ADJSCALES: DIFFERENTIATING BETWEEN SIMILAR
ADJECTIVES FOR LANGUAGE LEARNERS
Vera Sheinman and Takenobu Tokunaga
Department of Computer Science, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku Tokyo 152-8552, Japan
Keywords:
Adjective-scales, Patterns, Web-based, Second language learning, Gradation, Computational linguistics.
Abstract:
In this study we introduce AdjScales, a method for scaling similar adjectives by their strength. It combines
existing Web-based computational linguistic techniques in order to automatically differentiate similar adjec-
tives that describe the same property by strength. Though this kind of information is rarely present in most of
the lexical resources and dictionaries, it might be useful for language learners that try to distinguish between
similar words and that want to capture the differences from a single structure. Additionally, AdjScales might
be used by constructors of lexical resources in order to enrich them. The method is evaluated by comparison
with annotation on a subset of adjectives from WordNet by four native English speakers. The collected anno-
tation is an interesting resource by its own right. This work is a first step towards automatic differentiation of
meaning between similar words for learners.
1 INTRODUCTION
In the process of building their vocabulary, language
learners sometimes need to choose an appropriate
word to use from a set of near-synonymous words.
The subtle differences between words and the fact
that the semantics of near-synonyms between the na-
tive language and the second language usually over-
lap only partially make it all more difficult. Consider
for example the sentences, “This film is good”, “This
film is great”, “This film is superb”. All of these give
a positive evaluation of a film, but in which one and
under what circumstances will the film be perceived
by a native speaker of English as the best? How is the
learner to know?
A Linguistic Scale is a set of words of the
same grammatical category, which can be ordered by
their semantic strength or degree of informativeness
(Levinson, 1983). Linguistic scales are lexicalized
for various parts of speech. For instance, hsurprise,
startle, shocki is a verbal scale(Chklovski and Pantel,
2004).
Existing linguistic resources and dictionaries
rarely contain information on adjectives being part of
a scale, or being of a particular strength. Though,
this information may be deduced in some cases from
the word definition, such as “very small” for “tiny”
in WordNet (Miller, 1995), it is not always so, and
Head:
large
Attribute: size
enormous
huge
ANTONYM
similar
similar
Head:
small
smallish
tiny
similar
similar
Figure 1: Descriptive adjectives encoding in WordNet.
lacks the convenience of a single visual scale like
infinitesimal tinysmallsmallish.
Gradation is a related term describing variation
of strength between adjectives. (Fellbaum et al.,
1993) describes gradation as a semantic relation
organizing lexical memory for adjectives and pro-
vides six examples of gradation for attributes SIZE,
WHITENESS, AGE, VIRTUE, VALUE, and WARMTH.
For instance, the example gradation for SIZE is
hastronomical, huge, large, standard, small, tiny,
infinitesimali. According to Fellbaum, gradation
is rarely lexicalized in English, and thus it is not
encoded in WordNet. Adverbial expressions like
“slightly” or comparative expressions like “more” are
usually preferred. While agreeing with this claim, we
believe that having a method for grading adjectives
that are lexicalized is important and beneficial for
learners that struggle with similar adjectives. More-
229
Sheinman V. and Tokunaga T. (2009).
ADJSCALES: DIFFERENTIATING BETWEEN SIMILAR ADJECTIVES FOR LANGUAGE LEARNERS.
In Proceedings of the First International Conference on Computer Supported Education, pages 228-234
DOI: 10.5220/0001972602280234
Copyright
c
SciTePress
over, using the Web as a corpus, this information may
be extracted with less effort than before.
Descriptive adjectives describe a property and
tend to be scalar. WordNet encodes them in clus-
ters (adjective-sets). Two antonymous representa-
tive synsets (head-words) are linked to a noun they
describe (attribute). Each head adjective is linked
to similar adjectives. Relations between the similar
adjectives and differences between the SIMILAR con-
nections are not encoded. In the example encoding in
Figure 1, there is a clear difference between “small-
ish” that is slightly less small than “small”, and “tiny”
that is normally perceived to be smaller than “small”.
In this work, our objective is to identify such cases
and to provide this kind of distinction.
The similar adjectives in each adjective-set in
WordNet are not identical, and usually each synset
provides a nuance of meaning that differentiates it
from others. In addition to STRENGTH, there are oth-
ers, such as INFORMAL-LANGUAGE-OF relation that
holds between “teeny-weeny” and small”. Detect-
ing these kinds of relations is also important in the
context of lexical choice by learners. Gradation being
very central in adjectives, other possible relations are
left out of the scope of this work.
We introduce an automatic Web-based approach
to extract strength information for adjectives, AdjS-
cales, that incorporates recent advances in Natural
Language Processing. In choosing the suitable meth-
ods for this task, our goal was simple and freely ac-
cessible methods that do not require any special cor-
pora, parsing or tagging. The novelty of AdjScales is
in its automatic construction of adjective-scales from
several examples, in the language learner as the target
user, and in its evaluation. This work can contribute to
improving existing language resources, textbook au-
thoring, and tools for learners.
2 PROPOSED METHOD:
ADJSCALES
2.1 Pattern Extraction
Pattern extraction is a preparatory step for AdjS-
cales. Similarly to (Davidov and Rappoport, 2008),
we use pattern-extraction-queries of the form “a
b” to find patterns where a, b are seed words, and “
denotes a wildcard
1
. We extract binary patterns of the
1
denotes 0 or more terms that may appear in its place.
In practice, search engines, usually use the notation of for
a single-word, and we used the queries “a b”, “a b”, “a
b” for each pattern-extraction-query.
{bad, mediocre} {good, great}
Scale Relaxation
Scale Unification
Output
unified scale: bad mediocre good great superb
Scaling
Scaling
good greatsuperb
good
mediocre
bad
great
Input
ANTONYM
mediocre bad
Figure 2: General Illustration of the Proposed Method.
form
p = [prefix
p
x infix
p
y postfix
p
]
from the snippets of the query results returned by a
search engine
2
. Snippets are a good source for pat-
terns, because they contain the direct context of the
query text
3
. A pattern p can be instantiated by a pair
of words w
1
, w
2
to result in a phrase
p(w
1
, w
2
) = “prefix
p
w
1
infix
p
w
2
postfix
p
,
or similarly it can be instantiated by a word w
1
, and a
wildcard to result in a phrase “prefix
p
w
1
infix
p
postfix
p
to search for words cooccurring with the
word w
1
in a pattern.
Let’s consider an example pattern p
1
where
prefix
p
1
= φ, infix
p
1
= “if not”, and postfix
p
1
=
φ, if we instantiate it with the pair of words
(good, great) we will get a phrase p
1
(good, great) =
“good if not great”. Instantiating it with (, good) will
result in a phrase p
1
(, good) = if not good” that
can be used to search for items appearing on the left
side of the pattern p
1
with the word “good”.
If p(w
1
, w
2
) appears in snippets that are returned
by a search engine for a pattern-extraction-query, we
refer to it as p is supported-by (w
1
, w
2
).
Differently from (Davidov and Rappoport, 2008),
we choose the seed word pairs in a supervised man-
ner, so that seed
2
STRONGER-THAN seed
1
. For the
experimental settings described in this work we used
10 seed word pairs selected from the adjective scale
examples in (Fellbaum et al., 1993). The relation
STRONGER-THAN is asymmetric, therefore, we select
only the asymmetric patterns that are extracted con-
sistently so that the weaker word in each supporting
2
We use Yahoo search API (Yahoo Inc., 2008) through-
out the experiments described in this paper.
3
For the extraction purposes snippets are split into sen-
tences and are cleaned from all kinds of punctuation.
CSEDU 2009 - International Conference on Computer Supported Education
230
Final Structure:
INTENSE ADJECTIVES
Input: similar adjectives
{good, great}
Extension: {superb}
extend: get more similar adjectives
detect the head words
build a scale
good
great superb
Intermediate Structure:
good
great
superb
Figure 3: AdjScales Core.
pair is on the left side of the pattern (before the in-
fix words) or so that the weaker word is on the right
side of the pattern (after the infix words). If not all
the supporting pairs of words share the same direc-
tion, the pattern is discarded. We define the former
selected patterns as intense, and the latter as mild.
We select only the patterns supported by at least 3
seed pairs and we require a pattern instance by each
supportingpair to repeat at least twice in the sentences
extracted from the snippets to increase reliability. We
also require the patterns to be supported by adjec-
tives describing different properties. This constraint
is important, because patterns that are supported by
seeds that describe the same property tend to appear
in very specific contexts and are not useful for other
properties. For instance, [x even y amount] might be
extracted while supported only by seeds describing
the SIZE property, such as (huge, astronomical), (big,
huge), (tiny, infinitesimal).
To exclude patterns that are too short and tend to
be too generic, if pattern p is included in pattern q,
and both of them match the other requirements, we
select only the longer pattern, q.
2.2 Method Steps
AdjScales method, outlined in Figure 2, comprises
several steps listed below with Scaling (Section 2.2.5)
being its core. We divide the input adjectives into two
subsets in a Scale Relaxation (Section 2.2.2) step.
Then, the rest of the method is performed on each
of the subsets separately until the results are unified
in the final step of Scale Unification (Section 2.2.6)
outputing an adjective scale.
2.2.1 Input
AdjScales expects at least 2 similar adjectives as the
input. One adjective leaves the task of scaling open
for too many interpretations, while two adjectives
give a good clue on what scale is interesting for the
user. Similar adjectives for our purposes are adjec-
tives that describe the same property.
2.2.2 Scale Relaxation
According to (Hatzivassiloglou and McKeown,
1993), for adjectives, the total scale is commonly re-
laxed, so that elements of the scale can be partitioned
into several subscales. Consider the adjective scale
hcold, lukewarm, warm, hoti. It is not clear what is the
scale relationship between antonyms, such as “cold”
and “hot”. A total order by the relation of strength
within the subscale hlukewarm, warm, hoti is, how-
ever, evident.
In the Scale Relaxation step, AdjScales divides the
input into two antonymous subsets using antonymy
and similarity information from WordNet. If the in-
put words belong to the same adjective-set structure
in WordNet they are divided by their similarity to the
representative antonyms in the set. If the input words
all belong to the same subset they will remain in the
same set for the next steps. In other cases, if not all the
words appear in WordNet, or they are not encoded in
the same adjective-set structure, we currently assume
that the input words belong to a single subscale.
2.2.3 Extension
AdjScales attempts to provide the user with further
similar adjectives that do not appear in the input. Ad-
jectives that are encoded as SIMILAR to the input ad-
jectives in WordNet are added to the subset as an ex-
tension. For cases where WordNet is not applicable
no extension is currently performed.
2.2.4 Intermediate Structure
WordNet encodes adjectives by selecting the head ad-
jectives in each adjective-set and connecting the other
adjectives to them with similarity links. The rela-
tion between the head adjectives is antonymous. We
keep this type of encoding and call it an Intermediate
Structure. For cases where the input adjectives do
not appear in WordNet, we select the most frequent
words sharing the same context as others as the head
words. The Intermediate Structure allows us to reduce
the pairwise computations in the Scaling step. It also
allows the learner using the system to recognize the
most basic words in the scale.
ADJSCALES: DIFFERENTIATING BETWEEN SIMILAR ADJECTIVES FOR LANGUAGE LEARNERS
231
2.2.5 Scaling
The Scaling step depends only on availability of a
search engine that estimates page counts. For this
step, we refer to the set of patterns preselected by
Pattern Extraction (Section 2.1) as P. For each
pair (head-word, similar-word) from the Intermedi-
ate Structure, we instantiate each pattern p in P to
obtain phrases s
1
= p(head-word, similar-word) and
s
2
= p(similar-word, head-word). We estimate docu-
ment frequency, d f(s
i
), by using the estimated page
count hits returned by the search engine. We run
the resulting 2 phrases as 2 separate queries and
check whether d f (s
1
) > weight×d f(s
2
) and whether
d f(s
1
) > threshold. The higher the values are for
the threshold and weight parameters, the more reli-
able are the results, and the fewer there are. If p is
of the type intense, then a positive value is added to
the similar-word, otherwise if p is of the type mild
a negative value is added. When all the patterns are
tested, similar-words with positive values are classi-
fied as intense, while the similar-words with negative
values are classified as mild. Words that do not re-
ceive any points are classified as unconfirmed. For
each pair of words in the each one of the subsets (mild
and intense), the same procedure is repeated, creating
further subsets of mildest words that have the most
negative values within the mild subset, and most in-
tense words for the words with the highest positive
values within the intense subset. The information is
recored in a Final Structure that can be visualized
as a scale mildest words ··· least mild words
head-words least intense words · · · most
intense words.
To illustrate this process, consider the
example shown in Figure 3. Assume that
P = {p
1
= [x if not y]}. The Intermediate Struc-
ture in the example contains head-words={good},
and similar-words={great, superb}. We instan-
tiate s
1
= p
1
(good, great) = “good if not great”,
s
2
= p
1
(great, good). Choosing weight = 3 and
threshold = 100 pages, we run the queries s
1
,
s
2
. Google estimates d f (s
1
) as 353, 000 and
d f(s
2
) as 108
4
. p
1
is a pattern of type intense,
therefore a point will be added to the word
“great”. Similarly, d f(p
1
(good, superb) > 3 ×
d f(p
1
(superb, good)), and d f(p
1
(great, superb) >
3 × d f(p
1
(superb, great)). There are no
mild or unconfirmed words in this ex-
ample, resulting in the final structure:
{head-words={good},
intense words={great (-1) superb (1)}}
4
These figures are true for a search performed on 6th of
December, 2008 and might change slightly depending on
the date and the location of the search.
Figure 4: General pairwise agreement between AdjScales
compared to human subjects.
or simply good great superb.
2.2.6 Scales Unification
Subscales may be unified into a single adjective scale
simply by adding a link between the mildest words
on both sides. The internal STRONGER-THAN links
point from the mildest words towards the extremi-
ties. In fact, different properties are measured for each
subscale. For instance, the words “good”, “great”,
and superb” in our running example measure GOOD-
NESS, while their opponents bad” and “mediocre”
measure BADNESS. To present a unified scale of ad-
jectives that describe the property VALUE we reverse
the direction of links in one of the scales.
3 EVALUATION
For evaluation, we preselected 16 patterns (11 intense
and 5 mild) in the manner described in Section 2.1.
3.1 WordNet-based Corpus
We extracted descriptive adjective-sets
5
from Word-
Net 3.0 as the input to our system for evaluation of
the scaling step, and divided them into antonymous
subsets. Four native English speakers (2 Americans
and 2 British
6
), all male students from engineering
departments scaled the adjective subsets.
Some subsets in WordNet are too big to be
scaled by human subjects
7
, and they need prun-
ing. We downloaded snippets for queries of the type
5
The extracted adjective-sets comprise 757 head words
(645 distinct words) and 6,607 similar words (5,378 distinct
words).
6
We have observed no particular differences between
British and American subjects in the evaluation
7
The subset of “lean, thin” for attribute BODY WEIGHT
comprises 51 similar words, for instance.
CSEDU 2009 - International Conference on Computer Supported Education
232
Table 1: Subjects selections for WordNet adjectives.
#words mild #words intense
subject
1
137 358
subject
2
99 301
subject
3
89 290
subject
4
141 313
All subjects 22 163
p(head-word, ) and p(, head-word) for each pattern
p from the preselected patterns and for all the head-
words resulting in 625MB of data. If a word in an
adjective-set was not among the words appearing in
the wildcard slots in the extracted snippets, it was
pruned. The reasoning behind pruning is that cur-
rently our method cannot provide scaling decisions
for words that do not appear in any patterns, so, to
test this approach only the words that are potentially
applicable are considered. The final dataset for eval-
uation contained 308 subsets with 763 similar words
to be scaled.
Each subject scaled adjectives independently. For
each subset in the dataset, the subject was presented
with the head-words, attribute, and a set of similar
words. The head-words were fixed as neutral and
we asked the subjects to classify each similar word as
one of 5 types, compared to the head-words and to the
other words. When a word seemed stronger than the
head-words it was to be classified as intense or very
intense. When it seemed weaker than the head, it was
classified as mild or very mild. Words of similar in-
tensity to the head words were to be classified as neu-
tral. When not sure about a certain word or thinking
that it is not applicable for scaling, the subject classi-
fied it as not sure or not applicable, respectively.
We measure agreement between two subjects or
between AdjScales and a subject as follows. First,
general agreement is measured. If a word w in subset
s is selected as mild or as very mild by subject A (or
selected as mild by AdjScales), we will denote it as
w gen mild
A
(likewise for intense). Two subjects
A and B agree if w gen mild
A
w gen mild
B
or if w gen intense
A
w gen intense
B
. There
are many words that were undetermined (not sure, not
applicable, or unconfirmed for AdjScales), so it was
important to also measure the general disagreement
explicitly. For each two subjects A and B we measure
precision =
|gen mild
A
gen mild
B
|
gen mild
A
,
disagreement =
|gen mild
A
gen intense
B
|
gen mild
A
,
recall =
|gen mild
A
gen mild
B
|
gen mild
B
Table 2: Additional adjective scales.
precision disagreement recall
AdjScales 91.30% 8.70% 56.76%
for general agreement for words selected as generally
mild. Same notation holds for generally intense. Ta-
ble 1 shows the generally mild and the generally in-
tense selections made by the subjects, and the number
of selections all four of them agreed upon.
We averaged the pairwise agreement between the
subjects, and we averaged pairwise agreement of Ad-
jScales with each one of the subjects. Additionally,
we ran a baseline method that selected the most fre-
quently chosen classification, intense for all words,
and compared it in a similar manner. The compari-
son between the subjects, AdjScales
8
, and baseline is
shown in the chart in Figure 4.
We also compared AdjScales to the answers that
were generally agreed upon by all 4 subjects. AdjS-
cales disagreed with the four subjects consensus for
only one word. Additionally, to understand the finer
agreement on ordering adjectives on a scale, we mea-
sure order agreement. Subjects A and B agree on
order of a pair w
1
, w
2
if A and B both classified w
1
and w
2
as generally mild or as neutral and if both A
and B classified w
1
as milder than w
2
. They disagree
if A classified w
1
as milder than w
2
while B put them
in the inverse order. The same is true for the intense
side likewise. The subjects tend to agree on the order
between words (for 86.11% of word pairs within the
mild side, and for 88.74% of word pairs within the
intense side). AdjScales scores 86.11% and 70.20%
for order agreement for the mild and the intense sides
respectively.
3.2 Additional Adjective Scales
In independent experimental settings we requested
2 native English speakers and 3 non-native English
speakers to produce as many linguistic scales as pos-
sible. After the production step the subjects cross-
verified the results, and only the scales agreed upon
by all of them remained. We selected 9 of the scales
that were adjective scales for our dataset. We ex-
8
We used AdjScales parameters weight =
15, threshold = 20 set empirically. In order to re-
duce the search engine queries required for computation
of each scale, we grouped the queries of patterns into 4
subgroups unifying m patterns instances in each subgroup
by the operator OR:
p
1
(w
1
, w
2
) OR p
2
(w
1
, w
2
)OR . . . OR p
m
(w
1
, w
2
)
ADJSCALES: DIFFERENTIATING BETWEEN SIMILAR ADJECTIVES FOR LANGUAGE LEARNERS
233
tended this dataset by adding several adjective scales
from examples in the literature, such as 4 adjective
scales from a teaching resource (Cadman, 2008). In
the exercises in the suggested resource, several verb
and adjective scales are provided, where the students
are requested to order them by strength. The dataset
of this kind is less confusing than adjective-sets in
WordNet that are not organized as scalable-sets to be-
gin with.
We relaxed each of the scales in our dataset manu-
ally into 2 antonymous subsets, where there were two
antonymous components and performed scaling, re-
sulting in 21 subsets. We compared the scaling results
by AdjScales with the same parameters as in section
3.1 to the expected scales as shown in Table 2.
4 RELATED WORK
A major work in differentiation between near-
synonyms in computational linguistics by (Inkpen
and Hirst, 2006) provides a list of types of nuances of
meaning that need to be differentiated, such as stylis-
tic, attitudinal etc. Using automatic methods to dif-
ferentiate between near-synonyms is the objective of
our research, and in this sense this work is relevant
to ours. Language learners have difficulty in perceiv-
ing the differences among near-synonyms and adding
distinctions on these subtleties to existing language
resources is needed. Currently, we focus only on dif-
ferentiation of similar adjectives by strength. Adding
further types of differentiation is a much needed ex-
tension of our work in the future. Differently from
Inkpen and Hirst that use a collection of machine
readable dictionaries, we use the Web as a corpus.
(Hatzivassiloglou and McKeown, 1993) estab-
lished the first step towards automatic identification
of adjective scales. They provide an excellent back-
ground on adjectives and a general plan to identify
adjective scales, though, they focus only on cluster-
ing of similar adjectives.
Using patterns extracted from large corpora in or-
der to learn semantic relations between words is a
common approach in computational linguistics. The
pioneering work (Hearst, 1992) extracted hyponym
(IS-A) and meronym (PART-OF) relations. Further
studies (Chklovski and Pantel, 2004; Davidov and
Rappoport, 2008; Turney, 2008) intensively extend
this methodology, further relations are explored, su-
pervised and unsupervised methods are introduced.
Our work belongs to this school of relation extraction.
VerbOcean (Chklovski and Pantel, 2004),
explores fine-grained relations between verbs,
STRONGER-THAN being one of them. Their work is
very similar to ours in using lexico-syntactic patterns
extracted from the Web. Their selection of patterns is
manual, and it is based on training on 50 verb pairs,
with a total of 8 patterns selected for the STRONGER-
THAN relation. We utilize the asymmetry of the
STRONGER-THAN relation in a similar manner to
VerbOcean. We differ in our focus on adjectives and
in our evaluation procedure. VerbOcean, providing
differentiation between similar verbs, should be
considered in the context of language learners.
A large body of research (Turney and Littman,
2003; Popescu and Etzioni, 2005) has been conducted
in the field of opinion mining. An important dis-
tinction for opinion mining is semantic orientation
(positive, negative, or neutral) of words and utter-
ances. In this work we do not distinguish between
the positive or the negative senses of adjectives, but
rather make a more general distinction of the extent
of adjectival descriptive strength. We also have a dif-
ferent objective to provide linguistic distinction be-
tween synonymous adjectives for learners, while the
research in opinion mining concentrates on strength
of subjectivity and sentiment of words, and texts.
One of the main approaches in opinion mining is
extraction of semantic information from the Web, and
typically adjectives play a central role in understand-
ing opinion from texts. In these aspects, this field
is related to our work. According to (Turney and
Littman, 2003) semantic orientation of a word, in ad-
dition to its direction also comprises intensity, mild or
strong. They compute intensity in a combined com-
putation of the direction, using statistical association
with a set of positive and negative paradigm words.
OPINE (Popescu and Etzioni, 2005), a system for
product reviews mining ranks opinion words by their
strength as one of its subtasks. Both of these works
focus on detection of semantic orientation of words
and report on a very limited evaluation of ranking by
strength.
No previous work that we are aware of proposes
an automatic method to identify adjective-scales for
language learners.
5 DISCUSSION
We have presented AdjScales, a method to build ad-
jective scales from several examples of similar ad-
jectives using a state-of-the-art methodology of ex-
tracting relations using patterns over the Web. It is
simple, and the only required resource (for Scaling
step, which is the main focus of this work) is access
to a search engine. Overall, as can be seen from the
evaluation, AdjScales scales similar adjectives only
CSEDU 2009 - International Conference on Computer Supported Education
234
slightly less well than human subjects and much bet-
ter than the baseline
9
. It also performs quite well on
examples that seem more relevant in the context of a
language learner, although quite a few words still re-
main unconfirmed by the system (recall of only 56%
for the additional examples). There is only one dis-
agreement of the system with answers that all human
subjects agree upon, suggesting that in cases where a
scale is clear and thus suitable for learning, AdjScales
will be more accurate. The usefulness of the system
for learners is in the area of differentiation between
similar words using a simple structure (scale) to visu-
alize it.
A surprising observation from our experiments is
an unexpected asymmetry between the adjectives on
the mild side and the intense side of the head words
in WordNet. Subjects and AdjScales consistently se-
lected less words as mild, and also performed less
well for their mild selection. It may suggest that
WordNet structure or even language structure itself, is
such that there are many more words to intensify the
common head-words rather than weaken them. We
have also observed from analysis of the results that
some patterns perform better for mild words while
others do better in identification of intense words.
This direction will be further explored in the future.
Similar adjectives in general and adjectives in the
same adjective-set in WordNet differ in more than one
way. In many cases the subjects faced a difficulty
in scaling similar words that were presented to them,
because they were different in several aspects. This
suggests that the similar adjectives in adjective-sets
in WordNet are not necessarily in the same scalable-
adjectives set. We plan to study how to detect adjec-
tives that are on the same scale.
Some adjectives are much more intense than oth-
ers, while others are only slightly so. Estimating the
distances between the links on a scale seems to be an
interesting task that may be a useful visualization for
learners.
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