Question

to calculate the frequency of a note on a piano in the same octave or a different octave, the hertz (hz) of a particular note can be multiplied by the ratio of change in the octave between the original note and the new note to indicate the hz. when not starting at c major for an octave, the ratio would need to be calculated using a formula, and moving to the next octave results in a change of the next octaves note ratio by multiplying by 2. for example, moving from $b_5$ to $a_6$ on the next octave would be done with a ratio derived from: $\frac{a}{b}=\frac{\frac{5}{3}cdot2}{\frac{15}{8}} = (\frac{5}{3}cdot\frac{2}{1})cdot\frac{8}{15}=$ thus, a note at $b_5$ would be a different hz value on the next octave at $a_6$ based on the ratio multiplier as indicated above between the notes on different octaves (i.e., multiply the hz at $b_5$ by the ratio to find the frequency at $a_6$). what would the hz be at $a_6$ if the $b_5$ note is 270 hz?

Explanation:

Step1: Simplify the ratio formula

First, simplify $\frac{\frac{5}{3}\cdot2}{\frac{15}{8}}$. Multiply the numerator $\frac{5}{3}\cdot2=\frac{10}{3}$. Then, dividing by a fraction is the same as multiplying by its reciprocal, so $\frac{\frac{10}{3}}{\frac{15}{8}}=\frac{10}{3}\times\frac{8}{15}=\frac{80}{45}=\frac{16}{9}$.

Step2: Calculate the frequency of $A_6$

Multiply the frequency of $B_5$ (270 Hz) by the ratio. Let $f_{A_6}$ be the frequency of $A_6$ and $f_{B_5} = 270$ Hz. Then $f_{A_6}=f_{B_5}\times\frac{16}{9}$. Substitute $f_{B_5}=270$ into the formula: $f_{A_6}=270\times\frac{16}{9}=480$ Hz.

Answer:

480 Hz