## Exploring Measurement, Sequences, and Curves with Stringed Instruments

Students will measure lengths on stringed musical instruments, and discuss how the placement of frets on a fretted instrument is determined by a geometric sequence.

This lesson focuses on the investigation of geometric sequences by considering the placement of frets on a stringed instrument. A major component of this lesson is the measurement of distances between frets, and without correct measurements, the mathematics of the lesson will be lost. It is imperative to stress the importance of accurate measurements for this lesson, so you may wish to begin the lesson with a discussion of precision and accuracy.

As a first step, display the Pre-Activity Questions from the first page of the Overheads. Questions 2 and 3 are warm-up questions that allow students to review some of the skills that will be needed for the lesson. Question 1, however, will lead into a nice discussion about precision of measuring devices.

During the activities, students will be required to measure lengths on fretted musical instruments and calculate the ratios of consecutive lengths. Although there are advantages to students making their own decisions about which measurement system to use, a discussion about precision and accuracy might be useful in helping students choose a system so that their measurements are as accurate as possible and their calculations are less tedious. In general, the metric system is less tedious, less likely to result in measurement and calculation errors, and is the more precise than most U.S. measuring tapes. (A U.S. ruler with 1/16-inch precision, converted to metric, has precision of 1.6 mm. Comparatively, a metric ruler typically has 1-mm precision.)

Display the second page of the Overheads (which shows three rulers with various levels of precision) and discuss the difference between *accuracy* and *precision*.
Students should understand that precision refers to the smallest unit
that a tool will measure, while accuracy refers to the range of
possible values that might be described by a particular measurement.

*Precision* is the smallest unit of a particular
measuring tool. The actual length could be as small as the measured
length minus one-half the precision of the measuring tape, or as long
as the measured length plus one-half the precision of the tape. For
instance, a correctly measured length of 4½" inches, measured with a
measuring tape with ½-inch precision, could have an actual length of
4½ ± ¼ , or anywhere between 4¼" and 4¾". If the same length were
measured as 115 mm, with a metric measuring tape of precision 1 mm (and
accuracy of 0.5 mm), the actual measurement could be anywhere
between 339.5 and 340.5 mm. The first two rulers below demonstrate
precision and accuracy within the U.S. system; the first has
¼‑inch precision, whereas the second has 1/16‑inch precision. The third
ruler is a metric ruler with 1‑mm precision.

Students may convert between metric and U.S. measurements to determine which has greater precision; however, they might also "eyeball" the markings to determine which has greater precision. If the smallest marking on a U.S. tape is 1/8 inch, it is easy to see that 1 mm is the smaller measure, so the metric tape has greater precision. If the smallest marking is 1/16 inch, students may have to convert between metric and U.S. measurements to make the comparison. (However, they can probably still see the difference—a millimeter is smaller than 1/16 inch.)

Divide students into groups of four for the first activity, and distribute the To Fret Activity Sheet.

In this activity, students measure the distance from the bridge
to the nut and to each fret on an instrument. (See figure below.)
Two students can measure while the other students record information or
do calculations. Any instruments with frets will work, such as guitars,
ukuleles, and mandolins. Students should discover that the ratio of
consecutive lengths will be constant, regardless of the measuring
system or musical instrument. The ratio is approximately 0.94, meaning
that the distance from the (*n* + 1)st fret to the bridge is about 0.94 times the distance from the *n*th fret
to the bridge. Although measurements may differ from group to group,
the ratio should be the same for all groups. Slight variations will
occur due to measurement or calculation errors; other variations may
occur because of errors in the instrument itself, such as incorrect
bridge or fret placement.

After students complete the To Fret Activity Sheet, review the results as a class. Allow students to point out similarities and differences in what they found.

Rearrange students into new groups, and then distribute the Exact Ratio Activity Sheet. In the previous activity, students should have realized
that the ratio between lengths is approximately 0.94. For this
activity, students use the exact ratio of 2^{-1/12} to generate
a geometric sequence that gives the exact location of each fret. The
points of this geometric sequence can be approximated by an exponential
equation, and the *y*-intercept of the graph represents the scale length of the instrument.

The last row of the table in Question 5 on the Exact Ratio Activity Sheet reveals an interesting occurrence. Using the exact
ratio, the value of the calculated length from the 12th fret to the
bridge is *ar*^{12} = *a*(2^{-1/12})^{12} = ½*a*,
meaning that the 12th fret occurs exactly halfway between the nut and
the bridge. A string played "open" (that is, played without pressing
any fret on the instrument) is exactly one octave lower than the same
string played while pressing just behind the 12th fret. A string played
open is twice as long as a string played when the 12th fret is pressed,
so the open string vibrates at half the speed. (It is this very
occurrence that explains why the exact ratio is 2^{-1/12}.)

In the next lesson of this unit, *Fretting*, students will use what they’ve learned about the geometric sequence that occurs with frets to place frets on an instrument.

- Fretted chromatic instruments, such as guitars, violins, mandolins, and banjos
- Metric and U.S. rulers
- Graphing calculators
- Overheads
- To Fret Activity Sheet
- Exact Ratio Activity Sheet

**Assessment Options**

- Observe student groups working and their discussions. Are they getting the same ratio from fret to fret (with only minor differences)? Do they interpret those minor differences as results of measurement inaccuracies, or are they interpreting them as distinctly different ratios?
- Ask students individually to differentiate between
*precision*and*accuracy*. Do they comprehend that accuracy is based on correctness of measurement, the precision of the measuring tape, and quality of the measuring tape? (A cloth tape, for example may stretch and not give a very accurate reading. On some tapes, the end has "wiggle room" which may cause inaccuracies.)

**Extensions**

- Allow students to research the "Rule of Eighteen." This rule states that the distance to the 1st fret is one-eighteenth of the way from the nut to the bridge. The 2nd fret is then one-eighteenth of the way from the 1st fret to the bridge; the 3rd fret is one-eighteenth of the way from the 2nd fret to the bridge; and so on. Have students compare this approximation to the ratio that they discovered in this lesson. Do they think that the "Rule of Eighteen" will be accurate enough to produce a high-quality instrument?
- Have interested students interview a guitar maker to see how he or she designs the instrument and places the frets. In lieu of a local instrument maker, students can find information on the Internet or other sources about making guitars. Have them seek an answer to the question, "Why is it impossible to perfectly tune a fretted chromatic instrument?"
- Follow-up with a broader discussion about accuracy and precision. Ask the question, "How precise do we need to be? Is precision of 1 mm good enough to bore holes in building an engine, in measuring pattern pieces for a dress, to determine the thickness of a guitar string, or to measure the length of a guitar neck? For what activities would precision of 0.01 mm be necessary?"
- Move on to the next lesson,
*Fretting*.

**Questions for Students**

1. Describe the difference between a geometric sequence of the form a, ar, ar2, ar3, ..., arn, ..., and an exponential curve of the form y = arx.

[A geometric sequence is discrete, whereas an exponential curve is continuous. The geometric sequence y = arn represents a discrete function where the values of n correspond with the frets on the instrument; in this case, the value of n is always an integer. The continuous function y = arx models the continuously changing string lengths on a fretless instrument as the musician’s finger slides up or down the string.]

2. For graphs of the form y = arx, explain what the y-intercept represents on a stringed instrument.

[The y-intercept represents the length of a string from nut to bridge.]

3. Discuss what factors contribute to the distances between frets not forming a perfect geometric sequence.

[The biggest issue is precision. Although an instrument maker may be very good, the ability to place the frets correctly depends on the precision of the measuring tool. If a measuring tool with 1/16-inch precision is used, the error on each fret could be up to 1/32 inch. After 12 frets, the combined error could be as much as 12 × 1/32 = 3/8 inch, which is significant.]

**Teacher Reflection**

- Did students effectively compare differences and similarities between related geometric sequences and exponential functions?
- Were the high achievers challenged? Were the low achievers engaged?
- Were concepts presented too abstractly? Too concretely? How would you change them?
- What other content areas were integrate with the lesson? Was the integration successful?

### Fretting

### Learning Objectives

Students will:

- Determine the accuracy and precision of a measuring tool, and choose an appropriate measuring tool for an investigation.
- Discover the geometric sequence formed by the frets on a string instrument.
- Determine the ratio that occurs between frets.

### NCTM Standards and Expectations

- Analyze precision, accuracy, and approximate error in measurement situations.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP8

Look for and express regularity in repeated reasoning.