Bill Manaris : Fall 2009 / FYSM 117 Homework 3

Assigned Date: Monday, Oct. 26, 2009
Due Date: Monday, Nov. 2, 2009
Due Time: 9:00am

Last modified on November 08, 2009, at 07:09 PM (see updates)

This is a pair assignment. You may work with a partner.


This assignment focuses on:


  1. Read this article:

    Michael Beer (2008). "Mathematics and Music: Relating Science to Arts?" Mathematical Spectrum, 41(1):36-42.
  2. Use Audacity to generate two scales via sine waves: a just scale (based on harmonic overtones, i.e., naturally occurring oscillations), and an equal-temperament scale (based on balanced keyboard tuning, i.e., technology driven).
  3. For more information on just temperament and equal temperament (including a simple way to generate them, see Scales: Just vs Equal Temperament). Quoting from this:
    The "just scale" (sometimes referred to as "harmonic tuning" or "Helmholtz's scale") occurs naturally as a result of the overtone series for simple systems such as vibrating strings or air columns. All the notes in the scale are related by rational numbers. Unfortunately, with Just tuning, the tuning depends on the scale you are using - the tuning for C Major is not the same as for D Major, for example. Just tuning is often used by ensembles (such as for choral or orchestra works) as the players match pitch with each other "by ear."
    The "equal-tempered scale" was developed for keyboard instruments, such as the piano, so that they could be played equally well (or badly) in any key. It is a compromise tuning scheme. The equal tempered system uses a constant frequency multiple between the notes of the chromatic scale. Hence, playing in any key sounds equally good (or bad, depending on your point of view).
  4. Compare how they sound. Which scale do you prefer?
  5. Pick a scale and create a melody. The melody should be constructed with sine waves like the scales above.


The more you do, the more bonus points you earn...

  1. Create more scales. For instance try:
  2. Create a more interesting melody (use harmony, i.e., two or more voices). (Hint: If you select a song like "Row, row, row your boat", it is easy to add harmony, since you can reuse the same voice - just copy it and shift it in time.)
  3. Other ideas...


You will submit your assignment at the beginning of class on the due date. Bring your project on a standard CDROM disk (or a USB drive).

The CD (or a USB drive) should contain:

  1. The class, homework assignment, date, name of your composition, team names written on its surface (or an envelope).
  2. A readme file with:
    1. The class, homework assignment, date, and team names (at the top).
    2. Describe the two scales. Create a table with frequency ratios for each scale. Repeat for bonus work.
    3. Describe how the two scales sounded. Which one you liked better and why.
    4. Name the melody you selected, and which scale you picked to create it.
    • Format: MS word document (should be .doc and NOT .docx), or PDF, or text - a standard format that can be opened and printed.
  3. Your Audacity project file (call it Just_vs_Equal_Scales.aup).
  4. Three final mixed-down audio file (see Audacity's Export menu). The format should be MP3 or OGG. (For bonus wrok, use meaningful file names accordingly.)
    • justScale.mp3 (or .ogg)
    • equalScale.mp3 (or .ogg)
    • melody-name_scale-name.mp3 (or .ogg). For example, RowRowRowYourBoat_Equal.mp3.

Before submitting the CD make sure everything is readable from it.

Alternatively, you could submit everything on a memory stick (USB flash drive). Put it in an envelop with the class, homework assignment, date, name of your composition, team names written on its surface.


Your assignment will be graded based on how well you followed the above instructions.


  1. Michael Beer (2008). "Mathematics and Music: Relating Science to Arts?" Mathematical Spectrum, 41(1):36-42.
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