# Patterns in J.S. Bach

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#### Updated on Sep 21, 2011

Grade Level: High School; Type: Mathematics

### Objective:

Determine the mathematical patterns in JS Bach’s two movement preludes and fugues.

### Research Question:

• Is there a mathematical pattern in these compositions?
• Does the ratio of treble notes to bass notes in a measure repeat itself?
• Does the total number of notes in a measure repeat itself in a predictable way?

This experiment does demand a certain level of musical and mathematical knowledge. It's also helpful to know how to set up a spreadsheet in Excel or similar program.

### Materials:

• Computer with Excel, or comparable program installed
• Johann Sebastian Bach, Complete Preludes and Fugue for Organ. Bach’s Notebook for Anna Magdelina can also be used.

### Experimental Procedure

1. Identify the measure numbers in your sheet music by counting them off and labeling them: measure #1, measure #2, and so forth
2. Set up your spreadsheet. There should be a row for every measure and a column for the number of bass notes, treble notes and pedal lines, as well as a column for the ratio of treble notes to bass notes, the ratio of treble notes to pedal lines, and the ratio of bass notes to pedal lines. Be advised, there are no pedal lines in the Notebook for Anna Magdalena.
3. Count the number of treble notes, bass notes and pedal lines in every measure. Enter this data on your data sheet.
4. Create three bar graphs showing a) the number of treble notes per meausure, b) the number of bass notes per measure and c) the ratio between the two.
5. Examine the graph to determine a pattern. For example, is a measure with 16 treble notes more likely to be followed with a measure of 4 notes? Record your findings.
6. Examine the pedal lines as a function of the number of treble notes and as a function of the number of bass notes. Record what pattern, if any, you observe.

Terms/Concepts: treble notes, bass notes, pedal lines, ratio, function, repeating sequences