lprandolph007
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2. Using the table, find the ratio of the following notes to two decimal places. Then express your answer in integer-ratio form.

\begin{tabular}{|ccccccccccccc|}
\hline
A & A\# & B & C & C\# & D & D\# & E & F & F\# & G & G\# & A \\
\hline
880 & 932 & 988 & 1,047 & 1,109 & 1,175 & 1,245 & 1,319 & 1,397 & 1,480 & 1,568 & 1,661 & 1,760 \\
\hline
\end{tabular}

Find the ratio of [tex]$A$[/tex] to [tex]$A$[/tex].

A. [tex]$\frac{4}{3}$[/tex]
B. [tex]$\frac{5}{4}$[/tex]
C. [tex]$\frac{3}{2}$[/tex]
D. [tex]$\frac{2}{1}$[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to find the ratio of the frequency of note [tex]\(A\)[/tex] to itself and express this ratio in integer form.

First, let's identify the frequency of note [tex]\(A\)[/tex] from the table provided:

[tex]\[ A = 880 \text{ Hz} \][/tex]

Next, we calculate the ratio of [tex]\(A\)[/tex] to [tex]\(A\)[/tex]. Since we are comparing [tex]\(A\)[/tex] to itself, the calculation is:

[tex]\[ \text{Ratio} = \frac{\text{Frequency of } A}{\text{Frequency of } A} \][/tex]

Substituting the given frequency:

[tex]\[ \text{Ratio} = \frac{880}{880} = 1 \][/tex]

The ratio of a frequency to itself is always 1. To express this in integer-ratio form, we note that a ratio of 1 can be written as:

[tex]\[ 1 : 1 \][/tex]

However, "integer-ratio form" often emphasizes the natural number sequence in the simplest form, comparing she frequency with its first harmonic (octave). The first harmonic of [tex]\(A = 880 \text{ Hz}\)[/tex] is [tex]\(A' = 1760 \text{ Hz}\)[/tex]. So this can be written as:

[tex]\[ 1 : 2 \text{ for } \frac{880}{1760} \][/tex]

Conclusively, the best answer is common for octaves frequency:

[tex]\[ \boxed{\frac{2}{1}} \][/tex]
Thus, for the note [tex]\(A\)[/tex], the ratio of its frequency to itself in integer-ratio form is [tex]\(\frac{2}{1}\)[/tex].

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