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# Can Mathematical Patterns Be Found in Johann Sebastian Bach's Two-Movement Preludes and Fugues?

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Author: Julianne Blair Bochinski

### Purpose

To determine whether there are any mathematical patterns in the composition of Johann Sebastian Bach's two-movement preludes and fugues.

### Materials Needed

• Computer and printer
• Computer spreadsheet software for analyzing data and constructing graphs
• Sheet music: Johann Sebastian Bach, Complete Preludes and Fugues For Organ by Dover Publications, Inc., 1985

### Experiment

For each composition, the total number and frequency of notes, the number and frequency of notes in each line of music, and the number and frequency of notes in each measure will be analyzed, along with the ratios of the number of notes between the various lines of music.

### Procedure

1. Count the number of notes in the treble, bass, and pedal lines of each measure in each musical composition. Access the data-analyzing computer software program and enter this data. Use the software to calculate the total number of notes in each line of music and compose a graphical representation of your data. Print out your graph and visually inspect.
2. Count the number of measures in each musical composition and enter this data into your computer. Again, use the software to compose a graphical representation of your data and print for visual inspection.
3. Use the data entered into the computer to print out individual graphs of the number of notes in each of the treble, bass, and pedal lines of each measure for each work and check for patterns.
4. Use the computer data to calculate the ratio of the number of notes in the treble line of each measure to the number of notes in the bass line of each measure.Produce a graphical representation of these ratios and inspect for any patterns.
5. Use the computer data to calculate the total number of notes in each work and then take the square root of these numbers. Produce a graphical representation of this data and inspect for any patterns
6. Use the data produced in step 1 and calculate how many times a number appeared as the total number of notes in a measure, for each musical composition.Produce a graphical representation of these results.
7. Compare all graphs for results.

### Results

1. Was there a set ratio between the number of notes in any two lines of music? Does Bach use a certain number of notes in some measures more than others?
2. Did the number of measures in the minor key works vary more than the number of measures in the major key works?
3. Were you able to detect any mathematical patterns? Try to expand this experiment to determine if other mathematical patterning possibly exists.

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