Group Work: Pythagorean Tuning A musical system that is built entirely from 2:1 intervals (pure octaves) and 3:2 intervals (pure perfect fifths) is called a Pythagorean tuning system. These questions will walk you through its construction. (1) Start with Ab4 = 415.3047 Hz (this seems a little weird, but the reason will be clear later). Every other Ab in the system will be related to this one by a pure octave. What are the frequencies of Ab3 and Ab5? (2) As we have seen, there are 12 pitches within a given octave. Your goal is now to figure out the frequencies of the other 11 pitches in the octave. In a Pythagorean system, you can only “reach” the 11 other pitches in the octave by stacking up a certain number of pure fifths above the given pitch. If the resulting pitch is too high, then you need to bring it down the appropriate number of octaves so that it is within range. ‘ Thirteen pitches are listed below, on the left column of the table. In the second column, fill in the number of fifths you need to stack above Ab4 in order to reach the given pitch (without regard to octave). It will help to remember that on the keyboard, a perfect fifth encompasses seven semitones. In the third column, fill in the number of octaves you need to move the resulting stacked-fifths pitch down in order to get it into the proper range. Based on your second and third columns, fill in the factor by which you must multiply the frequency of Ab4 in order to get the frequency of the pitch in the next column. Write in that frequency in the last column. Pitch Ab4 Number of fifths stacked above or below Ab4 0 Number of octaves down to proper register 0 Multiplicative factor Frequency (3/2)0 /20 1.0000 415.3047 Hz 440.0000 Hz A4 Bb4 B4 C5 = 72 C#5 D5 Eb5 1 (3/2)1 /20 1.5000 0 E5 F5 F#5 G5 G#4 (3) Generating the rest of the pitches in the system is now simply a matter of adding (or subtracting) pure octaves from the 12 pitches that you’ve generated. In this system, almost any “perfect fifth” (an interval of seven pitch classes) will be pure (i.e., will embody a frequency ratio of 3:2). But some fifths will be out of whack. Looking at the first 12 pitches you generated in the table, which ones won’t have a pure fifth above it in the Pythagorean system? What is the frequency ratio between that frequency and the pseudo-fifth above it? How close is it to the pure 3:2 ratio? (Write the answer as a product of powers of 2 and 3, and convert it to a decimal.) (4) Suppose you were not limited to just 12 pitches to the octave, but could go on stacking fifths as long as you’d like. How far would you need to go in order to achieve a closed musical system? In other words, how many pitches per octave would you need in order to have a system made up completely of pure fifths and pure octaves? Justify your answer!