In the name of the Moon, I will teach you music math! ✨ Let's be brave and careful, and do this step-by-step like a magical music detective. We'll use middle C = 261.63 Hz as our starting star.
Part 1 — Octave limits & splitting the string
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What are the lower and upper limits for a C octave?
An octave is a 1:2 ratio. That means doubling the frequency goes up one octave, and halving goes down one octave.
If we use middle C = 261.63 Hz:
Lower limit (one octave below) = 261.63 ÷ 2 = 130.815 Hz (about 130.82 Hz)
Upper limit (one octave above) = 261.63 × 2 = 523.26 Hz
So the C octave that sits with middle C as the low C runs from 261.63 Hz up to 523.26 Hz. If you want the octave centered on middle C, the frequencies one octave below and above are 130.815 Hz and 523.26 Hz respectively.
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If a string playing middle C (261.63 Hz) is shortened to 2/3 of its length, what is the new frequency?
Important physics idea: frequency is inversely proportional to string length. If the string length becomes 2/3 of the original, the frequency becomes 1/(2/3) = 3/2 times the original.
Algebra:
f_new = f_old × (1 ÷ (2/3)) = f_old × (3/2)
So f_new = 261.63 × (3/2) = 261.63 × 1.5 = 392.445 Hz
Rounded to two decimals: 392.45 Hz. This is the pitch a perfect fifth above middle C (that note is G).
Part 2 — Build the Pythagorean scale (using 3/2 fifths and moving octaves)
Quick rule reminder (Pythagoras method): To go up by a perfect fifth multiply by 3/2. If the result is higher than the C-octave top (523.26 Hz), divide by 2 to bring it down an octave. If the result is below the C-octave bottom, multiply by 2 to bring it up.
Start: C = 261.63 Hz
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C → G
G = C × (3/2) = 261.63 × 1.5 = 392.445 Hz → round to 392.44 Hz
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G → D
D_raw = G × (3/2) = 392.445 × 1.5 = 588.6675 Hz
588.6675 Hz is above the C octave top (523.26 Hz), so divide by 2 to bring it into the C octave:
D = 588.6675 ÷ 2 = 294.33375 Hz → round to 294.33 Hz
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D → A
A = D × (3/2) = 294.33375 × 1.5 = 441.500625 Hz → round to 441.50 Hz
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A → E
E_raw = A × (3/2) = 441.500625 × 1.5 = 662.2509375 Hz
662.2509... Hz is above the C octave top, so divide by 2:
E = 662.2509375 ÷ 2 = 331.12546875 Hz → round to 331.13 Hz
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E → B
B = E × (3/2) = 331.12546875 × 1.5 = 496.688203125 Hz → round to 496.69 Hz
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C → F (special instruction)
The worksheet tells us F is 2/3 below C (so use 2/3 of C):
F_raw = C × (2/3) = 261.63 × 0.666666... = 174.42 Hz (approx)
174.42 Hz is below the C octave bottom (261.63 Hz), so multiply by 2 to bring it into the C octave:
F = 174.42 × 2 = 348.84 Hz
Put the notes in ascending order inside the C octave (C up to the next C):
- C = 261.63 Hz
- D = 294.33 Hz
- E = 331.13 Hz
- F = 348.84 Hz
- G = 392.44 Hz
- A = 441.50 Hz
- B = 496.69 Hz
- C (one octave above) = 523.26 Hz
Ta-da! ✨ You used algebra (multiplying by 3/2 or 2/3 and sometimes dividing or multiplying by 2 to stay inside the octave) to build the Pythagorean C scale. Every step was like casting a musical spell: multiply for fifths, fix octaves when needed, and write the final frequencies.
If you want, we can show the same work with more decimal places, draw the chain of fifths on paper, or hear how each frequency sounds. Shall we transform these frequencies into sounds next? 🌙🎵