On the relationships between finite fields, linear recursive sequences, and 12-tone Western music.
Ref: Hermann Helmholtz, “On the Sensations of Tone” (1863)
Consider the keyboard of a piano. We are concerned here with the organization of the white keys and the black keys with respect to their unusual layout on the keyboard. For reference, consider the letters A1 B1 C1 D1 E1 F1 G1 A2 B2 C2 D2 E2 F2 G2 … to be a linear recursive sequence. Note the presence of the black keys, such that the keyboard can be extended to the sequence A A# B C C# D D# E F F# G G# A etc. Note that there is no B# or E#.

The tones produced by the various keys on the piano are related by multiplicative powers. For example, if A1 produces a tone at 440 Hz, A2 will produce the second harmonic at 880 Hz, A3 the 4th harmonic at 1760 Hz, etc.

For a variety of reasons that are captured in Bach’s “Well Tempered Clavier” (1722) and explained by Helmholtz, it is common to tune a piano so that adjacent notes, including the black keys (i.e., the sharps and flats) are tuned to be at frequencies such that the ratios of the frequencies of any two adjacent notes (including the black keys) are the twelfth root of 2 apart. This guarantees that A3 = 2A2 = 4A1, B3 = 2B2 = 4B1, etc. (i.e., the octaves are perfectly tuned with respect to each other).

For your exam, answer the following questions. Work with others as you desire, but acknowledge their assistance. Use any books or web sources that you please.
1. Compute the twelfth root of 2.
2. Compute the frequency of C1, assuming that A1 = 440 Hz. Remember to include A# when counting the number of times 440 is to be multiplied by the twelfth root of 2. There is no B#.
3. Compute the frequencies of C2, C3, G2, G1, E3, E2, and E1. What are the relative frequencies of C1, G2, and E3?
Since C1, G2, and E3 are harmonically related, they form a “chord” that sounds “in tune”. For the convenience of piano players whose hands are not gigantic, this chord is typically played as C1, E1, and G1, with the keys depressed either simultaneously or in any arbitrary sequence. The chord can be inverted in many different ways, as well, such as by playing E1, G1, and C2.
4G LTE cells phones also use chords, wherein the tones of each note are called subcarriers, and the subcarriers are deliberately chosen to be “anharmonic”, so as to not be in tune with each other. They are thus separable by decoding algorithms (i.e., FFT’s) in your smart phone. When this is done with music, the sounds are referred to as being “atonal”. Note that the amplitudes of LTE subcarriers are functions of time, as they are produced using amplitude modulation.

4. Given this, what is the word “piano” short for (in Latin), and what does the full name piano-forte mean in English? How does this relate to amplitude modulation vis-à-vis a constant-envelope instrument such as a harpsichord (whose keys are like “triggers”, in that each key produces a sound of the same amplitude no matter how hard or soft the key is pressed).
5. Take the sequence of white keys, ABCDEFGABCDEFG, and decimate by 4 in both the forward and backward directions. Start with F in the forward direction to get FCG… and start with B in the reverse direction.
This yields the circle of fifths, which looks suspiciously like a 2’s complement numbering system or like the unit circle in the complex number domain when used to show a “modem constellation”. When looking at the “key signature” in a piece of music, if there are four sharps (ask a friend if you don’t know what this means), they will ALWAYS be written in the order FCGD… For flats, they are always written in the reverse order, BEA…

6. What does this have to do with the famously confusing homework question about “reversing the taps” in order to make a sequence go backwards?
7. What is the ratio of frequencies of any two white keys on a piano that are five white keys apart (i.e., C-G. D-A, etc., where the starting key is numbered as 1, and the ending key is numbered as 5). Remember to take into account the sharps when computing how many times to compute xn, where x = 21/12. (Hint, any white keys that are five white keys apart are always 8 keys apart when taking into account the relevant black keys. These constraints are why the black keys alternate 2-3-2-3 etc. ) Note the similarity between the factors of xn in a generating function for an LRS and the number of times one multiplies by 21/12 here.
8. Take the C-major chord/triad E1G1C2 and rewrite it in terms of the factors xn, where E1 = x0 = 1, G1 = x2, etc. Then, rewrite this as the sequence represented by a generator function to produce the sequence 101001. What shift register and corresponding generator function can be used to produce this sequence? What is the order of the shift register, and what is its repeat length with respect to the number of white keys in an octave on a piano?
Problems 9 and 10 are optional.
9. What is “movable do solfege”, where “do” is pronounced “doe”, as in “female deer”? (cf. Wikipedia)
10. The “Sound of Music” is about the Trapp Family Singers, who opened a ski lodge in Stowe, Vermont. Many decades ago, the Baroness von Trapp’s staff chewed out the instructor’s brothers at this resort because there were:
a) riding their cream-colored ponies, b) climbing all of the mountains, c) parked in someone else’s space.

On the relationships between finite fields, linear recursive sequen...