I am having great difficulty understanding Hewitt\'s ConceptualPhysics 10th Edit

Question

I am having great difficulty understanding Hewitt's ConceptualPhysics 10th Edition, Chapter 21, Problem 4: Starting with a fundamental tone, how many harmonics arethere (a) between the first and second octaves? (b) between the second and third octaves? I've read numerous chapters and so far I can only conclude thefollowing: 1. A tone consists of a fundamental frequency (which is thelowest frequency and determines the pitch) 2. A tone may also contain partial tones which provide thetimbre (clarinet as opposed to piano) 3. These partial tones (harmonics?) are whole number multiplesof the fundamental frequency (e.g. if fundamental frequency = 262Hz, then the second harmonic would be 524 Hz, third harmonic wouldbe 786 Hz etc.) I can't seem to find any information though as to therelationship of octaves and harmonics. I understand the concept ofoctaves (I have some basic understanding of music - I playkeyboards and read music). Any help on this would be greatly appreciated. Many thanks inadvance. Regards Tony I am having great difficulty understanding Hewitt's ConceptualPhysics 10th Edition, Chapter 21, Problem 4: Starting with a fundamental tone, how many harmonics arethere (a) between the first and second octaves? (b) between the second and third octaves? I've read numerous chapters and so far I can only conclude thefollowing: 1. A tone consists of a fundamental frequency (which is thelowest frequency and determines the pitch) 2. A tone may also contain partial tones which provide thetimbre (clarinet as opposed to piano) 3. These partial tones (harmonics?) are whole number multiplesof the fundamental frequency (e.g. if fundamental frequency = 262Hz, then the second harmonic would be 524 Hz, third harmonic wouldbe 786 Hz etc.) I can't seem to find any information though as to therelationship of octaves and harmonics. I understand the concept ofoctaves (I have some basic understanding of music - I playkeyboards and read music). Any help on this would be greatly appreciated. Many thanks inadvance. Regards Tony

Explanation / Answer

Middle C is 262 Hz 1 octave above middle C is 524 Hz 2 octaves above middle C is 1048 Hz 3 octaves above middle C is 2096 Hz f1= 262, f2=524, f3=786, f4=1048,   f5=1310,   f6=1572,  f7=1834,   f8=2096 Hz. so, there are 4 harmonics in the first two octaves: f1, f2, f3, and f4. and there are 5 harmonics in the second and thirdoctaves:    f4, f5, f6,   f7,and f8. The wording of the question is not 100% clear to me asto its intent.   Do we count all harmonics, including theones at the ends of the octaves as I have done above, or do we onlycount the harmonics "between" the starting and endings, in whichcase we would get two harmonics for the first answer, and threeharmonics for the second answer.

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