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