A musical work consists of a segment of time, framed by its beginning and ending moments. In between these two boundaries, the perception of musical form is shaped by changes of content. A listener understands formal divisions based on differences and similarities between musical elements as present in time. The introduction of a new section, for example, can only be perceived as being in fact new, if it doesn’t repeat what was heard immediately before.
Each section of a work occupies a portion of the overall duration. The sequence of ratios derived from the duration of each individual section to the total time length provides the formal rhythm of a work, the highest level of temporal hierarchy in music.
Formal rhythm, as an algorithm, can be either explicit, if it subordinates musical processes (top-down approach), or, implicit, when the algorithm is itself the consequence of musical processes (bottom-up approach).
In works where formal rhythm is composed from a top-down approach, musical processes are conditionally selected to fit predetermined lengths of individual sections. Phillip de Vitry (1291-1361), Johannes Ciconia (ca. 1370-1412), Nicolas Grenon (ca. 1380-1456), John Dunstaple (ca. 1390-1453) and Guillaume Du Fay (1397-1474), all composed using algorithms to control the time proportion between sections. Du Fay’s motet Nuper rosarum flores, written for the consecration of the cathedral of Florence in 1436, has its formal rhythm governed after the proportion 6:4:2:3, a symbolic reference to the biblical measurements of Solomon’s temple, as described in the Old Testament (Wrigth 1994). Such deliberate concern with time proportions in early Western music, has no other historical parallel until the twentieth century. It is not until Béla Bartók (1881- 1945) that form in music is again conceptualized with acute temporal intentionality. In the first movement of Contrasts (1938), for example, Bartók segments time according to a continuos proportion, the golden section (Lendvai, 1983), aligning the recapitulation precisely when the amount of time left to end the movement (b) compared to the time already used since the beginning of the movement (a), compared to the duration of the entire movement (c) exhibit a permanent ratio (a/b=b/c).
A much later example, John Cage’s (1912-1992) “square root form”, used in, among other works, Imaginary Landscape No. 1 (1939), proposes an organizational principle in which “the whole having as many parts as each unit has small parts, and these, large and small, in the same proportion” (Harrison 1971).
Karlheinz Stockhausen (1928-2007), in his opera Licht (Stockhausen, 2001), has pursued, for each of the seven week days that comprise the cycle, a compositional process with a systematic architecture of time, pre-composing, within the detail of seconds, the duration of each section.
When the formal rhythm of a work derives from a compositional bottom-up approach, the act of composing parallels prose writing. When writing prose, one searches and follows through trial and errorto the next sentence, never anticipating how many phrases and paragraphs will suffice to communicate an idea in mind. Similarly, a composer working from a bottom-up approach does not exercise any direct control over the time proportions in a work.
Instead, the number and length of individual sections follows subjective reactions to the shaping of materials leaving the formal rhythm of a work as an unaccounted consequence of phrase lengths, number of phrases, and thematic design.
Independently of which compositional approach is followed, top-down or bottom-up, the traditional representation used to describe formal designs, such as “ABA” for the sonata-allegro design, can only inform us about the flow of similarities. That is, in this particular case of an “ABA” design, we expect an initial section (A) to be followed by a second section different in content (B), followed by a third and last section that is different from the previous and similar to the first (A). Most frequently, one assumes that since the recapitulation is the reiteration of material previously presented, both the recapitulation and the exposition sections should have the same length, where the symmetrical properties of the formal design directly translate to non-retrogradable formal rhythms (i.e., palindromes). However, the description of formal designs is temporally completely uninformative. From the description of the formal design it is impossible to discern how the length of the B section compares to the A section leading to the assumption that the reiteration of A equals its first appearance. The first movement of Mozart’s piano sonata in C Major KV. 545, for example, sums up to a total of 73 measures of 4/4 meter which equal to 292 beats displaying a straightforward design. The exposition (A), the development (B) and the recapitulation (A) last respectively 112, 52 and 128 beats. The ratios between each of these sections and the total amount of beats dictate a formal rhythm that can be simplified to 28/73, 13/73, 32/73. By notating these simplified ratios in traditional note values (Fig. 1), with 1/73 represented as a sixteenth- note, it becomes clear that, since the exposition and the recapitulation differ in length,symmetrical properties in formal thematic designs are not expected to carry over to formal rhythm.
Each section of a work occupies a portion of the overall duration. The sequence of ratios derived from the duration of each individual section to the total time length provides the formal rhythm of a work, the highest level of temporal hierarchy in music.
Formal rhythm, as an algorithm, can be either explicit, if it subordinates musical processes (top-down approach), or, implicit, when the algorithm is itself the consequence of musical processes (bottom-up approach).
In works where formal rhythm is composed from a top-down approach, musical processes are conditionally selected to fit predetermined lengths of individual sections. Phillip de Vitry (1291-1361), Johannes Ciconia (ca. 1370-1412), Nicolas Grenon (ca. 1380-1456), John Dunstaple (ca. 1390-1453) and Guillaume Du Fay (1397-1474), all composed using algorithms to control the time proportion between sections. Du Fay’s motet Nuper rosarum flores, written for the consecration of the cathedral of Florence in 1436, has its formal rhythm governed after the proportion 6:4:2:3, a symbolic reference to the biblical measurements of Solomon’s temple, as described in the Old Testament (Wrigth 1994). Such deliberate concern with time proportions in early Western music, has no other historical parallel until the twentieth century. It is not until Béla Bartók (1881- 1945) that form in music is again conceptualized with acute temporal intentionality. In the first movement of Contrasts (1938), for example, Bartók segments time according to a continuos proportion, the golden section (Lendvai, 1983), aligning the recapitulation precisely when the amount of time left to end the movement (b) compared to the time already used since the beginning of the movement (a), compared to the duration of the entire movement (c) exhibit a permanent ratio (a/b=b/c).
A much later example, John Cage’s (1912-1992) “square root form”, used in, among other works, Imaginary Landscape No. 1 (1939), proposes an organizational principle in which “the whole having as many parts as each unit has small parts, and these, large and small, in the same proportion” (Harrison 1971).
Karlheinz Stockhausen (1928-2007), in his opera Licht (Stockhausen, 2001), has pursued, for each of the seven week days that comprise the cycle, a compositional process with a systematic architecture of time, pre-composing, within the detail of seconds, the duration of each section.
When the formal rhythm of a work derives from a compositional bottom-up approach, the act of composing parallels prose writing. When writing prose, one searches and follows through trial and errorto the next sentence, never anticipating how many phrases and paragraphs will suffice to communicate an idea in mind. Similarly, a composer working from a bottom-up approach does not exercise any direct control over the time proportions in a work.
Instead, the number and length of individual sections follows subjective reactions to the shaping of materials leaving the formal rhythm of a work as an unaccounted consequence of phrase lengths, number of phrases, and thematic design.
Independently of which compositional approach is followed, top-down or bottom-up, the traditional representation used to describe formal designs, such as “ABA” for the sonata-allegro design, can only inform us about the flow of similarities. That is, in this particular case of an “ABA” design, we expect an initial section (A) to be followed by a second section different in content (B), followed by a third and last section that is different from the previous and similar to the first (A). Most frequently, one assumes that since the recapitulation is the reiteration of material previously presented, both the recapitulation and the exposition sections should have the same length, where the symmetrical properties of the formal design directly translate to non-retrogradable formal rhythms (i.e., palindromes). However, the description of formal designs is temporally completely uninformative. From the description of the formal design it is impossible to discern how the length of the B section compares to the A section leading to the assumption that the reiteration of A equals its first appearance. The first movement of Mozart’s piano sonata in C Major KV. 545, for example, sums up to a total of 73 measures of 4/4 meter which equal to 292 beats displaying a straightforward design. The exposition (A), the development (B) and the recapitulation (A) last respectively 112, 52 and 128 beats. The ratios between each of these sections and the total amount of beats dictate a formal rhythm that can be simplified to 28/73, 13/73, 32/73. By notating these simplified ratios in traditional note values (Fig. 1), with 1/73 represented as a sixteenth- note, it becomes clear that, since the exposition and the recapitulation differ in length,symmetrical properties in formal thematic designs are not expected to carry over to formal rhythm.
