Division Operation

The point of this section is to make some of the mathematics as clear as possible: How exactly does Music Algorithms produce its results in terms of mapping sequences of numbers to pitches?

In general, the Division Operation takes an interval of numbers (like a chunk of a sequence) and maps this interval to a range of pitches. The mapping can be thought of as a transformation in terms of “shrinking” (mapping a wider interval of numbers to a narrower range) or “stretching” (mapping a narrower interval of numbers to a wider range). Let’s illustrate both:

Stretch

1. Consider the first eight Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, 13). The interval defined by these numbers goes from 0 – 13, so the interval width is 13 { That is, 13 minus 0, or the maximum value minus the minimum value in general }
2. Suppose we want to map those eight numbers in [A] to a range of pitches that goes from 20 – 35, and note that the width of the pitch range is 15 { 35 minus 20 }
3. Here’s how the Fibonacci Numbers in [A], get mapped to pitch range in [B]
   1. The lowest Fibonacci Number goes with the lowest Pitch.
   2. The highest Fibonacci Number goes with the highest Pitch
   3. The other Fibonacci Numbers must fit proportionally between the lowest and highest pitches. Here’s how the end result would look:
      Fibonacci Numbers: 0 , 1, 1, 2, 3, 5, 8,13   …become mapped to…
      Pitches: 20,21,21,22,23,25,29,35
   4. What follows below is a step-by-step analysis of how just one of the Fibonacci Numbers (8), got mapped to the Pitch of { 29 } in this example:

1. 1. Let’s first look at the Fibonacci Numbers:
      • (8-0) ÷ (13-0) ~ 61.5%
      • So, we know that 8 is 61.5% of the distance from 0 to 13.
   2. Now let’s look at the Pitches.
      • The distance from 20 to 35 is 15.
      • 61.5% of 15 is ~ 9.23
      • 20 + 9.23 = 29.23, we’ll truncate to 29
        • To check, note that (29.23-20)/(35-20) ~ 61.5%
        • So, we know that 29.23 is located ~ 61.5% of the distance from 20 to 35
   3. Thus, the Fibonacci Number (8) is approximately the same proportional distance from (0) to (13) as its pitch {29} is from {20} to {35}.

Shrink

1. Consider the ninth through the sixteenth Fibonacci numbers (21, 34, 55, 89, 144, 233, 377, 610). The interval defined by these numbers goes from 21 – 610, so the interval width is 589 { 610 minus 21}.
2. Suppose we want to map those eight numbers in [A] to a range of pitches that goes from 25 – 40, and note that the width of the pitch range is 15 { 40 minus 25 }.
3. Here’s how the Fibonacci Numbers in [A], get mapped to pitch range in [B]
   1. The lowest Fibonacci Number goes with the lowest Pitch.
   2. The highest Fibonacci Number goes with the highest Pitch
   3. The other Fibonacci Numbers must fit proportionally between the lowest and highest pitches. Here’s how the end result would look:
      Fibonacci Numbers: 21,34,55,89,144,233,377,610   …become mapped to…
      Pitches: 25,25,25,26, 28, 30, 34, 40
   4. What follows below is a step-by-step analysis of how just one of the Fibonacci Numbers (144), got mapped to the Pitch of { 28 } in this example:

1. 1. Let’s first look at the Fibonacci Numbers:
      • (144-21) ÷ (610-21) ~ 20.1%
      • So, we know that 144 is 20.1% of the distance from 21 to 610.
   2. Now let’s look at the Pitches.
      • The distance from 25 to 40 is 15.
      • 20.1% of 15 is ~ 3.02
      • 25 + 3.02 = 28.02, we’ll truncate to 28
        • To check, note that (28.02-25)/(40-25) ~ 20.1%
        • So, we know that 28.02 is located ~ 20.1% of the distance from 25 to 40
   3. Thus, the Fibonacci Number (144) is approximately the same proportional distance from (21) to (610) as its pitch {28} is from {20} to {35}.