Course and Section __PH 101-001__________      Name ___________________________________


Date ____May 1, 2006_______________                         __________________________________________

Sound of Music

  1. Designing a pleasing chromatic scale

    Background:

    Music (at least traditional Western music) is made up of (more or less pure) tones of specific frequencies chosen from a discrete set, called a scale. The simplest scale is a well-tempered chromatic scale in which the ratios of successive frequencies are all the same, say r>1.  If we pick one frequency from the scale (call it the "tonic") and denote it by f0, then the others are rf0,  r2f0,   r3f0,   etc. [We can also go downward from the tonic, . . .r-2f0,   r-1f0.]
    1. Make a geometric series in Excel:  Open an Excel worksheet.  Label it "Chromatic Scale" in cell A1.  Type the label "ratio" in cell A4, then type a ratio in cell B4, say 1.1.  Type a tonic frequency in Hertz, maybe 1000, in cell B6 (you can label it "tonic" in cell A6).  In cell B7, type "=B6*B$4", meaning it's the tonic times the ratio.   Click and drag the lower right corner of this cell downward about 10 rows to produce the geometric series.  In the first copied cell (B8),  B6 will change to B7; the $ tells Excel not to change the 4 in B4.
    2. Notice that there's nothing special about these frequencies.  It is found that a scale sounds better if some of the frequency ratios are the ratios of small integers, like 2/1, 3/2, 3/1, 4/3, etc.  The simplest of these is 2/1 (this relationship is called an octave), so we want to make sure the chromatic scale contains this ratio.  The easiest way to do this is to set r=2 (i.e, replace the 1.1 in cell B4 by 2).  However, this scale has only one note per octave, and we'd like more, say N notes.  Let row 3 give this number N (write "# notes per octave" in A3 and 1 in B3).  Put 2 in C3, and extend this to N = 20 by selecting B3 and C3 and dragging the lower right corner to the right.  We will get N notes per octave if  rN = 2.  So replace the r = 2 in cell B4 by r = 21/N.  [Here N = 1, but use B3 so you can drag it for different values of N.  What Excel function can you use to take 2 to a power?]  Then drag B4 to the right to get all the values of r = 21/N.  You can also copy the rest of column B (the frequencies) to the other columns to get all the chromatic scales.  [This can be done by selecting and dragging, but you need to copy it once to keep Excel from thinking you want to increment the tonic.]
    3. Now we've ensured that the ratio 2/1 is present.  The next simplest (and presumably, next most pleasant) is 3/2.  Which of the scales comes the closest to containing this ratio? [Hint: look for tonic*3/2 in the scales.]
      N = _______________.  What is the percent deviation from 3/2? ____________________
    4. Which of the scales comes closest to the ratio 4/3?  N = _______________, deviation(%) = _____________.
    5. How about 5/3?   N = _______________, deviation (%) = ______________.
      Turn in a printout of the Excel file with these 3 best ratios circled.
    6. [Optional, if you happen to know:]
      Which chromatic scale does Western music use? N = _______________

  2. Intensity questions:
    1. What is meant by “Intensity?”




    2. What is the intensity level, β, for a sound of intensity 10-6 W/m2?




    3. What is the intensity of a sound wave which has β = 40 dB?




    4. If sound A is louder than sound B by 5 db, how much more intense is A than B…that is, what is IA/IB?