We have established that a heard metric structure may have several possible notations, and that, from the standpoint (hearpoint) of the listener, it is the tempo of the structure as a whole that determines which of its pulse streams will be sensed as carrying the beat. With triple metric structure, tempo is especially important in determining notation. If the triple stream is either faster or slower than the beat it will be conducted using a duple or quadruple pattern.
In the 17th century in western Europe, before tempo indications like “Adagio,” “Andante,” and “Allegro” and the metronome were in common use, note values and time signatures did suggest general tempos that arose from a variety of sources—dances, military music, church styles and the like. While some of these conventions persisted, composers felt less and less bound by them. 4/4 could be accompanied by a “Molto Adagio” or a “Vivacissimo” indication, changing the speed of the quarter note radically. Metronome markings like quarter note = 50 or = 152 might further specify this wide range of tempos, while the relative note value relationships endure (two eighth notes = one quarter note in a given tempo). Understanding the metric structure and notation of a specific piece, then, requires some knowledge of the conventions at the time the music was composed, as well as a critical ear that can make fine discriminations in the interactions of accent and pattern.
Another way tempo affects metric structure has to do with the number of levels we are able to detect and the nature of the levels at the extremes (the fastest and slowest levels). First, let’s expand our definition of metric structure —the set of tappable pulses—a bit. Tapping, after all, is a matter of digital dexterity. Musicians will differ in their ability to tap at very fast tempos. It is plausible that we can hear faster pulses than some of us can tap; adding these to our metric structures will give us a fuller comprehension. Now, let us imagine a duple metric structure consisting of 9 levels. It is impossible for any one of us to hear or tap all these levels; the representation of it in Example 52 attempts to show why:
Let’s call level 5 the beat. As we listen for the slower pulse streams, it will get more and more difficult to distinguish the equal spans of time between pulses that group the beats at level 5. The fuzzier dots as you proceed up from level 4 to level 1 are an attempt to represent, not the fact that the pulses themselves are difficult to hear, but that where they might be expected to be heard is impossible to tell in the absence of faster streams. If the beats at level 5 are one second apart, then the slower pulse streams will be as follows: level 4—2 seconds apart, level 3—4 seconds, level 2—8 seconds, and level 1—16 seconds apart.
Try performing starting with level 5, moving up to higher (slower) levels as far as possible. You will soon reach the conclusion that as pulses have more and more time between them, their regularity gets increasingly more difficult to hear and perform.
Perceiving the faster pulse streams in this hypothetical metric structure will be just as difficult, for a different reason, however. As we listen for the faster pulse streams, it will get more difficult to distinguish the equal spans of time that divide the beat at level 5. If the beats at level 5 are again one second apart, then the faster pulse streams will be as follows: pulses at level 6 — 1/2 a second, level 7—1/4 of a second, level 8 —1/8 of a second, and level 9 pulses will be 1/16 of a second apart. The spans of time at level 8 are hardly distinguishable; the spans at level 9 are indecipherable.
Again, try performing starting with level 5, this time moving down to lower (faster) levels as far as possible. You will soon reach the conclusion that as pulses have less and les time between them, their regularity gets increasingly more difficult to hear and perform.
From these experiments in performance we learn that the very quality of measuredness, periodicity, or regularity we sense in a multi-leveled metric structure fades as we listen for pulse streams at the extremities.*
*This characteristic of metric structure and its relation to the fifteenth- and sixteenth-century idea of the tactus is discussed in Lerdahl and Jackendoff, A Generative Theory of Music, Chapter 2 [Cambridge: MIT Press, 1983]. Indeed, this text is based on some key ideas consistent with this important text.