MT 007.02 / SL 266.01
Ideas in Mathematics: The Grammar of Numbers

Exercise: Interval arithmetic

In this exercise you will use the musical interval proportions and nomenclature from our lectures to test the construction of a 'natural' diatonic musical scale on c.
Remember that in order to add intervals you multiply the fractions which represent the proportions characteristic of each interval. In order to find the difference between, i.e. to subtract, intervals, you divide the fractions. You can find examples of both processes below.

A. Given the following proportions for the adjacent component intervals of a diatonic scale starting on c,

• c - d 9:8 (major second)
• d - e 10:9 (minor second)
• e - f 16:15 (semitone)
• f - g 9:8 (major second)
• g - a 10:9 (minor second)
• a - b 9:8 (major second)
• b - c' 16:15 (semitone)

  establish and 'prove' by addition the proportion that characterizes each of the following intervals. Wherever the result corresponds to an interval proportion which we have identified already, or which you may be aware of from other sources, add that interval label to your solution.

  1. c - e
  2. c - f
  3. c - g
  4. c - a
  5. c - b
  6. c - c

  E.g. (study this!)
  c - e = (c - d) + (d - e) => (9:8) x (10:9) = 90:72 = 5:4 = major third

  B. After you have answered A, answer the following:

  1. How many major thirds (5:4), in addition to c-e, can you find in this scale?
     In each case 'prove' the intervals.
     What combination of components characterizes a major third (do-mi)?
  2. How many minor thirds (6:5) can you find in this scale?
     Is d-f a minor third? Comment.
     What combination of components characterizes a minor third (la-do')?
  3. How many fourths (4:3) can you find in this scale?
     Are f-b or a-d (transposing the a down an octave) pure fourths? Comment.
     What combination of components characterizes a fourth (do-fa)?
  4. How many fifths (3:2) can you find in this scale?
     Are d-a or b-f pure fifths? Comment.
     What combination of components characterizes a fifth (do-sol)?

  An example of adding intervals:

  major third plus minor third
  = (5:4) x (6:5) = 30:20 = 3:2 = fifth

  An example of subtracting intervals:

  octave less a fourth
  = (2:1) div (4:3) = (2:1) x (3:4) = 6:4 = 3:2 = fifth

   

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