly, the whole-tone scale is the first mode. If we start with the set P0  {C D E F# G# A#} and transpose it up one-half step, we get the new set P1 {C# D# F G A B}. If we try and transpose the new set P1, we end up with {D E F# G# A# C} -- which is precisely the same as P0 starting on a D instead of a C! See the Wikipedia article Modes of Limited Transposition for a good overview on this topic. I also made a separate post which goes over many of the modes.

Like his predecessor Claude Debussy, Messiaen used parallel motion as a way of harmonization. However, Messiaen applied parallel motion to his set of modes instead of just diatonic, whole-tone, and pentatonic scales. We will use a descending, harmonized scale that appears frequently in part 2 of the 1st movement as an example on how this works.

Figure 1: Thrown back and forth between the brass section and solo piano, this scale in mode 3.1 appears 16 times in part 2 of the 1st movement and is harmonized using parallel motion. When Messiaen finds something he thinks is interesting, you are going to hear it again, and again, and again.
To start, lets look at and listen to a simple one-octave scale in mode 3.1.

Figure 2: Mode 3.1

Notice that the top eight notes on the descent are the same as the top voice in the previous figure. Messiaen constructs chords by adding voices -- layers of the same scale, but starting in different places.

Figure 3: A bit messy to read, but it illustrates the idea of adding voices.

Figure 4: The final result with a total of 6 voices.

It is interesting to see how many major chords are embedded within the harmonies. On the top staff, we can see that the first chord is a C major chord, the second chord is a B major chord, the fourth chord is an A-flat major chord, the fifth chord is a G major chord, the seventh chord is an E major chord, and the eighth chord is an E-flat major chord! The bottom staff doesn't contain any major chords, but we can recognize that the third and sixth chords are diminished.


Messiaen's music is filled with non-retrogradable rhythms. To "retrograde" essentially means to go back in time, or to reverse. Applied to music, this would mean reading something from right-to-left instead of the usual left-to-right. Thus, a rhythm that is non-retrogradable is one that does not change when it is read backwards. This is the musical equivalent of a palindrome (i.e. "wow" or "Rise to vote, sir").

Figure 5: Six examples of non-retrogradable rhythms. See below for audio clips of each.
Example 1:

Example 2:

Example 3:

Example 4:

Example 5:

Example 6:


One of Messiaen's most frequently used techniques is the augmentation and diminution of a rhythm. This involves repeating some base rhythm and on each repetition adding or subtracting a value.
Figure 6: An example of rhythmic augmentation (top) and diminution (bottom) using the base rhythm 2-1-2. Click on image to enlarge.

For the example of rhythmic diminution (bottom line), notice how the last iteration (2-1-2) cannot possibly be subtracted from without removing the middle note (we would have 1-0-1). There seem to be many cases in Messiaen's compositions where this "triggers" some other event. For instance,  in the 3rd movement when the maracas can no longer be contracted any further (see Figure 8 in the "Rhythmic Characters" section), the ondes-Martinot enters with wailing glissandi.


For the trivial case of rhythmic augmentation and diminution, the base rhythm is a single note (i.e. a sixteenth note to which a sixteenth note is added upon each iteration). Then we have what Messiaen refers to as a scale of chromatic durations.
Figure 7: An example of a scale of chromatic durations in sixteenth notes, ascending from a value of 1 to 8.

A dramatic example of scales of chromatic durations happens three times in the eighth movement. One group (mostly brass, lower winds, and percussion) plays an ascending scale, a second group (higher winds and strings) plays a descending scale, and a third group plays constant eighth notes -- all simultaneously! It is the sonic equivalent of being drawn and quartered.


In his scores, Messiaen calls this "rhythmic personnages" which approximately translates to "rhythmic characters". The idea here is that he has a set of three base rhythms -- let's label them A, B, and C (which he often labels them as in his scores). These rhythms are repeated several times in that order -- A, B, C. On each repetition, he alters one by augmenting it by a given value. Likewise, he alters another by diminishing it by the same value. The third rhythm remains constant.

There are several places where this happens during the Turangalila-Symphonie (not to mention the rest of his output), but there is a particularly clear example in the middle of the 3rd movement. Rhythm A is played by the maracas and is diminished by one 16th note, rhythm B is played by the woodblock and remains constant, and rhythm C is played by the bass drum and is augmented by one 16th note.

Figure 8: An example of "Rhythmic Characters" occurs at rehearsal no. 6 in the 3rd movement. Notice that rhythm B is an example of a non-retrogradable rhythm. Click on image to enlarge.


Messiaen also has a melodic counterpart to the technique of rhythmic augmentation/diminution which he calls "Agrandissement Asymetrique". The English translation "Asymmetric Aggrandizement" isn't a good descriptor of the technique (nor does it really make sense), so I am going to call it "Systematic Transposition". For the technique of rhythmic augmentation/diminution, recall how a rhythm is repeated and various notes have values either added or subtracted upon each iteration. Systematic transposition does the same thing, but with transposition of pitches up or down an interval. For instance, let's say that our "melody" is three A's. Now, let's say that the first 'A' will be transposed up 1/2-step on each iteration, the second 'A' will be transposed down 1/2-step on each iteration, and the third 'A' will remain constant.

A simple melody consisting of three A's. Notes highlighted in green are transposed up 1/2-step on each iteration, notes highlighted in red are transposed down 1/2-step on each iteration, and the third note remains the same. The numbered brackets indicate each iteration. Click on image to enlarge.

 As a more complicated example, let's try and apply the same technique to the first two bars of Mary Had a Little Lamb and see how it evolves.

Systematic transposition applied to the first two bars of Mary Had a Little Lamb. Notes highlighted in green are transposed up 1/2-step on each iteration, notes highlighted in red are transposed down 1/2-step on each iteration, and notes that aren't highlighted remain the same. The numbered brackets indicate each iteration. Click on image to enlarge.

Monday, June 27, 2016

Messiaen's Turangalila-Symphonie: An Informal Analysis (Motivic Index)

The Turangalila-Symphonie is filled with a number of recurring melodies, chords, and rhythms which span all ten movements. Messiaen identifies four of them in the score -- the "Statue Theme", the "Flower Theme", the "Love Theme", and the "Chord Theme". There are also a couple of other recurring ideas which will be discussed here.

The notion of recurring themes also plays a prominent role in his major piano work Vingt Regards sur l'Enfant-Jesus (tr. Twenty Contemplations on the Birth of Jesus), written just a few years earlier. In fact, both works contain a "Love Theme" and a "Chord Theme" which are similar.


This theme is described by the composer as brutal and oppressive. In its original form, it is played by the trombones in 3rds. Messiaen grew up in Grenoble, France and was impressed by the awe-inspiring, imposing, and almost terrifying presence of the towering French Alps. Many of his more terrifying themes are clearly influenced by this environment.

The "Statue Theme" in the most primitive of its many guises (also referred to by the composer in the score as the 1st cyclic theme).

The "Statue Theme" plays a particularly important role in the 5th movement, where it is transformed into a frenetic cosmic dance (shown in my analysis of the 5th movement).

As heard in the 1st movement:

As heard in the 4th movement (looming brass entrance at around 15 seconds into the clip):

As heard in the 5th movement:

As heard in the 7th movement (around 6 seconds into the clip):

As heard in the 8th movement (occurs as a brass canon at around 10 seconds into the clip and then plainly 1 minute in) :


The antithesis/foil of the ominous "Statue Theme". It is quiet and gentle. Messiaen likens it to a blossoming flower. Out of the four cyclic themes identified by the composer, it seems to play a significantly smaller role. The first two measures are played by two clarinets and the third measure is played by flute/bassoon. It appears most frequently in the 8th movement.

The "Flower Theme" (also referred to by the composer in the score as the 2nd cyclic theme).


Messiaen says that this is the most important of the four cyclic themes he identifies. While the theme itself doesn't occur until the slow 6th movement, variations of the theme play significant roles in the 2nd and 4th movements. It is played both quietly (in the 6th movement) and with full-force (in the 8th and 10th movements). Usually accompanied by garish and colorful harmonies.

The full "Love Theme" is heard in the 6th, 8th, and 10th movements (also referred to by the composer in the score as the 3rd cyclic theme).
As heard in the 6th movement:

As heard in the 8th movement as a fragment:

As heard in the 8th movement as an "explosion" (begins at 11 seconds in the clip):

As heard in the 10th movement (begins at 18 seconds in the clip):

A variation of the "Love Theme" heard in the 2nd movement, played by the saccharine sounds of strings and ondes-Martinot.

Another variation of the "Love Theme", played passionately in the second trio of the 4th movement.


This is the most elusive of the four cyclic themes identified, but it is probably the most common. It is a simple chain of four highly dissonant chords, each containing seven pitches.

The "Chord Theme" (also referred to by the composer in the score as the 4th cyclic theme). It is usually (but not always) played by the piano.

The "Chord Theme" seems to form a harmonic basis of the work and likely appears in every movement, but I have only been able to locate it in the 2nd, 4th, 7th, 8th, and 10th movements.


One of Messiaen's many musical fixations was Indian rhythms (often referred to as "talas", although I admit that I have been using that term too loosely). He collected a series of ancient Indian rhythms and often used them as what he called a "rhythmic pedal" -- that is, a series of repetitions played over other material. Three talas which he seemed to be particularly fond of are the Ragavardhana, the Candrakala, and the Lakskmica. Not only that, but he specifically liked the result when the three were juxtaposed in that order, hence the title of the theme. It occurs in several of his early works, including Quatour pour le Fin du Temps (tr. Quartet for the End of Time), Vingt Regards sur l'Enfant-Jesus (tr. Twenty Contemplations on the Birth of Jesus), and Visions de l'Amen (tr. Visions of the Amen).

The Ragavardhana-Candrakala-Lakskmica Juxtaposition plays a major role in Messiaen's music.

Like his student, the maverick German composer Karlheinz Stockhausen, Messiaen had a strong dislike of steady pulses. This rhythm is used to keep a beat, however unsteady. Within the Turangalila-Symphonie, the theme occurs in the fourth, fifth, seventh, and tenth movements (and perhaps more that I have yet to find). The Ragavardhana and Lakskmica also appear independently in the first movement (shown in my analysis of the 1st movement).

As heard in the 4th movement (played clearly by woodblock starting at 4 seconds into the clip):

As heard in the 5th movement (very difficult to pick out, but it played by the cymbal and triangle):

As heard in the 7th movement (again very difficult to pick out among the cacophony, but it is played by the piano where the two hands are in a rhythmic canon):

As heard in the 10th movement (played clearly by woodblock and cymbal from the start):

Thursday, June 23, 2016

Messiaen's Turangalila-Symphonie: An Informal Analysis (Mvt. V -- Joie du Sang des Etoiles)

Before even listening this movement, the title "Joie du Sang des Etoiles" ("Joy of the Blood of the Stars") makes it clear as to what kind of music we are about to hear. In volume II of Messiaen's seven-volume Traite de Rythme, de Couleur, et d'Ornithologie, the composer freely admits is excessive and over-indulgent, yet he argues that this is necessary to convey a sense of unbridled joy and enthusiasm.

Structurally, the movement can be divided into three parts and a coda. Part 1 introduces the main theme, part 2 is a rhythmic development of the "Statue Theme", and part 3 is basically a reappearance of part 1. A piano cadenza (based off of the "Statue Theme", of course) links part 3 and a short coda. Part 2 is the most complex and will be divided into four subsections which we will call 2A, 2B, 2C, and 2D.

Blogger hates me and won't show the score right-side up (yes, I tried uploading an upside-down image as well and that didn't work either), but this is the first two pages of part 2A (with my notes). Try right-clicking "view image" to see if it will display properly.


The movement opens up immediately with the main theme of the movement, which is to be repeated many times (a music critic I know likes to say "If you hear something you like in this symphony, don't worry -- you'll hear it again!").

At first listen, the melody seems to be a little bit cheesy -- unabashedly tonal in the key of D-flat major, lots of "vulgar" major chords with added 6ths, and an over-the-top orchestration. However, upon closer inspection it becomes clear that the melody is actually a transformation of the "Statue Theme"! By transposing the brutal "Statue Theme" and placing it on a rhythm in 3/16 time, he turns it into a frenetic cosmic dance. The first audio clip below shows how the "Statue Theme" is harmonically transformed (we hear the original theme followed by its transformation). The second clip is as it is heard in the fifth movement. From here on, we will refer to this transformed theme as the "Joyous Theme".

PART 2A (Rehearsal Nos. 14-21)

While part 1 is a lot of fun, part 2 is where things get interesting and where Messiaen uses his arsenal of compositional techniques -- rhythmic characters, retrograde, systematic transposition, non-retrogradable rhythms, and more. Click here for a diagram of the processes in part 2. An audio clip of the beginning of the part is below.

Note: For the diagrams in this section, one minor (thin) grid line is equal to one sixteenth note. Since it is in 3/16 time and there are three minor grid lines per major (thick) grid line, each major grid line represents one measure.

Try and make a few observations on your own before reading any further (part of the fun of this piece is finding new patterns). In particular, what do you notice between the top part (rehearsal numbers 14-21) and the bottom part (rehearsal numbers 25-32)? Look carefully -- paying attention to the violin and ondes-Martinot parts makes it more obvious. Also, can you find any examples of rhythmic characters? Any examples of non-retrogradable rhythms? 

Cells 1, 2, & 3: Brass Groups "A", "B", and "C"

Instruments: 4 French horns, 3 trumpets in C, 3 trombones, Chinese cymbal, suspended cymbal, glockenspiel, celesta

Cells 1, 2, and 3 are the heart of this section. This is a rhythmic development of the "Statue Theme" (the original theme, not the transformed theme demonstrated above) using the idea of rhythmic characters. We notice two things immediately. The most obvious is that the cells sound in the same order each time (i.e. first we have the green block, then the blue block, and then the red block). The other thing that we notice is that on each iteration, Messiaen changes the durations of the green and blue blocks (representing cells 1 and 2, respectively). For instance, follow the green blocks and count the length of the black spaces in each one. On its first appearance we have 4-1-4. Next we have 5-2-5, then 6-3-6, etc. The rhythm is expanded by one sixteenth note on each iteration. If we do the same for the blue blocks, we can see that we start at 8-4-8, then go to 7-3-7, then 6-2-6, etc. The red blocks more or less remain constant.

The collective brass parts (left) juxtaposed with their respective rhythmic diagrams (right). Click on image to enlarge.

Part 2A ends after group B cannot possibly be contracted any more (the smallest duration is one sixteenth note) triggering group C to seemingly panic in the form of a long non-retrogradable rhythm, leading into part 2B.

Cells 4 & 5: Woodwinds, Strings, and Piano

Instruments: Piccolo, 2 flutes, 2 oboes, English horn, 2 clarinets, bass clarinet, 2 bassoons, 2nd violins (divisi), violas (divisi), solo piano

Cell 4 is the "Statue Theme" played by a combination of woodwinds and strings. The first two notes are played by piccolo, two flutes,  two oboes, and two clarinets. The following two notes are played by divisi 2nd violins and violas. The last three notes are played by 2 oboes, English horn, 2 clarinets, bass clarinet, and 2 bassoons. On each iteration the set of pitches is transposed down a half-step except for the last two notes, which remain at the same pitches. This is an example of what we will call "systematic transposition".

The "Statue Theme" undergoing systematic transposition in cell 4. Click on image to enlarge.

Cell 5 is the "Statue Theme" played by the solo piano and runs just one sixteenth note behind cell 4 (this can clearly be seen on the diagram -- look at the purple and yellow blocks). On each iteration the set of pitches is transposed up a half-step except for the first two notes, which remain at the same pitches.

The "Statue Theme" undergoing a different systematic transposition in cell 5, lagging one sixteenth note behind cell 4. Click on image to enlarge.

Cells 6 & 7: Solo violin and ondes-Martinot

Instruments: 1st violin solo, ondes-Martinot

In an unusual orchestration decision, Messiaen has a single violinist playing its own line among a densely-scored section. He pairs it with the ondes-Martinot and the two instruments toss a series of trills and glissandos back-and-forth (actually, the violin has discrete notes -- not glissandos -- but they serve the same purpose as the glissandos in the ondes-Martinot).

While there isn't much to discuss here, it does provide a framework for the entire section. Furthermore, it makes it a little bit clearer that parts 2A and 2C are nearly mirror images of each other.

PART 2B (Rehearsal Nos. 21-25)

In part 2B (referred to in the diagram as "first glimpse of main theme") we hear a return of the "Joyous Theme" accompanied by what appears to be chaos in the extreme upper and lower registers of the orchestra.

Cell 1: The Statue Theme

Instruments: Piccolo, 2 flutes, bass clarinet, 3 bassoons, celesta, glockenspiel, solo piano, cellos, basses

The "Statue Theme" is played frenetically in repeated groups of seven 16th-notes at a dynamic of forte (fortissimo for the solo piano). Upon each repetition, the two outermost-notes on each end are transposed down a half-step (leaving the central three notes untransposed). On the sixth repetition, the entire cycle begins again with all of the notes (including the central three notes) transposed up a half-step.
The first six repetitions of the "Statue Theme" in cell 1, showing a systematic transposition. After repeating itself five times, the cycle starts again with all of the notes transposed up a half-step. Click on image to enlarge.

In case you are still having trouble believing that this is actually the same "stone-cold" theme that first appeared in the 1st movement, try listening to this slowed-down audio clip to convince yourself:

The orchestration is particularly clever in that Messiaen uses the upper and lower registral extremes of the orchestra. The use of high instruments (piccolo, flutes, celesta, glockenspiel, right-hand piano) contributes to a "glitter" effect, whereas the use of low instruments (bass clarinet, bassoons, left-hand piano, cellos, basses) gives the heavy feeling of frenetic, joyous stomping.

Cell 2: The Joyous Theme

Instruments: 2 oboes, English horn, 2 clarinets, 4 French horns, 3 trumpets in C, 3 trombones, tuba, ondes-Martinot, violins, violas

Cell 2 re-introduces the main "Joyous Theme" of the 5th movement. Another interesting effect is obtained here. While Messiaen has the accompanying "Statue" theme in group 1 playing forte/fortissimo, he instructs the main theme in group 2 to play piano up until the 17th measure of the section, giving a feeling of order emerging from chaos. This section is particularly interesting because the composer has the original "Statue Theme" playing in cell 1 at the same time that the transformed theme is being played in cell 2.

The return of the "Joyous Theme" in the context of E major. Click on image to enlarge.

Cell 3: The Râgavardhana-Candrakalâ-Lakskmîça Juxtaposition

Instruments: Percussion (triangle, suspended cymbal)

Cell 3 is the most elusive of the three. It contains the triangle and suspended cymbal quietly and repeatedly tapping out the Râgavardhana-Candrakalâ-Lakskmîça juxtaposition.

The Râgavardhana-Candrakalâ-Lakskmîça juxtaposition, notated in the context of 3/16 time. Click on image to enlarge.

PART 2C (Rehearsal Nos. 25-32)

Part 2C is mostly the retrograde (i.e. "reverse") of part 2A, although it is transposed up 1/2-step. However, you refer to the diagram of part 2 it can be seen that Messiaen adds a second layer in the brass section which plays exactly what was played in part 2A. This is going on as the original layer is playing its retrograde.

Parts of the score eight measures before part 2B and eight measures after part 2B, clearly showing that part 2C is the retrograde of part 2A (though it is transposed up 1/2-step). Click on image to enlarge.

The idea of literally retrograding a score is not a new idea that Messiaen invented, even though it fits perfectly into his style of composition. A famous example of this occurs in the central section of Alban Berg's Chamber Concerto for Violin, Piano, and 13 Wind Instruments (1923). In the slow 2nd movement, he puts the ensemble in reverse. Berg even instructed the publisher of the concerto to arrange the pages around this section so that the central axis runs through the center of the page, thus clearly displaying the compositional device!

PART 2D (Rehearsal Nos. 32-38)

This is more or less the same as part 2B except it is in a different key and it is transposed mid-way through. All three of the layers of part 2B are present here. This leads into a transition to part 3.


Part 3 is a return of the "Joyous Theme" as it was played in part 1. After being repeated twice, we hear two ecstatic swoops from the ondes-Martinot which leads into a short (yet impressive) impressive piano cadenza. The movement ends with a return of the "Statue Theme" and a massive chord -- D-flat major with an added sixth.

These saccharine indulgences in which Messiaen partakes makes it a little clearer as to why his student Pierre Boulez once called the symphony "brothel music". Of course, one can't help but wonder what kind of brothels Boulez frequented (though I think it is best not to dwell on it, don't you?).

Note: In this movement, all audio recordings either ones that I generated using Finale or from Hannu Lintu's recent (and outstanding) recording of the piece with the Finnish Radio Symphony Orchestra.