Math / Algebra

QuestionIf D has a frequency of 292 cps , find the frequencies of the following notes: a. an octave and six half-steps above D $292 \times 2=584$ b. two octaves and nine half-steps above $D$ $24+q=33 \quad 292 \times 1.05946^{33}$ c. an octave below D $1964.144707$

Studdy Solution

STEP 1

What is this asking? Given the frequency of D is $292$ cps, we need to find the frequencies of other notes relative to D using octaves and half-steps. Watch out! Remember, an octave doubles the frequency, and each half-step multiplies the frequency by the twelfth root of $2$ , which is approximately $1.05946$ .

STEP 2

1. An octave and six half-steps above D
2. Two octaves and nine half-steps above D
3. An octave below D

STEP 3

Alright, so we're starting with D at $292$ cps!
An octave above D doubles the frequency, so that's $292 \cdot 2$ .

STEP 4

Then, we go six half-steps higher.
Each half-step multiplies the frequency by $1.05946$ , the twelfth root of 2.
So, we multiply the octave frequency by $1.05946$ six times, which is the same as multiplying by $1.05946^6$ .

STEP 5

Putting it all together, the final frequency is $292 \cdot 2 \cdot 1.05946^6$ .
Let's calculate that! $292 \cdot 2 = 584$ , and $1.05946^6 \approx 1.41421$ .
Multiplying these gives us $584 \cdot 1.41421 \approx 825.98$ , which is approximately $826$ cps.

STEP 6

Two octaves above D means we double the frequency twice.
So, we multiply the initial frequency of $292$ cps by $2$ two times: $292 \cdot 2 \cdot 2 = 292 \cdot 4$ .

STEP 7

Now, we add nine half-steps.
Each half-step multiplies the frequency by $1.05946$ .
Nine half-steps means we multiply by $1.05946$ nine times, which is $1.05946^9$ .

STEP 8

Combining everything, the final frequency is $292 \cdot 4 \cdot 1.05946^9$ .
First, $292 \cdot 4 = 1168$ .
Then, $1.05946^9 \approx 1.68179$ .
Finally, $1168 \cdot 1.68179 \approx 1964.14$ , which is approximately $1964$ cps.

STEP 9

An octave below means we halve the frequency.
So, we divide the initial frequency $292$ cps by $2$ .

STEP 10

Calculating this gives us $292 / 2 = 146$ cps.
That's it for this one!

STEP 11

a. An octave and six half-steps above D: approximately $826$ cps b. Two octaves and nine half-steps above D: approximately $1964$ cps c. An octave below D: $146$ cps

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QuestionIf D has a frequency of 292 cps , find the frequencies of the following notes: a. an octave and six half-steps above D $292 \times 2=584$ b. two octaves and nine half-steps above $D$ $24+q=33 \quad 292 \times 1.05946^{33}$ c. an octave below D $1964.144707$

Studdy Solution

STEP 1

STEP 2

1. An octave and six half-steps above D
2. Two octaves and nine half-steps above D
3. An octave below D

STEP 3

Alright, so we're starting with D at $292$ cps!
An octave above D doubles the frequency, so that's $292 \cdot 2$ .

STEP 4

Then, we go six half-steps higher.
Each half-step multiplies the frequency by $1.05946$ , the twelfth root of 2.
So, we multiply the octave frequency by $1.05946$ six times, which is the same as multiplying by $1.05946^6$ .

STEP 5

Putting it all together, the final frequency is $292 \cdot 2 \cdot 1.05946^6$ .
Let's calculate that! $292 \cdot 2 = 584$ , and $1.05946^6 \approx 1.41421$ .
Multiplying these gives us $584 \cdot 1.41421 \approx 825.98$ , which is approximately $826$ cps.

STEP 6

Two octaves above D means we double the frequency twice.
So, we multiply the initial frequency of $292$ cps by $2$ two times: $292 \cdot 2 \cdot 2 = 292 \cdot 4$ .

STEP 7

Now, we add nine half-steps.
Each half-step multiplies the frequency by $1.05946$ .
Nine half-steps means we multiply by $1.05946$ nine times, which is $1.05946^9$ .

STEP 8

Combining everything, the final frequency is $292 \cdot 4 \cdot 1.05946^9$ .
First, $292 \cdot 4 = 1168$ .
Then, $1.05946^9 \approx 1.68179$ .
Finally, $1168 \cdot 1.68179 \approx 1964.14$ , which is approximately $1964$ cps.

STEP 9

An octave below means we halve the frequency.
So, we divide the initial frequency $292$ cps by $2$ .

STEP 10

Calculating this gives us $292 / 2 = 146$ cps.
That's it for this one!

STEP 11

a. An octave and six half-steps above D: approximately $826$ cps b. Two octaves and nine half-steps above D: approximately $1964$ cps c. An octave below D: $146$ cps

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Studdy Solution

STEP 1

STEP 2

1. An octave and six half-steps above D2. Two octaves and nine half-steps above D3. An octave below D

STEP 3

Alright, so we're starting with D at 292292292 cps!An octave above D doubles the frequency, so that's 292⋅2292 \cdot 2292⋅2.

STEP 4

Then, we go six half-steps higher.Each half-step multiplies the frequency by 1.059461.059461.05946, the **twelfth root of 2**.So, we multiply the octave frequency by 1.059461.059461.05946 six times, which is the same as multiplying by 1.0594661.05946^61.059466.

STEP 5

STEP 6

Two octaves above D means we double the frequency twice.So, we multiply the **initial frequency** of 292292292 cps by 222 two times: 292⋅2⋅2=292⋅4292 \cdot 2 \cdot 2 = 292 \cdot 4292⋅2⋅2=292⋅4.

STEP 7

Now, we add nine half-steps.Each half-step multiplies the frequency by 1.059461.059461.05946.Nine half-steps means we multiply by 1.059461.059461.05946 nine times, which is 1.0594691.05946^91.059469.

STEP 8

STEP 9

An octave below means we halve the frequency.So, we divide the **initial frequency** 292292292 cps by 222.

STEP 10

Calculating this gives us 292/2=146292 / 2 = 146292/2=146 cps.That's it for this one!

STEP 11

a. An octave and six half-steps above D: approximately 826826826 cps b. Two octaves and nine half-steps above D: approximately 196419641964 cps c. An octave below D: 146146146 cps

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1. An octave and six half-steps above D
2. Two octaves and nine half-steps above D
3. An octave below D

Alright, so we're starting with D at $292$ cps!
An octave above D doubles the frequency, so that's $292 \cdot 2$ .

Then, we go six half-steps higher.
Each half-step multiplies the frequency by $1.05946$ , the twelfth root of 2.
So, we multiply the octave frequency by $1.05946$ six times, which is the same as multiplying by $1.05946^6$ .

Two octaves above D means we double the frequency twice.
So, we multiply the initial frequency of $292$ cps by $2$ two times: $292 \cdot 2 \cdot 2 = 292 \cdot 4$ .

Now, we add nine half-steps.
Each half-step multiplies the frequency by $1.05946$ .
Nine half-steps means we multiply by $1.05946$ nine times, which is $1.05946^9$ .

An octave below means we halve the frequency.
So, we divide the initial frequency $292$ cps by $2$ .

Calculating this gives us $292 / 2 = 146$ cps.
That's it for this one!

a. An octave and six half-steps above D: approximately $826$ cps b. Two octaves and nine half-steps above D: approximately $1964$ cps c. An octave below D: $146$ cps