Tempo modulation deserves its own special post, if at least only to fully explain the math used to calculate new tempi.
To review, tempo modulation involves a new grouping of some subdivision of the pulse. For example, a quarter-note pulse can be divided into four sixteenth notes. If those sixteenth notes are are accented in groups of three (rather than four), you can create a new tempo based on the groupings of three sixteenth notes. Now, let’s say that the original tempo is quarter = 90 bpm. The new tempo will be 90(4/3) = 120 bpm. ((90 times the ratio 4:3))
The formula can be stated in a simple, easy to remember way:
original_tempo * (original_grouping_number / new_grouping_number) = new_tempo
In the example above, the original tempo is 90. The original grouping number is four (for four sixteenths to a beat). The new grouping number is three (since three sixteenths comprise the new beat).
(***Note, this formula is the same as the one in the book on p. 130, but is just expressed as one equation.)
In Elliott Carter’s Canaries, there is a section where tempo modulation is used to slow down. At measure 47 the tempo is quarter note = 120 bpm. The quarter is divided into sixteenth notes (original grouping = 4). At the same time there is a pattern of accents on every fifth sixteenth note through measure 49 (new grouping = 5). The modulated tempo at m. 50 is 96 bpm.
120 * (4/5) = 96
The textbook uses an excerpt from Carter’s String Quartet no. 1 as an example with multiple simultaneous tempi. With a passage like this the equation changes only in how we label the groupings. You have to find a common rhythmic subdivision between two parts (a common factor), and then you can compare the groupings in the ratio of standard_grouping_number to new_grouping_number. The somewhat difficult part is that the common factor will be different for each part.
The cello is the only instrument playing at the indicated tempo of 120 bpm, and it is playing quarter notes. The other parts use some common subdivision of the quarter note, but accented in some way other than four. The second violin part is playing at a rate of 5 sixteenth notes (the new grouping is 5). The cello has four sixteenth notes to a beat (standard number = 4).
120 * (4/5) = 96 bpm
The first violin is moving at a very slow rate of speed. The triplet-eighth subdivsion stands out in the second measure. Since quarter notes can divide into 3 triplet eighth notes, the standard grouping is 3. Counting triplet eighth notes returns 10 triplet eighth notes per pitch (the new grouping).
120 * (3/10) = 36 bpm
When the viola enters it has quarter note triplets. Since multiple quarter-note triplets cannot fit within one beat we have to expand the reference group. Three quarter-note triplets occur during the time of 2 regular quarter notes. To fit within our formula, we can think of there being three triplet eighth notes in a quarter note beat (the standard grouping), and each beat in the viola last two triplet eighths (the new grouping).
120 * (3/2) = 180 bpm
Let’s take one more example. Suppose you have a piece with a tempo of 100 beats per minute, with a quarter note getting one beat. Say the bassoon is playing quarter notes. The clarinet is playing at a tempo lasting 3 sixteenth notes. The oboe is playing at a rate of 5 sixteenth notes. The flute is playing at a rate of 7 sixteenth notes. Combining all the tempi gives a 7:5:4:3 relationship (flute:oboe:bassoon: clarinet), reordered because of the numbers. But we can relate each to the basson.
100 * (4/7) = 57.14 (flute)
100 * (4/5) = 80 (oboe)
100 * (4/3) = 133.3 (clarinet)
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