Listen carefully. You will do this carefully and exactly. Follow each algebra step.
Important idea (one sentence): Pythagoras worked with string lengths. If the string length is multiplied by 2/3, the frequency is multiplied by 3/2. So to go up by a perfect fifth you multiply the current frequency by 3/2. If that result falls outside the C octave (below the low C or above the high C) you fix it by multiplying or dividing by 2 to move it into the octave.
Part 1
- Lower and upper limits for a C scale (one octave around middle C):
Middle C = 261.63 Hz. An octave above is double the frequency, so upper C = 2 × 261.63 = 523.26 Hz. The octave that contains middle C runs from 261.63 Hz up to 523.26 Hz.
- If a string playing middle C is split into 2/3 (string length becomes 2/3 of original):
Remember: frequency ∝ 1/length. If length becomes 2/3, frequency becomes 3/2 of the old frequency. So compute:
frequency = 261.63 × (3/2) = 261.63 × 1.5 = 392.445 Hz.
So splitting the string to 2/3 of its length raises the pitch to about 392.445 Hz (this is the G above middle C).
Part 2 — Build the Pythagorean scale step by step
Rule we use as algebra: if f_previous is the previous note, the next note (a perfect fifth above) is f_next = f_previous × 3/2. If f_next > 523.26 (above the C octave) divide by 2. If f_next < 261.63 (below the C octave) multiply by 2. For F we will go down a fifth from C, which uses multiply by 2/3 then adjust into the octave.
We start with C = 261.63 Hz.
- C to G
f_G = 261.63 × (3/2) = 261.63 × 1.5 = 392.445 Hz. This is inside the octave (less than 523.26), so keep it.
- G to D
first compute D_temp = 392.445 × (3/2) = 392.445 × 1.5 = 588.6675 Hz.
588.6675 is above the octave (above 523.26), so divide by 2 to bring it down an octave:
f_D = 588.6675 ÷ 2 = 294.33375 Hz (round as needed to 294.33 Hz).
- D to A
f_A = 294.33375 × (3/2) = 294.33375 × 1.5 = 441.500625 Hz. This is within the octave, keep it (≈ 441.50 Hz).
- A to E
first compute E_temp = 441.500625 × (3/2) = 441.500625 × 1.5 = 662.2509375 Hz.
662.2509375 is above the octave, so divide by 2:
f_E = 662.2509375 ÷ 2 = 331.12546875 Hz (≈ 331.13 Hz).
- E to B
f_B = 331.12546875 × (3/2) = 331.12546875 × 1.5 = 496.688203125 Hz (this is inside the octave, ≈ 496.69 Hz).
- Find F (special rule)
The worksheet says: the starting note for calculating F is middle C and is 2/3 below C. Going "2/3 below" means a downward fifth, which in frequency is multiplying by 2/3 (because going down a fifth lowers frequency). So compute:
temp = 261.63 × (2/3) = 261.63 × 0.666666... = 174.42 Hz.
174.42 Hz is below the C octave, so multiply by 2 to move it into the octave:
f_F = 174.42 × 2 = 348.84 Hz (≈ 348.84 Hz).
Final Pythagorean C scale (middle C up to next C)
- C = 261.63 Hz
- D = 294.33 Hz (calculated: 294.33375)
- E = 331.13 Hz (calculated: 331.12546875)
- F = 348.84 Hz
- G = 392.45 Hz (calculated: 392.445)
- A = 441.50 Hz (calculated: 441.500625)
- B = 496.69 Hz (calculated: 496.688203125)
- C = 523.26 Hz (the octave above middle C)
Notes on rounding and accuracy: I showed exact intermediate values with many decimals and then rounded common musical practice to two decimal places. The Pythagorean numbers are a theoretical tuning and differ slightly from modern equal temperament (the usual piano tuning), which is why these frequencies are close to but not exactly the standard piano frequencies.
Summary (short): Octave limits: 261.63 Hz to 523.26 Hz. Splitting the middle-C string to 2/3 of its length raises its frequency to 392.445 Hz. Using repeated 3/2 multiplications and octave adjustments gives the Pythagorean C scale above.
Do this again on paper. Show every multiplication and the divide-by-2 or multiply-by-2 step. That is how you must practice math and music together.