1. [1pt] In this homework we will find frequencies for notes of an A major scale
ID: 1601740 • Letter: 1
Question
1. [1pt] In this homework we will find frequencies for notes of an A major scale. The note names for this scale are A, B, C#, D, E, F#, G#, A. The # symbol means "sharp." (C# is the black key just above the white C key on the piano keyboard.) If we start at A4 = 440 Hz, then the pitch of B4 can be found by multiplying 440 by the square of the twelfth root of two, since B is two half steps above A and each half step raises the pitch by the 12th root of 2. Do this calculation and see if you get the frequency given in the Appendix E of the online textbook. Now, suppose you are trying to reproduce an A major scale from the early 18th or 19th century, when A4 was perhaps tuned to 426.5Hz. What is the correct frequency (in Hz) for C#5 (assuming we want an equal tempered scale)?
Explanation / Answer
The scale was originally conceived of as a 7 note scale, with the notes A, B, C, D, E, F, G. However, these 7 notes are not equally distributed throughout the octave. Most of these pitches are a whole step above the previous one, but there is only a half step between the B and C, and between the E and F.
But sometimes, we want to move around where this half step occurs. For example, if we were playing in the key of G, we want a half step between the F and the G, but not between the E and the F. The solution is to bump up the F's pitch by a half step, which makes it a whole step higher than the E, and just a half step under the G. This "bumped up" higher version of F, we call F. A sharp always refers to raising the pitch by a half step, and a flat always refers to lowering the pitch by a half step. This is true regardless of whether the resulting pitch is a white or black key on the keyboard.
From this, you can see that a B, for example, is a half step higher than a regular B. But you'll notice there is already a key on the keyboard that sounds a half step higher than B -- we usually call it C, but B is also a perfectly valid name for that note, in the proper context (for example, the key of C would contain a B -- this occurs in Beethoven's Moonlight Sonata). Similarly, the note B can be called a C in the proper context (such as in an A minor chord). Granted, these don't come up very often, because writing in a key that requires them means reading/playing a lot of sharps or flats, which can often be tricky, but they do show up when needed.
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