Two musical notes played at the same time produce harmony. The closest harmony is produced by frequencies with the greatest GCF. A,C,and C sharp have frequencies of 220,264,and 275, respectively. Which pair of these notes produces the closest harmony?
1 * 275 = 275.
5 * 55 = 275.
11 * 25 = 275.
1 * 264 = 264.
2 * 132 = 264.
3 * 88 = 264.
4 * 66 = 264.
6 * 44 = 264.
1 * 220 = 220.
2 * 110 = 220.
4 * 55 = 220.
5 * 44 = 220.
10 * 22 = 220.
220 and 275 have the GCF:
GCF = 55.
To determine which pair of notes produces the closest harmony, we need to find the pair of frequencies with the greatest greatest common factor (GCF).
Let's calculate the GCF of all possible pairs of frequencies:
1. GCF(220, 264)
- Prime factors of 220: 2 × 2 × 5 × 11
- Prime factors of 264: 2 × 2 × 2 × 3 × 11
- Common prime factors: 2 × 2 × 11 = 44
- GCF(220, 264) = 44
2. GCF(220, 275)
- Prime factors of 220: 2 × 2 × 5 × 11
- Prime factors of 275: 5 × 5 × 11
- Common prime factors: 11
- GCF(220, 275) = 11
3. GCF(264, 275)
- Prime factors of 264: 2 × 2 × 2 × 3 × 11
- Prime factors of 275: 5 × 5 × 11
- Common prime factors: 11
- GCF(264, 275) = 11
Based on the calculations, we can see that the pair with the greatest GCF (closest harmony) is A (220 Hz) and C sharp (275 Hz), with a GCF of 11.
Therefore, the pair of notes that produces the closest harmony is A and C sharp.