When two pure tones with frequencies f1 and f2 are sounded simultaneously, combination tones are produced at frequencies of (f1 + f2) and (|f1 - f2|). In this case, the two pure tones are Cs and Gs from the Pythagorean diatonic scale, which have frequencies of 256 Hz and 384 Hz, respectively.
a) The frequencies of the three combination tones are:
- (f1 + f2) = 640 Hz (Cs + Gs)
- (|f1 - f2|) = 128 Hz (Gs - Cs)
- (2f1 - f2) = 128 Hz (Cs + Gs - 2Cs)
b) To find the notes on the Pythagorean scale to which these tones belong, we need to compare their frequencies to those of the Pythagorean diatonic scale. The Pythagorean diatonic scale is based on a ratio of 3:2 between adjacent notes, which means that the frequency of each note is 3/2 times the frequency of the previous note. Starting from Cs (256 Hz), the frequencies of the Pythagorean diatonic scale are:
- Ds: 288 Hz (256 Hz x 3/2)
- Es: 324 Hz (288 Hz x 3/2)
- Fs: 341.33 Hz (324 Hz x 81/80)
- Gs: 384 Hz (341.33 Hz x 3/2)
- As: 432 Hz (384 Hz x 3/2)
- Bs: 486 Hz (432 Hz x 3/2)
- Cs: 512 Hz (486 Hz x 81/80)
Comparing the combination tones to the Pythagorean diatonic scale, we can see that:
- 640 Hz is between Gs (384 Hz) and As (432 Hz)
- 128 Hz is between Cs (256 Hz) and Ds (288 Hz)
- 128 Hz is between Cs (256 Hz) and Ds (288 Hz)
Therefore, the combination tones belong to the notes Gs-As, Cs-Ds, and Cs-Ds, respectively.