J.D. Flood and D.W. Follet 1
Explorations of Intervals: Numbers and Consonance
Introduction
The Pythagoreans were the earliest known music theorists and they used monochords to explore the relationships between music and mathematics. As you work through the activities to get a feeling for the relationships between string lengths and intervals, keep in mind two important characteristics that determine the note sounded by a string. For a fixed tension, the longer the string, the lower the note. For a fixed length, the tighter the string, the higher the note.
You will be exploring the relationship between musical intervals and string lengths. In your text, Johnston describes an interval as the degree to which one note sounds higher than another. Two special intervals in Western music are the octave and the fifth. Many people, on hearing the notes making up these intervals, agree that there is a certain element of sameness about them. Pythagoras and his followers developed a method for generating scales using these two intervals.
The two exercises below will explore Pythagorean methods for generating scales and give you practice listening to musical intervals. Keep these goals in mind as you work through the activities:
Goals
- To understand how the ratios of lengths are related to musical intervals and to demonstrate these relationships on a monochord.
- To understand that ratios of small numbers are related to what we experience as consonance.
- To be able to predict the locations of the bridge to form the notes in a pentatonic scale and play a pentatonic scale on the monochord.
Equipment
The apparatus you will use is a dual string sonometer – it’s like a double monochord. The sonometer comes with adjustable bridges that allow you to alter the length of a string and tuning pegs that allow you to adjust the tension on the string.
Exercise 1: Simple ratios and perfect intervals
1. Hang five kilograms from the end of each string on the sonometer (the 1 kg hanger plus 4kg additional). Tune both strings to middle C (C4). We’ll give you the pitch. Pluck or bow to hear the unison between the two strings. Choose one string to be your reference “unstopped” string for this entire exercise. Now put a bridge under the other string at about the middle. This is now your “stopped” string. Pluck or bow each half of the string to compare the two notes, and adjust the position of the bridge until the notes produced by each half are in unison.
For Exercise 1 you will need to tilt the bridge so that it does not lift the stopped string. You will need to hold the bridge in place while you read the lengths of the stopped string segments.
2. Before moving the bridge on the stopped string, compare your measurements to the lengths of the unstopped string and each part of the stopped string in the table provided on Page 2. (The long and short segments of the stopped string should be the same length at this point.) If your measurements differ from those in the top row of the table consult with Dr. Follet or Dr. Flood.
3. Now adjust the position of the bridge until the longer part of the string sounds a perfect fifth above the unstopped string, and record the lengths of both parts of the string in the table. You should also record the interval relationship between the two segments of the stopped string.
4. Repeat step three for a perfect fourth and a major second. Put the larger number on top when calculating ratios.
Interval you hear(Unstopped to long segment) / Unstopped string length
(cm) / Stopped string,
long segment
(cm) / Ratio of unstopped length to long stopped length / Stopped string, short segment
(cm) / Interval you hear
(long segment to short segment) / Ratio of long segment to short segment
Octave / 97.6 / 48.8 / 2/1 / 48.7 / unison / 1/1
Perfect fifth / 97.6
Perfect fourth / 97.6
Major second / 97.6
5. What ratio (not in decimal form) seems to correspond to a perfect octave? A perfect fifth? A perfect fourth? Keep in mind that you have two kinds of comparisons – between the unstopped string and the long stopped length, and between the long and short segments of the stopped string. Write your answers below.
6. What ratio (not in decimal form) seems to correspond to a major second? Is this a consonant interval? What does this tell you about numerical ratios associated with consonant intervals? Consider you results from the table.
7. To see how intervals are affected by the tension in the string, tighten both strings until the pitch goes up about a semitone. Keep the two strings in unison. Now insert the bridge so that the ratio of the unstopped string length to the longer segment of the stopped string is 3/2. What is the interval between the two segments of the string?
Based on your observation, does the pitch of the unstopped string affect the relationship between the length ratios and perfect intervals?
Exercise 2: Generating a Pentatonic scale using the Pythagorean method
On page 7 of your text Johnston describes a Pythagorean note generating procedure used to produce scales.
Take an existing ratio and multiply or divide it by 3/2. If the number you get is greater than 2 then halve it; if it is less than 1 then double it.
As you saw in Exercise 1, the simplest length ratio is 2/1, corresponding to the octave, with the next being 3/2 for the perfect fifth. You will use the Pythagorean note generating procedure to find the ratios required to produce a pentatonic scale.
Before starting the calculations of the length ratios let’s briefly review the data you collected in Exercise 1. You found that the ratio of the lengths of the unstopped string to the stopped segment was 2/1 for the interval of an octave. This came from the observation that the segment sounding the octave is one-half as long as the open string. The segment sounding the fifth is 2/3 as long as the unstopped string. The string length goes as the inverse, or reciprocal, of the ratios you found in Exercise 1. This will help you determine the bridge locations you need to play the scale.
At this point the collection of notes we have corresponds to ratios of 1/1, 3/2 and 2/1, the unison, fifth and octave respectively. We get the first new note by dividing 1/1 by 3/2.
Since this gives 2/3, which is less than 1, we must multiply by 2 to get a number between 1 and 2.
Our new note corresponds to the ratio 4/3, which we found to be a perfect fourth in exercise 1.
We can generate another note from the fifth by multiplying by 3/2.
To get a note in the same octave as the others (with a length ratio between 1 and 2) we divide by two to get the ratio 9/8.
We now have a four note scale as noted below.
Part A: 19/8 4/33/2 2, (or in decimal form 1.00 1.1251.331.502.00)
Now apply the procedure (both ways – multiplication and division) to the ratio 4/3. Show your work below.
Make a list of the notes (as in Part A above) including the two new notes you generated. This should be the pentatonic scale. Compare your results to the stave at the top of page 8 in your text. If you don’t have the same set of ratios check your calculations with Dr. Follet or Dr. Flood before continuing.
Pentatonic Scale
Playing the pentatonic scale
As noted above, the ratios of the string lengths you need to play the notes are the reciprocal of the ratios you found above. Using this information along with your results above, determine the string lengths, in centimeters, you need to play the scale if the open string is 97.6 cm long. Show your work and a summary list of the string lengths below.
Play the scale for your lab partners and the Drs. F.