The Euclidean Distance in Diphthongs – R Graph and Code

Representing and plotting a distance in F1/F2 graph, in terms of the Euclidean distance, is relatively easy in R. This post shows one of the ways of achieving that. First, we provide a sample data, which consists of F1 and F2 values for two diphthong targets.  Then, draw the diphthong positions with their starting and ending targets, and, finally, calculate the distance. This R code does most of the F1/F2 calculations and drawing.

Data sample (Formants in a Diphthong)

First, a data sample.

    ascii   ipa     f1      f2
1  aw_l_1 ɑʊl_1 900.96 1600.10
2  aw_l_2 ɑʊl_2 373.61 1082.59 

    ascii   ipa     f1      f2
1  aw_l_1 ɑʊl_1 823.07 1542.39
2  aw_l_2 ɑʊl_2 411.39 1405.78

These are the values for the first two formants in /ɑʊ/, as measured in a group of 15 female ESL student and one RP speaker (also female). Number 1 in the notation marks the first vowel target, 2 the second (thus, aw_l_1 is /ɑ/ and aw_l_2 is /ʊ/), while the “l” marks a long diphthong. The two targets will be the starting and the ending of a line, and the line’s length is expressed by the Euclidean distance.

The Euclidean Distance

Euclidean distance is a metric distance from point A to point B in a Cartesian system, and it is derived from the Pythagorean Theorem. Thus, if a point p has the coordinates (p1, p2) and the point q = (q1, q2), the distance between them is calculated using this formula:

distance <- sqrt((x1-x2)^2+(y1-y2)^2)

Our Cartesian coordinate system is defined by F2 and F1 axes (where F1 is y-axis), and the metric distance refers to the distance from one diphthong target to another. The vowel targets, corresponding to A and B points are defined by the F1/F2 values in Hertz for a particular vowel. In our example above, A and B  are rows 1 and 2, while the values are F2 and F1 frequencies.

Plotting in R

The third step in the process is plotting, so we could see the graphical representation of the distance. We can do that by:

  1. Drawing the F1/F2 “coordinate system”.
  2. Drawing the vowels in A and B positions, and connecting them with a line.
  3. Drawing the arrows showing the direction of pronunciation and placing the IPA symbols.
An example looks like this:
Diphthongs drawn on F1/F2 plot
The English diphthongs as pronounced by the ESL students and a native RP speaker.
The diphthong  /ɑʊ/ is plotted in the lower right corner of the graph. Here are the Euclidean distances for that diphthong (in both variants):
          RPSpeaker  ESLStudents
aw_l ɑʊl  738.86     433.75
aw_s ɑʊs  816.08     471.60
The R code used to plot the graph can be found here.

Published by

Romeo Mlinar Author

6 thoughts on “The Euclidean Distance in Diphthongs – R Graph and Code”

  1. Dear all,
    How could we modify the script to add a third measurement point? Namely, at 25%, 50% and 75% of the vowel duration?

    Thank you ever so much,


  2. Hello

    thanks for your post. It is really interesting.

    I have a question. What was your criteria of dividing a diphthong into two vowels. For example, you mentioned
    /ɑʊ/. What was your criteria of dividing /ɑ/ from /ʊ/? Did you manually, looking spectrum and formants and hearing sound file, divide each vowel from each speakers?

    Thanks a lot!


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