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.
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