On the relation between types and tokens of Japanesemorae
Katsuo Tamaoka

1Type and token frequencies of Japanese mora

The fundamental set of Japanese sounds, which are described in phonologicalscripts of two types of kana symbols: hiragana or katakana, consists of 101
basic kana including the special sound of the nasal /N/ (written in hiragana as
S and in katakana as S). Each Japanese kana symbol fundamentally corresponds to a speciﬁc moraic sound. For example, a kana symbol in hiragana
or in katakana represents a moraic sound /ka/ consisting of consonant /k/and vowel /a/. Using the word printed-frequency database created by Amanoand Kondo (2000, and 2003 for CD-ROM version)1 , Tamaoka & Makioka(2004) studied the frequency (token frequency) and the context of Japanesemorae, consisting of words in which they occur (type frequency).
Since type frequency represents here the property of cotextuality, i.e. thenumber of different contexts of a mora, and token frequency represents thefrequency, the present study proposes the hypothesis that there is some relationship between these two properties known from synergetic linguistics (cf.
Köhler 1986, 2002). Köhler tried to express this hypothesis by means of astraight line and obtained a signiﬁcant correlation coefﬁcient signaling theexistence of a linear relationship. However, the ﬁtted straight line is merely aﬁrst exploratory approximation because the lower frequency values of thecomputed straight line are negative, which is empirically impossible. The
present study therefore tries to ﬁnd a more adequate expression of this re1. A very large lexical corpus of 341 771 morphemes was established from newspapers
containing 287 792 797 morphemes, all of which were taken from the Asahi Newspaper
printed from 1985 to 1998. At present, this is the largest and the most up-to-date lexical
database created from calculating the word frequency of occurrence in Japanese written
texts. There are some coding errors in this database. For instance, Tokyo, written in two
kanji as !, is phonologically stored as /higasikyoR/ (/R/ refers to a long vowel)
instead of /tokyoR/. In addition, a long vowel /R/ is not clearly distinguished in thedatabase. Thus, I did not include the sounds of the long vowel /R/ and the geminate
sound /Q/ in this study.

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Katsuo Tamaoka

lationship.
The present study starts from the general theory of language laws (Wimmer & Altmann, 2005) stating that the majority of relationships between linguistic variables can be expressed by means of the relation between theirrelative rates of change, i.e. between dy/y and dx/x. Here y is frequency(tokens), x is cotextuality (types). This assumption is reasonable because linguistic variables cannot attain inﬁnite sizes and their changes tend to slow
down depending on the already attained value. Thus we postulate a very simple relation
dyb= c+dx(1)
yxwhere b is the proportionality coefﬁcient and c is interpreted either as a kindof “noise” or disturbance, or it represents the ceteris paribus condition, i.e. a
constant state of all other possible factors. The solution of this equation isy = axb ecx(2)where a = eC with integration constant C.
This relationship follows directly from the respective part of the synergetic
control cycle joining polytexty with frequency. Since in synergetic linguistics
one considers the variables usually in their logarithmically transformed form,
we speak about L-polytextuality and L-frequency. In the cycle presented below (Köhler, private communication, cf. Köhler 1986; 2002) one can seethat polytextuality affects frequency proportionally, with an additive constantcoefﬁcient of usage requirement L-Appl (“communicative relevancy” of thegiven meaning); at the same time it affects the frequency by means of an exponential operator with a constant C, which is assumed to represent the meansign complexity (or length). The cycle is a new variant in Köhler’s theory,
allowing a non-logarithmic, “plain” effect of one variable on another, using
the exponential operator.
Thus, the formula becomesln(F) = R ln(Appl) + B ln(PT ) −C exp(ln(PT )),i.e.
ln(F) = R ln(Appl) + B ln(PT ) −C(PT )from which it follows that
F = Appl R PT B e−C(PT ) .

On the relation between types and tokens of Japanese morae

139L-ApplRL-Polytextuality

L-FrequencyBexp–C

Figure 1: The relationship between polytextuality and frequency in general

Since ApplR can, in the framework of a synchronic study, be considered aconstant, for example, A, while PT = x, and F = y, we can obtainy = Axb e−cx
which is identical to the above solution of the differential equation. Iterative
computing of this curve yieldedy = 26.57366832x1.31502554 exp(−0.0000125937521x) .

(3)

The determination coefﬁcient is R2 = 0.93, indicating that relationship (2)can be preliminarily accepted. The observed and the computed values areshown in Table 1 . The computation has been performed in logarithmic transformation, thus R2 holds for the logarithmic values in the given interval.
Though in many cases the deviation is considerable, the variability of token
frequencies is so enormous that it renders R2 very high and the F-test verysigniﬁcant.

2Conclusion

The problem of type and token frequencies can in this case be reduced tothat of cotextuality and frequency, a relationship that has been very lightly
studied in quantitative linguistics. In any case, the result conﬁrms Köhler’s140

Katsuo Tamaoka
4.4e+007

3.52e+007

2.64e+007

1.76e+007

8.8e+006

00

18000

36000

54000

72000

90000

Figure 2: Relation between types and tokens of Japanese morae

assumption. In the majority of linguistic cases, frequency is considered as anindependent variable but in the case of Japanese morae it is exactly the otherway round. This can be explained by the fact that frequency, as most of theother linguistic quantities in the synergetic control cycle, has direct as wellas indirect effects on other variables and is, at the same time, a dependent
variable, e.g. depending on polytextuality.

References
Amano, Shigeaki; Kondo, Tagahisa
2000
Nihongo-no goi tokusei. Dai-7-kan. [Lexical properties of Japanese.
Volume 7.] Tokyo: Sanseido.
2003

Nihongo-no goi tokusei. Dai-2-ki CD-ROM-ban. [Lexical propertiesof Japanese. The second volume of CD-ROM version.] Tokyo: Sanseido.

Ferrer i Cancho, Ramon; Solé, Richard V.; Köhler, Reinhard
2004“Patterns in syntactic dependency networks.” In: Physical Review, E69, 051915.
Köhler, Reinhard
2002Korpuslinguistische Untersuchungen zur quantitativen und systemtheoretischen Linguistik. [http://ubt.opus.hbz-nrw.de/volltexte/2004/]

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2005

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“Synergetic linguistics”. In: Köhler, Reinhard; Altmann, Gabriel; Piotrowski, Rajmund G. (Eds.), Quantitative Linguistics. An International
Handbook. Berlin: de Gruyter, 760–774.
Tamaoka, Katsuo; Makioka, Shogo
2004“Frequency of occurrence for units of phonemes, morae, and syllables
appearing in a lexical corpus of a Japanese newspaper.” In: Behavior
Research Methods, Instruments & Computers, 36(3); 531–547.
Wimmer, Gejza; Altmann, Gabriel
2005“Uniﬁed derivation of some linguistic laws”. In: Köhler, Reinhard;
Altmann, Gabriel; Piotrowski, Rajmund G. (Eds.), Quantitative Linguistics. An International Handbook. Berlin: de Gruyter, 791–807.

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Katsuo Tamaoka

Table 1: Types (cotextuality) and tokens (frequency) of Japanese morae in a corpus#Types#Tokens(3)#Types#Tokens(3)

945
69
69
82
83
113
117
147
170
175
183
201
207
400
504
551
700
738
1122
1165
1166
12161343
1814
1898
2034
2084
2183
2442
2633
2937
3024
3125
3208
3480
3623
3769
4139
4929
5175
5317
5464
5569
5736
5965
6249
6260
6422
7088
8336
8670

95
511
3304
3087
125380
16016
14646
15343
29907
53284
10406
50769
15494
59424
33782
162751
166750577939446886214108182551
188263
199251
19284792920699590041135693704450756115412241254559984460164645493677921381384455571482387
1591698
272185320419282858438
11049984484240
1177824
138499377448222009223
2771906
232894525494994064902
1644763477.813964.816953.576953.57
8724.038864.0913294.87
13916.47
18781.27
22731.02
23612.77
25039.87
28321.35
29436.0869829.51
94506.13
106199.38
145211.16
155590.11
268616.69
282083.06
282397.6
298240.76
339317.3
500865.36
531023.82
580621.35
599085.76
635993.97
734627.19
809153.65
930614.9
965975.25
1007342.14
1041580.88
11552741215911.6
1278397.76
1439209.44
1792936.88
1905609.9
1971142.4
2039337.01
2088262.07
2166436.68
2274317.01
2409134.24
2414376.53
2491787.48
2813332.09
3427819.93
3594413.22

8688
8694
8857
8951
9007
9174
9263
9566
10011
10038
10491
11507
11707
12158
12180
12822
12919
13622
13922
13925
14103
14413
14435
14873
15529
16836
17622
17737
18207
18898
19482
20442
21459
22195
22624
22763
22853
23598
23656
25431
27131
27384
27551
28364
31183
38359
40585
43982
67643
81463
84624

374088125773864303939
121058622064874118162275899457778892749654
14757618
16200663554252306768153068176791877
11037767
12594714
34322783777201
29275524767153
19326674
78589316823149
655023181275909953427
11996376
10498607
6149909580819969471073972466
14053377
79320869271734
9612206
3607169
28899891
6116529
17102180
16442465
12154255
6289027
992621217211261
16518016
20530065
43985426
35719268
39052254

3603412.66
3606409.18
3688009.24
3735140.99
3763249.39
3847166.66
3891956.70
4044762.26
4269991.40
4283689.66
4513869.51
5032502.51
5134888.29
5366028.81
5377309.08
5706765.14
5756559.53
6117584.83
6271623.81
6273169.18
6364539.70
6523625.18
6534915.86
6759522.22
7095418.24
7762131.57
8160955.32
8219151.77
8456526.23
8804141.22
9096493.22
9574022.17
10075369.91
10435061.19
10643439.60
10710752.90
10754271.59
11112889.60
11140687.70
11981880.94
12769738.39
12885432.49
12961581.65
13329688.48
14571893.89
17480465.76
18306022.84
19494944.36
25488953.21
27348656.72
27630516.29