As promised, but a little late, here’s a compilation of links to review for the Midterm Exam.

Text: Chapters 1 – 6

Links:

• Twilight of the Tonal System
• Integer Notation
• Scale Formations
• Chords and Verticalities
• Melody and Voice Leading
• Harmonic Progression and Tonality
• Rhythm and Meter: Bartok, Additive Meter and Syncopation
• Rhythm and Meter in Post-Tonal Music
• Tempo Modulation and Polymeter
• Quartet for the End of Time, Mov’t I: Isorhythm
• Quartet for the End of Time, Mov’t VI: Added Values and Nonretrogradable Rhythms

You should also review pieces we’ve covered in class. I will include some analysis of pieces that we have talked about in class, which will assume that you know things like composer and piece title.

• Debussy, Sunken Cathedral
• Debussy, Prelude to the Afternoon of a Faun
• Bartok, Mikrokosmos – Diminished Fifth
• Bartok, Music for Strings, Percussion, and Celesta
• Varese, Density 21.5
• Ives, The Cage
• Ruth Crawford Seeger, String Quartet, Mov’t III
• Ligeti, Galamb Borong
• Carter, Canaries (from 8 Pieces for Four Timpani)
• Bartok, Mikrokosmos, Numbers 115 (Bulgarian Rhythm), 133 (Syncopation)
• Messiaen, Quartet for the End of Time (Movements I and VI)

Movement VI, Dance of the Furies, deals with additional rhythm and meter topics: added values, nonretrogradable rhythms, and augmentation/diminution of rhythmic patterns.

Added Values

Added rhythmic values occur in otherwise simple rhythmic and metric patterns by adding a short additional note, dot, or rest. Example 6-11 in the textbook has the opening portion of the movement, with added values marked with a +. Here are the first three measures in the piano (all parts play the same line in unison or octaves), with added values marked by an arrow.

Notice that Messiaen does not use a meter signature at all for this movement. The beat is still clearly indicated by beaming, so that the performers know what beats are being stretched by the added values.

Nonretrogradable Rhythms

Nonretrogradable rhythms have the same pattern of rhythmic durations when played forward or backward (retrograde). You can think of them as a rhythmic palindrome. Towards the middle of the movement (also shown in Ex. 6-12 in the text) Messiaen employs a series of nonretrogradable rhythms (per measure). Three of those measures are shown below, with the middle/mirror note indicated by a rectangle.

Nonretrogradable rhythms eliminate rhythmic directionality within small gestures. The overall passage involves an acceleration of rhythm that works against the lack of direction in individual measures.

Augmentation and Diminution

Rhythmic augmentation and diminution is not a new development in post-tonal music. One can find examples dating back to Bach and earlier. The explicit use in Messiaen’s work is worth pointing out, as it utilizes more proportions than one usually finds in tonal music.

From the first measure of the above example to the second, the rhythmic values have all been increased by 1/3 – one third of the value from the first measure is added to produce the value in the second measure. The dotted quarter is comprised of 3 eighth notes. One third of this duration is one eighth note. Added together, augmenting, you end up with four eighth notes – a half note. The same process is true for the dotted eighth, but now you are dealing with the addition of a sixteenth note. The proportion from measure one to measure two is 3:4. Note: the proportion is different depending on which measure serves as the reference point. The values in measure two are a third longer than the values in measure one (4/3). The values in measure one are a quarter shorter than the values in two (3/4). (Fun with proportions!)

From the second measure to the third, each duration is being diminished to 1/4 its previous value (1:4). The values in measure four are five times the values in measure three. (5:1). You can also compare values from any two measures, since all are multiples of eighth notes and sixteenth notes. Measure one to measure four is 3:5. Measure three to measure one is 1:3.

We’re using Quatuor pour la fin du temps (Quartet for the End of Time) 1940 – 41, by Olivier Messiaen (1908 – 1992), as the basis for several concepts introduced in chapter 6 (Rhythm and Meter). This post will focus on the first movement, “Liturgie de Cristal,” as an example of isorhythm.

You may have studied isorhythm in the context of 14th- and 15th-century music – notably, isorhythmic motets. You can also find isorhythmic procedures in other 20th- and 21st-century composers, and in the music of India. An isorhythm features a repeating rhythmic pattern and an independent repeating pitch pattern. Messiaen referred to the repeating rhythmic pattern as a rhythmic pedal.

The cello and piano both feature independent isorhythms. The cello has the shortest repeating pitch pattern (5 pitches), along with a 15-duration rhythmic pattern. The pitch pattern in the cello is C4, E-natural4, D4, F#4, Bb3. The rhythmic pattern is substantially longer at 15 note values, but having the rhythmic pattern be a whole-number multiple of the pitch pattern means that the rhythmic pattern will always start with the beginning of the pitch pattern. The rhythmic pattern lasting for a total duration of 16.5 beats, the pattern will alternate starting off the beat, and on the beat.

The piano isorhythm is less synchronized than the one in the cello, and it has a longer pitch pattern than rhythmic pattern. The rhythmic pattern is 17 note values in length, starting with the three quarter notes in succession, and the duration of the entire pattern is 13 beats. The pattern always starts on the beat but in different part of the measure each time. The pitch pattern is comprised of 29 chords. The number of chords in the pitch pattern and the number of rhythmic values in its pattern are both prime numbers. These two patterns will not start together for 17 x 29 repetitions (493), which is much longer than the duration of the movement. This lack of synchronization, along with the lack of synchronization between the two isorhythms, contributes to a static feeling for the movement, and a feeling of incompleteness.

To sum up the two isorhythms:

• cello: 5 pitches, 15 rhythmic values, total duration of 16.5 beats.
• piano: 29 chords, 17 rhythmic values, total duration of 13 beats.

The violin and the clarinet both have the performance indication of “comme un oiseau” (like a bird). These two instruments are providing music that resembles bird calls. Along with Messiaen’s devout Catholicism, he was greatly influenced by the music of birds, going as far as transcribing their calls and incorporating them into his works.

Last year’s post walks you through the basics. Look it over and make sure you understand the formula and how to use it for both tempo modulation (metric modulation to some) and polytempo. Just to include it here and reiterate:

The formula can be stated in a simple, easy to remember way:

original_tempo * (original_grouping_number / new_grouping_number) = new_tempo

For the ratio to work, the rhythmic value of both the original and new groupings must be the same. In other words, compare groupings of sixteenth notes to sixteenth notes, eighth notes to eighth notes, etc.

If you are trying to solve for polytempi, you will always compare to subdivisions of the stated reference tempo (the given metronome marking). You can always subdivide the reference tempo into the necessary rhythmic subdivisions to compare to the other tempi. In the Carter String Quartet No. 1, the reference tempo is for the quarter note in the cello part. The quarter note can be divided into sixteenth notes to compare to the second violin, into triplet eighths to compare to the first violin, etc.

If you are ever confused about which grouping goes first in the ratio, consider whether you are speeding up or slowing down. If you are speeding up, the smaller number will come second, making the ratio greater than one and increasing the original tempo via multiplication. If you are slowing down, the smaller number will be first, making the ratio smaller than one and decreasing the original tempo via multiplication.

I want to include a chapter 6 clean-up to make sure all are clear about the major concepts. This post from last year summarizes the topics very briefly, except for polytempo and tempo modulation.

Friday we covered two Bartok Mikrokosmos pieces: No. 115, Bulgarian Rhythm; and No. 133, Syncopation.

Additive Rhythm and Additive Meter (“Bulgarian Rhythm”)

Bulgarian Rhythm is an example of additive meter. The 5/8 meter is expresses as both (3+2)/8 and (2+3)/8. Additive rhythm is related to some folk rhythmic practices (African drumming, for example), where there is a basic pulse that is never subdivided, so all rhythms are some multiple of the basic pulse. Additive meter takes this idea and organizes it into a meter with strong and weak beats.

Additive meters will always have unequal beat lengths. As an example, consider the difference between 4/4 and 8/8. Both meters have the same number of eighth notes, but 8/8 specifically implies that the eighth notes will be grouped into beats of 3+3+2 eighth notes (or some order of 3, 3, and 2), instead of 2+2+2+2.

Bulgarian Rhythm uses meter as one an organizing aspect of form. The piece is basically an ABA, with a short coda. The A and A’ section (mm. 1 – 8; x – x) uses a 3+2 additive groupings, while the B section switches to 2+3 groupings. The final two phrases comprise a coda or closing section, where a measure with 5 eighth notes in 3+2 grouping is followed by a measure with quarter note followed by dotted quarter note (2+3 grouping). The coda combines the additive grouping patterns from both sections.

We also talked about the tonal organization (pitch centricity), how it was established, and how it corresponded to the sections.

Syncopation (“Syncopation”)

Syncopation as a concept involves accenting weak parts of any meter, which could mean accenting off the beat, or accenting weak beats within a measure. Accents can be either dynamic or agogic (time-based, duration).

The Bartok Syncopation uses rhythmic accents to create additive grouping that differ from the notated meters. The result of these groupings are perceived meters: audible groupings that we hear as metrically organized. The opening 5/4 measure is grouped into perceivable 10/8 meter, (3+3+2+2)/8. What the listener hears is very similar in effect to Bulgarian Rhythm.

The frequently changing time signatures – often a change every measure – are another characteristic of post-tonal music.