CHAPTER 5

4- AND 6-CYCLE SYSTEMS

5.1 THE EXISTENCE AND NATURE OF 4-CYCLE SYSTEMS

Among the world's languages the occurrence of 4-cycle numeral systems is comparatively rare. Schmidt says "the quaternary system forms the numeral above 'four' by composition: 5 = 4+1, 7 = 4+3, 8 = 4+4 (or 2x4), 9 = (2x4)+1, 16 = 4x4. In this consequent type, however, it is but seldom met, for example in California with the Salina and traces of it with the Chumash."[1] Eells, in discussing the Amerindian numeral systems indicates that "fairly well defined quaternary systems reaching to eight may be found among the Montaignais of the far north, the Foxes of Wisconsin, the Iowas and Missouris of the Plains - but they find their best and fullest development into true systems in various California tribes."[2] Dixon and Kroeber, in their article on Californian numeral systems, confirm this: "Counting by fours is a striking feature of Californian languages."[3] Beeler also documents the existence of 4-cycle systems among the Ventureño of California.[4] Generally speaking, the 4-cycle system has not been documented in any other region except Africa where only one or two instances of numerals being compounded such that 8 = 2x4, 9 = (2x4)+1, and 16 = (2x4)x2 have been recorded. Zaslavsky cites "the Haku language of Central Africa"[5] as having a numeral system with these features although the system is not purely 4-cycle having, for instance, 7 = 6+1 and a distinct word for 10. Beeler[6] remarks that there is a vestigial trace of a 4-cycle in Proto Indo-European in that the numeral 8 is the dual form of 4; in the Indo-European languages as a whole, however, there is no sign now of 4-cycle features.

Eells, in discussing how 4-cycle systems might arise, says that "perhaps a few tribes, for reasons best known to themselves, did not use their thumbs in counting."[7] He also speculates, as does Schmidt,[8] that the number four held special ritual signifance for many Indian tribes. Dixon and Kroeber, however, dismiss this latter view: "it is probably not connected to any extent with ritualism ... the Californian Indians are distinctly unritualistic."[9] They also indicate that the Californian 4-cycle systems often have terms which refer to fingers, implying that they are a type of digit tally and they add that "it does not follow that because people count by their fingers they count by fives."[10] This view is supported by the data reported below.

5.2 4-CYCLE SYSTEMS AMONG THE NAN LANGUAGES

Among the NAN languages of the New Guinea region, 4-cycle systems are rare: only about 10 or 11 languages possess numeral systems which exhibit, in whole or in part, features which are clearly 4-cycle in nature. These languages are located mainly in two geographical areas: the northern coast of New Guinea and the central highlands of PNG (see Map 15). Two languages, Vanimo and Rawo, both of which belong to the Sko Phylum, and which are spoken in the West Sepik Province (PNG), had their numeral systems recorded by Friederici who published his data in 1913.[11] According to this report, both systems were 4-cycle. The data are given below in Table 50 .[12]

Table 50

Numerals of Vanimo and Rawo as Recorded by Friederici in 1913.

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Vanimo Rawo

______________________________________________________

1 opa opa

2 yumono yemeno, yumunu

3 enu eno, ino, inomumu

4 no no, nu-o

5 no meneau no meu

6 no meneyu no meyu

7 no menehendu no meno

8 nuyu noyo

9 nuyu meneau no yo meu

10 nuyu meneyu, moti no you meyu

_____________________________________________________________________

In both of these systems we have three monomorphemic numerals 1 to 3 while 4 is a "hand" morpheme no- or nu- (which normally has possessive suffixes). The numbers are tallied on the fingers of one hand excluding the thumb. With Rawo, for example, we have:

5 no me u , that is 4 + 1 (u is a suppletive form of 1)

6 no me yu, that is 4 + 2

7 no ne 'no, that is 4 + 3

8 no yo, that is 4x2

9 no yo me u, that is (4x2) + 1

10 no yo me yu, that is (4x2) + 2

The Vanimo data indicate an alternative for 10, moti, a distinct term, and data recently collected by the author indicate that the Vanimo system no longer uses the (4x2) + 2 construction for 10 and has instead a secondary 10-cycle.

Further to the east, in the East Sepik Province (PNG), the Island dialect of Boiken, spoken in the Tarawai and Walis Islands, also has a 4-cycle system.[13] The remaining six dialects of Boiken, however, all of which are located on the mainland, have 5-cycle systems. Adjacent to the Boiken region and located on the mainland coastal region are the Mountain Arapesh who have an unusual numeral system first described by Fortune in 1942. This has two monomorphemic numerals 1 and 2; 3 is 2+1 and the system thus possesses a "quasi" 2-cycle. There is a distinct term for 4, nybat, which, interestingly, means "dog". Fortune says that "counting from four to twenty-four proceeds on two roots for numerals ... as follows: four one, four two, four three, biogu nybat (two fours or eight), eight one, eight two, eight three, biogu atut (three fours or twelve [the nybat being understood]), twelve one, twelve two, twelve three, bigi biogu (two eights or sixteen), and so on to biogu atuga biogu or twenty and from there similarly to anauwip or twenty-four."[14]

To the west of the Vanimo-Rawo region, in Irian Jaya, Nafri is spoken.[15] This is a member of the Sentani Family and its numeral system shows both 4- and 5-cycle features. It has four monomorphemic numerals 1 to 4 while 5 contains a "hand" morpheme me; 6 has a 5+1 construction. The compounds for 8, 9, and 10 each show a 4-cycle construction and are, respectively, 4x2, (4x2)+1, and (4x2)+2. None of the other members of the Sentani Family exhibit traces of a 4-cycle in their numeral systems. There are, however, two other languages adjacent to Nafri which possess numeral systems showing 4-cycle features; these are Ormu and Yotafa, both of which are Oceanic AN and which are discussed in the next section.

The only other coastal or island NAN language which shows some trace of a 4-cycle in its numeral system is Yele, spoken on Rossel Island in the Milne Bay Province (PNG). While the numerals 1 to 10, and the decades and hundreds, show no trace of a 4-cycle, we find, however, that the thousands, used for counting shell money, do. These are given below in Table 51.[16]

In the sequence given in Table 51 there is an "x+n" construction for the numerals 5,000 to 8,000 where x is mwa- and n takes, respectively, the values 1,000 to 4,000. Normally, for a 4-cycle system, 8,000 would have a multiplicative construction (4,000x2) rather than an additive one. The 4-cycle construction does not continue after 8,000 and 10,000 has the surprising construction (9,000)(8,000). Armstrong, who studied the Rossel Islanders in the 1920s, says that this is explained in a legend about a mythological being, Wonajo, who invented counting and "having counted up to 9,000 grew weary, and, unable to think of a fresh word for 10,000 adopted the novel, if unmathematical, device of using in juxtaposition the words for the last two thousands."[17]

Table 51

The Yele Numerals 1000 to 10000

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Yele

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1000 yili

2000 dwong

3000 teme

4000 dab

5000 mwa yili

6000 mwa dwong

7000 mwa teme

8000 mwa dab

9000 mwa di

10000 mwa di mwa dab

_____________________________________________________________________

The remaining 4-cycle systems are all located in the PNG highlands and several of these are remarkable and unique. Two standard 4-cycle systems, however, are exhibited by Wiru and the various dialects of Kewa, both languages spoken in the Southern Highlands Province (PNG). The numerals are given in Table 52 below.

Table 52

4-Cycle Systems of Wiru and Kewa

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Wiru Kewa

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1 ondene pameda

2 takura laapo

3 tebolo repo

4 tuyono, lu-u ki

5 lu ke ondene kode, kina kode

6 lu ke takura kode laapo

7 lu ke tebolo kode repo

8 lu-u takura ki laapo

_____________________________________________________________________

Both systems have the numeral 4 containing a "hand" morpheme, yono in Wiru and ki in Kewa. The use of the "hand" morpheme, for both systems, indicates that the fingers but not the thumb are tallied. The nature of the Kewa system becomes apparent when we consider the semantics of the numbers 5 to 12, the data deriving from an article by the SIL linguists Karl and Joice Franklin and given in Table 53.

Table 53

The Semantics of the 4-Cycle System of (East) Kewa

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5 kode the thumb

6 kode laapo two thumbs, i.e. one hand and two thumbs

7 kode repo three thumbs, i.e. one hand and three thumbs

8 ki laapo two hands

9 ki laapo na kode two hands, one thumb

10 ki laapo kode laapo two hands, two thumbs

11 ki laapo na kode repo two hands, three thumbs

12 ki repo three hands

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The Wiru and Kewa 4-cycle systems are not the only means of enumeration for these language groups: both possess body-part tally systems, that for Kewa having a 47-cycle.

Three unusual 4-cycle systems which employ a similar principle of enumeration are exhibited in the languages Enga (Mai dialect), Hagen (Kaugel dialect), and Mendi (Angal Heneng dialect) which are spoken, respectively, in the Enga, Western Highlands, and Southern Highlands Provinces (PNG). Typically, these systems are such that counting proceeds initially from 1 to 8. Beginning at 9, there comes into play a means of proceeding by fours which employs what I have termed "cycle units". In counting a particular sequence of four, for example 13 to 16, the cycle unit for that sequence is a word or phrase, perhaps a mnemonic, which is employed to form the compounds for 13 to 16 in the following way:

13 "cycle unit" + 1

14 "cycle unit" + 2

15 "cycle unit" + 3

16 "cycle unit completed"

In enumerating the next sequence of four, a different cycle unit is used but the same constructions are retained. To illustrate this, Table 54 shows enumeration of the sequences 13 to 16 and 17 to 20 for each of the three languages.[18]

Table 54

Examples of Cycle Units in the 4-Cycle Systems of Enga, Angal Heneng, and Kaugel

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Enga Angal Heneng Kaugel

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1 mendai pombor telu

2 lapo kap talu

3 tepo tep yepoko

13 mapunya mendai moklaopun pombor malapungga telu

14 mapunya lapo moklaopun kap malapungga talu

15 mapunya tepo moklaopun tep malapungga yepoko

16 mapunya gato tu moklaopu malapu

17 yupunya mendai supun pombor supungga telu

18 yupunya lapo supun kap supungga talu

19 yupunya tepo supun tep supungga yepoko

20 yupunya gato tu supu supu

_____________________________________________________________________

For the Enga data shown, the cycle unit for the sequence 13 to 16 is mapun(ya) which means "sweet potato", while that for the sequence 17 to 20 is yupun(ya) which has the gloss "ground, earth".[19] Bowers and Lepi indicate, for the Kaugel system, and the same is true for the other two, that "unlike English or Pidgin numeration, when a set of four has been completed, the next three items are considered parts of the following named set rather than addition to the just-completed set, for example supungga talu '2 of 20' i.e., 18."[20] The enumeration in fours terminates at a different value for each system: Enga at 60, Kaugel at 32, and Angal Heneng at 48. In the case of Kaugel, 32 does not constitute a secondary cycle: the term for 24, tokapu, is used to form compounds for larger numbers so that 48 = 24 x 2, that is tokapu talu, and 72 = 24 x 3, that is tokapu yepoko.

This type of 4-cycle system is not documented in the literature of numeral systems for any other region in the world. Its uniqueness implies that the principle of the cycle unit as a means of enumeration is an independent and localized invention. It seems likely, however, that the 4-cycle nature of the systems has the same basis as other 4-cycle systems which exist in this region, that is treating the "hand" as four fingers. Bowers and Lepi, indeed, suggest that, for the Kaugel system, "the everyday system was probably derived from finger counting."[21] Both numerals 4 and 8, for Kaugel, contain a "hand" morpheme ki. Similarly, the number 4 in Enga, kitomende, also contains a "hand" morpheme ki, a feature shared by Wiru and Kewa, discussed earlier, as well as by the Melpa dialect of Hagen in which 8 contains a "hand" morpheme.

5.3 4-CYCLE SYSTEMS AMONG THE AN LANGUAGES

Only four AN languages possess numeral systems which exhibit 4-cycle features. Two of these, Wogeo and Bam, both of which belong to the Manam/Kairiru chain of the North New Guinea Cluster,[22] are spoken in the Schouten Islands in the East Sepik Province (PNG). The data for these systems are given in Table 55.[23]

Table 55

Two 4-Cycle Systems of Oceanic AN Languages

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Wogeo Bam

_____________________________________________________

1 ta tini

2 ru ru

3 tol tuol

4 kwik kiki

5 kwik bo koba kikik be kubua

6 kwik ba rago kiki be areg di ru

7 kwik be tol kiki be areg di tuol

8 kiki ru kiki ru

_____________________________________________________________________

The Bam system is regular and shows clearly the pattern: 6 = 4 + 2, 7 = 4 + 3, and 8 = 4 x2. While the numerals 1 to 3 are recognisable as reflexes of the POC numerals, that for 4 is not. It is not known whether, as is the case for the NAN 4-cycle systems, counting is done on the fingers but not the thumb of the hand. The word for "hand" in Bam is lima, identical to the POC numeral, which does not appear in the number 4; if it did this might lend some semantic weight to the idea that counting is done on the fingers only.

Two other Oceanic AN languages have numeral systems which possess 4-cycle properties. These are Ormu and Yotafa, both located on the north coast of Irian Jaya adjacent to Nafri, a NAN language with a 4-cycle system discussed in the previous section. The data for these are given in Table 56.[24]

Table 56

4-Cycle Systems of Two Oceanic AN Languages in Irian Jaya

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Ormu Yotafa

___________________________________________________

1 nitji tei

2 rohi ros, roti

3 toru tor(u), ossor

4 awa au(a)

5 ore-nitje-ma meniam

6 man-rohi-ma mandosim, ma-roti-ma

7 samecho-don-rohi-ma tamecho-none-roti-ma

8 don-rohi-ma none-roti

9 nen-rohi-fraja-nitje-ma none-roti-fraja-tei

_____________________________________________________________________

Both of these systems have irregularities and display both 4- and 5-cycle features. In each case there is a distinct term for 5, and 7 has an "x+2" construction. However 6 contains a 2 morpheme and both 8 and 9 have constructions which suggest the operation of a 4-cycle. As both of these languages do not possess systems with a 10-cycle which we might expect because both are daughter languages of POC, it is not known whether the 4-cycle properties apparent in their numeral systems is a result of influence by neighbouring 4-cycle NAN systems, that is those of Nafri and of Vanimo and Rawo to the east.

5.4 SUMMARY OF 4-CYCLE SYSTEM DATA

There is a total of 10 NAN languages (11 if we include Yele) and 4 AN languages which have numeral systems displaying 4-cycle features.[25] Map 15 indicates the distribution of these which is restricted to two main regions: 1) the northern New Guinea coast and islands, and 2) the central highlands of PNG. No 4-cycle numeral systems are found in Island Melanesia, Polynesia, or Micronesia, although counting objects in groups of four is quite common in these regions. Only several of the systems display "pure" 4-cycle features as described by Schmidt, that is 5 = 4+1, 7 = 4+3, 8 = 4+4 or 2x4, and 9 = (2x4) + 1. As indicated above, Ormu and Yotafa display both 4- and 5-cycle features while Mountain Arapesh and the Melpa dialect of Hagen both have a primary 2'-cycle as well as a secondary 4-cycle. Yele exhibits a 4-cycle structure for the thousands only while the Enga, Kaugel, and Angal Heneng systems have their unique 4-cycle sequences beginning at 9.

The evidence is clear that some 4-cycle systems arise due to the practice of treating the four fingers, but not the thumb, as one "hand". This is certainly the case for the NAN languages Vanimo and Rawo and those of the central highlands. There is, however, no lexical evidence suggesting that this is also true for the four AN systems. While it would appear that counting on the fingers in this way may well be a localized innovation for the central highlands languages, it is not at all clear whether there is a similar common thread connecting the nine 4-cycle systems of the northern coast and islands, that is whether these are largely small groups of 4-cycle counters which have developed their systems independently or, as seems more likely, whether the 4-cycle system was developed in one location and was then diffused, possibly by trading, along the northern seaboard.

5.5 THE EXISTENCE AND NATURE OF 6-CYCLE SYSTEMS

6-cycle, or "senary", systems are comparatively rare and the literature indicates their existence in only two major regions: Africa and North America, as indeed was found in the case of the 4-cycle systems. Schmidt notes that "it has a rather limited dispersion in north-west Africa, e.g., in the Huka, the Bulanda, the Apko."[26] In her 1915 article on African numeration, Schmidl indicates several languages which use the principle of "composition with 6" and her map of the distribution of various systems shows some seven locations where this occurs.[27] Zaslavsky, quoting Schmidl's material, gives several instances of systems with 6-cycle properties in which, for example, 12 = 6x2 and 24 = 6x4. The Balante (Schmidt's "Bulanda") have numerals 7 to 12 compounded as 7 = 6+1, 8 = 6+2, and so on.[28]

In North America, 6-cycle systems appear to be limited to a few examples in the languages of California. Beeler published in 1961 an account of the Wintun, Nomlaki, Patwin and Maidu systems, each of these being members of California Penutian.[29] None of these systems is "pure" 6-cycle in nature, that is in a "pure" system we would expect 7 to 11 expressed as "6+n" where n takes the values 1 to 5 respectively, and 12 = 6x2. They do, however, exhibit some 6-cycle features in that 7 = 6+1, 8 = 6+2, and 9 = 6+3. Gamble, similarly, in his analysis of Chunut, also located in California, indicates vestigial traces of a 6-cycle numeral system.[30]

5.6 6-CYCLE SYSTEMS IN THE NAN LANGUAGES

I have no evidence of the existence of 6-cycle numeral systems among the AN languages of New Guinea and Oceania. There is, however, evidence that such systems exist in a small group of NAN languages which are located in two regions on the south coast of New Guinea (see Map 15). Three languages, Kimaghama, Riantana, and Ndom, are all spoken on Kolopom Island adjacent to the south coast of Irian Jaya, immediately to the west of the Irian Jaya/PNG border. The numeral systems of each of these languages possesses a primary 6-cycle; Kimaghama has a secondary 20-cycle while Ndom has a (6, 18, 36) cyclic pattern. The data for these systems are given in Table 57.[31]

Table 57

Three Examples of 6-Cycle Numeral Systems of the NAN Languages

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Kimaghama Ndom Riantana

__________________________________________________________

1 növere, nubella sas mebö

2 kave thef enava

3 pendji ithin pendö

4 jando thonith wendö

5 mado meregh mata

6 turo, ibolo-nubella mer törwa

7 iburo-növere mer abo sas mebö-me

8 iburo-kave mer abo thef enava-me

9 iburo-pendji mer abo ithin pendö-me

10 iburo-jando mer abo thonith wendö-me

11 iburo-mado mer abo meregh mata-me

12 - mer an thef törwa-me

_____________________________________________________________________

Although these three languages are related, each belonging to the Kolopom Sub-Phylum-Level Family, the numerals of each system differ markedly from those of the others. Nevertheless, each exhibits a 6-cycle structure, that of Ndom being the most regular. In addition to the data shown, Ndom has distinct terms for 18 and 36; 72 and 108 are compounds of the latter, that is 72 = 36x2 and 108 = 36x3.

To the east of Kolopom Island, on the Irian Jaya/PNG border, two other languages are located, both of which possess 6-cycle systems. These are Tonda and Kanum which belong to the same Sub-Family of the Trans-Fly Stock. Although the data for Tonda are incomplete, so that it is uncertain how the numerals 7 to 12 are constructed, nevertheless we do have 12 = 2x6 and 18 = 3x6. The data for Kanum are more complete: there are distinct terms for 1 to 6 and the numerals 7 to 12 have an "x+n" construction where n takes, respectively, the values from 1 to 6. The alternative construction for 12 is 2x6, 18 is 3x6, and 24 is 4x6. There is a distinct term for 36, as we found with Ndom, and this forms a secondary cycle with 72 = 2x36. These various systems discussed here form the most fully displayed 6-cycle systems to be found anywhere in the literature. The data for Kanum and Tonda are given in Table 58.[32]

Table 58

The 6-Cycle Systems of Kanum and Tonda

_____________________________________________________________________

Kanum Tonda

__________________________________________________________

1 namper nabi

2 jempoka yalmbe

3 juau yala

4 eser hasar

5 tampui tambui

6 tarawo trawa

7 pesmeri-emper -

8 pesmeri-jalmpö -

9 pesmeri-jela -

10 pesmeri-eser -

11 pesmeri-tampui -

12 pesmeri-tarawo yalmbe trawa

18 juau-tarawo yala trawa

36 nimpe -

72 jempoka-nimpe -

_____________________________________________________________________

It is interesting to note that while the existence of 6-cycle systems has been reported by the various authors cited earlier, none of these has speculated on why such systems should exist and whether there is some anatomical basis for them as we have found for the 5-, 10-, and 4-cycle systems. From the five languages discussed above we have no semantic evidence which might suggest an answer. There is, however, a special counting system employed by the Mountain Arapesh which offers a clue to the origin of 6-cycle systems. The Mountain Arapesh possess a 4-cycle system which was discussed in the previous section; they also possess another unusual system which has a (3,6) cyclic pattern. The procedure used in this type of counting has been described by Fortune; after counting to 3, the count proceeds: "three one, three two, anauwip or six. Then repeat to a second anauwip or twelve. This is supposed to be the count on the hands, five fingers and a thumb joint as well, to make six for each hand."[33] Thus, for this particular system, the 6-cycle does arise from finger tallying. We have found, therefore, that tallying on the hand gives rise not only to 5-cycle variants, but also to 4-cycle systems in which the thumb is excluded from the count, and to 6-cycle systems in which the thumb joint augments the four fingers and the thumb.

5.7 SUMMARY OF 6-CYCLE SYSTEM DATA AND GENERAL COMMENTARY

The situation regarding 6-cycle systems in New Guinea is similar to that pertaining in Africa and North America, that is that such systems are rare and are only found in a few locations. There are five NAN languages which possess systems with a primary 6-cycle; three of these are located on Kolopom Island in southern Irian Jaya and two are located near the south end of the Irian Jaya/PNG border. Apart from these two neighbouring regions, 6-cycle systems do not exist anywhere else in New Guinea or Oceania, and in particular are not found in any of the AN languages. As indicated above, one other NAN language, Mountain Arapesh, has a special counting system which has a primary 3-cycle and which terminates on a count of 6 (and thus does not actually have a secondary 6-cycle). The notable feature of the 6-cycle systems located in southern New Guinea is that they are the most fully formed 6-cycle systems found anywhere in the literature.

In concluding this four chapter survey of the various types of numeral systems and tally methods which exist in New Guinea and Oceania, it is clear that these possess a considerable, but not unlimited, diversity. Evidence has been presented to indicate the existence of numeral systems which have primary cycles of 2, 4, 5, 6, and 10. The special Mountain Arapesh system has a primary 3-cycle and there are two other systems, those of Bumbita Arapesh and Wasembo, which appear to have features of a secondary 3-cycle.[34] The secondary cycles of the numeral systems range through 3, 4, 5, 10, 18, 20, 36, 60, to 100. The system of the Melpa dialect of Hagen, which has a "hand" morpheme for 4, also has a distinct word for 8 and a consequent 8-cycle. Despite this diversity of cyclic structures, it is important to note that there is no evidence that any language in the region under consideration possesses a numeral system with either a primary or secondary cycle of 7 or 9, in the first decade, or of 11, 12, 13, 14, 15, 16, 17 and 19 in the second decade.

Tallying on the fingers of the hand (and, by extension, the toes) gives rise to the common "digit-tally" system with a primary 5-cycle and a secondary 20-cycle. However, we have also found that digit-tallying can give rise to both 4- and 6-cycle systems depending, respectively, on whether the thumb is omitted from the tally or whether the thumb joint augments it. There is no evidence, however, that "pure" 2-cycle systems have their genesis in digit-tallying. Tallying on body parts other than digits appears to be unique to New Guinea and Australia. Of the 40 body-part tallies documented for New Guinea, the least complete cycle is 18, the greatest complete cycle is 68, and the modal cycle is 27. We therefore have a considerable range of unusual cyclic structures, each of which has its genesis in parcelling out the body for tallying. There is evidence that such tallies are, or were, used for calendrical purposes, in particular for establishing when a certain feast should occur during a cycle of ceremonial feasts. Language groups which possess body-part tallies almost invariably possess, in addition, a relatively simple numeral system, that is either a 2-cycle variant or, as we find in the Southern Highlands Province (PNG), a 4-cycle system. The body-part tallies functioned as ordinal devices, using the name of a body part to indicate the position of a point in an ordered sequence of points; the names of the body parts were not generally used (in noun phrases, say) for stating the number of objects in a set and for this the numeral system was used and which thus had an independent purpose and existence.