Challenging Music/Pysics question, can anyone answer it?

Question

In a room, you notice that there are two lights that "buzz" at two different pitches that you hear makes an interval of a minor third. You play middle C on the (in-tune) piano in the room, and hear that it makes an interval of a major tenth with the lowest "light buzz".

a) What are the frequencies of the buzzes? (Assume Pythagorean tuning.)
b) What are the notes of the buzzes?
c) What is the quality of the chord that these three pitches make?
d) What note would you need to make it into a minor seventh chord and what is the frequency of that note? (The root does not have to stay the same)

Answer 1

A tenth from middle C ( C4 ) could be Eb5 (above) or A3 (below). So the light buzzes could be the notes A3, C3 or Eb5, G5. Since the problem says there are three pitches, we'll go above: C4, Eb5, G5. That's a minor triad (the i chord in C). To make a minor seventh, add B5 or B6.

Answer 2

Pythagorean tuning...how annoying. This question would be much nicer if it were in equal temperament. Can't have everything, I suppose.

Let's assume the Pythagorean tuning is based on middle C (261.626 Hz). The notes, ratios, and frequencies are therefore:
C - 1 - 261.626
Db - 256/243 - 275.622
D - 9/8 - 294.329
Eb - 32/27 - 310.075
E - 81/64 - 331.120
F - 4/3 - 348.835
Gb - 729/512 - 372.510
G - 3/2 - 392.439
Ab - 128/81 - 413.434
A - 27/16 - 441.494
Bb - 16/9 - 465.113
B - 243/128 - 496.681
C - 2/1 - 523.252

Now we can address some of the questions.

Let us call the frequencies of the two noises X and Y, with X being the lower-pitched of the two. We know that middle C to X is a major tenth (an octave plus a major third). X must be the E that is an octave and a major tenth above middle C. The frequency ratio of a major tenth is (81/64 * 2) = 2.53125. Therefore:
X = 2.53125 * (middle C)
X = 2.53125 * 261.626 Hz
X = 662.241 Hz

The other buzz is a minor third above X, or a minor third above E, which makes it the G that is an octave and a fifth above middle C. The frequency is therefore
Y = (3/2 * 2) * (middle C)
Y = (3/2 * 2) * (261.626 Hz)
Y = 784.878 Hz

The notes are C, E, G, which makes it a C major chord.

If you add the A below middle C, it will become an A minor seventh chord. The frequency of that note is
A = ((27/16) / 2) * (middle C)
A = ((27/16) / 2) * (261.626 Hz)
A = 220.747 Hz