CHAPTER 2

TALLIES AND 2-CYCLE SYSTEMS

2.1  THE EXISTENCE AND NATURE OF 2-CYCLE SYSTEMS

Schmidt, in an article on "Numeral Systems" in the 1929 edition of the Encyclopaedia Britannica, says "there is no language without some numerals; the notion of unity and plurality is expressed at least in the formation of 'one' and 'two', though 'two' is often equal to 'much', thus concluding a numeration that has only just started."[1]  The type of numeration in which we have two, or perhaps three or four, numerals which are not compounded to form larger numbers and of which the largest is the limit of counting, Schmidt terms "systemless" and notes that it is doubtful that it really exists "as it is mostly reported of peoples that are vaguely known."[2]  In this study, I shall adopt the view, consistent with Schmidt's, that the basic criterion that must be met in order for a set of distinct numerals to qualify as a "counting system" is that there must exist some method for forming composite number words which extend the limit of counting beyond the numeral of greatest magnitude.  The simplest such system is that for which we have the numerals "one" and "two" and all larger numerals are compounds of these so that, for example, 3 is a compound of "two" and "one", 4 is a compound of "two" and "two", and 5 is a compound of "two", "two", and "one", and so on.  This type of system is known variously in the literature as the "pure 2-system",[3] the "pair system",[4] and the "pure 2-count".[5]  In Salzmann's terminology, the system has a frame pattern (1, 2), a cyclic pattern (2), and an operative pattern which may be summarized as (3 = 2 +1, 4 = 2 + 2, 5 = 2 + 2+ 1, ...).  As we shall see subsequently, there are other systems which have a primary cycle of 2 but which have a secondary cycle as well.  However in our search for the existence of "pure" 2-cycle systems we will require that the system has at least: 3 = 2 + 1, 4 = 2 + 2, and 5 = 2 + 2 + 1.  This requirement does not of course preclude the possibility of a secondary cycle of magnitude greater than 5, e.g. 10, and we should bear this in mind if the data are insufficient to determine whether or not this is the case. 

Seidenberg[6] lists four geographical locations where "pure 2-counting" (defined as I have given above) is found: South America, Australia, South Africa and New Guinea.  For South America, Schmidt notes that "it is found among the ethnologically oldest tribes - the Fuegian tribes",[7] which are located near Tierra del Fuego at the southernmost tip of the continent.  Conant, in his book The Number Concept ,[8]  lists seven examples of 2-cycle systems ("binary scales") from South America, different from those given by Schmidt.  Similarly, Seidenberg[9] provides nine examples from South America, all of which derive from Kluge,[10]  and three of which are identical with those given by Conant.  Between the various sources cited, a total of about 18 language groups are listed as possessing 2-cycle systems.  By contrast, for the entire African continent, the same sources indicate that only the counting system of the Bushmen, located in southern Africa, possesses a 2-cycle system.  Schmidl[11] gives an example of a system from one of the Bushmen dialects: this appears to have three numerals 1, 2, 3, the larger numerals being compounds of the first two so that 4 = 2 + 2, 5 = 2 + 2 + 1, 6 = 2 + 2 + 2, and so on until 10 is reached.

In Australia, there is evidence of widespread use of the 2-cycle system.  Schmidt says that it is found "among the tribes ethnologically the oldest - the Kurlin-Kurnai of the south-east, the Narrinyeri of the south; several of them count up to 'ten' in this manner".[12]  Conant gives 35 examples of "Australian and Tasmanian number systems", each of which shows evidence of a 2-cycle.[13]  Only six of the examples, however, give data beyond the numeral 4 and thus it is impossible to establish whether the remaining 29 systems given have secondary 5-cycles or whether they are "pure" 2-cycle systems.  Kluge[14] has an extensive collection of numeral lists of Australian languages taken mainly from sources published in the nineteenth century.  Of the several hundred lists only a few give data beyond the first four numerals.  About 25 numeral lists clearly indicate examples of 2-cycle systems with 3 = 2 + 1, 4 = 2 +2, and 5 = 2 + 2 + 1.  Those lists which give only the first four numerals include about 150 which clearly indicate the existence of a 2-cycle in which we have either 3 = 2 +1 or 4 = 2 + 2, or both.  It is quite clear from the data presented in Kluge's manuscript that other types of counting systems exist in Australia; some of these will be discussed in the next chapter.  It would nevertheless appear to be the case that variants of the 2-cycle system predominate in Australian language groups and that these take one of the three forms: (a) (1, 2, 2+1, 2+2), or (b) (1, 2, 3, 2+2), or (c)  (1, 2, 2+1, 4).  Only a small proportion of these types combine the numerals 2 and 1 to form numerals larger than 5.  Both Dawson[15] and Kluge[16] give examples of type (b) where a secondary 5-cycle operates (so that 6 = 5 + 1, 7 = 5 + 2, etc.), and Kluge also has many examples where type (a) also has a secondary 5-cycle.  There are, then, language groups that do possess a "pure" 2-cycle system; there is a larger number of groups, however, that possess variants as given above, and these include systems with a (2, 5) cyclic pattern.

2.2   2-CYCLE SYSTEMS IN NEW GUINEA

Various 2-cycle systems are found in mainland New Guinea, the Torres Strait islands, and the island of New Britain to the east of the mainland.  Nowhere are they found in the rest of Island Melanesia, Polynesia or Micronesia.  As was found in the case of the Australian languages, it is possible to distinguish several variants of the 2-cycle system. Similar variants are also possessed by New Guinea language groups and each of these will be discussed separately below.  Altogether I distinguish five types:

(a)  the "pure" 2-cycle system which has two basic numerals, 1 and 2, all other larger numerals being compounded from these two so that no secondary cycle exists,

(b)  the (2, 5) or (2, 5, 20) type which is such that the first four numerals follow the 2-cycle pattern: 1, 2, 2+1, 2+2, but subsequently a secondary 5-cycle comes into play so that, for example, 6 = 5+1, 7 = 5+2, and 8 = 5+2+1, and so on.  In some cases this sequence is continued until a tertiary 20-cycle comes into operation.

I have characterized two variants, (c) and (d), of the 2-cycle system as "modified" or "quasi" 2-cycle systems:

(c)  This type is such that we have at least three distinct numerals 1, 2, and 4; the numeral 3 is compounded as (2+1).  This system usually has a secondary 5-cycle and occasionally a tertiary 20-cycle.  To distinguish this variant the notation 2'-cycle is used and the cyclic patterns of the system are expressed as (2', 5) or (2', 5, 20).

(d)  The other "quasi" 2-cycle system is such that we have at least three distinct numerals 1, 2, and 3; the numeral 4 is compounded as (2+2).  This system, too, usually possesses a secondary 5-cycle and occasionally a tertiary 20-cycle as well.  The type (d) variant is distinguished by the notation 2"-cycle and the cyclic patterns of the system are expressed as (2", 5) or (2", 5, 20).

(e)  It was noted that the type (c) system, defined above, usually has a secondary 5-cycle.  There is a single variant of the 2'-cycle system, however, which instead possesses a secondary 4-cycle so that the system may be represented as: 1, 2, 2+1, 4, 4+1, 4+2, 4+2+1, and so on.  This variant, type (e), has a tertiary 8-cycle so that  the sequence continues: 8, 8+1, 8+2, 8+2+1, 8+4, 8+4+1, and so on.  The cyclic pattern of this type is expressed as (2', 4) or (2', 4, 8).

2.2.1  Type (a): The "Pure" 2-Cycle.

It is recalled that the total number of Papuan (NAN) languages in the counting systems data-base is 430; the total number of Austronesian (AN) languages in the data-base is 453, of which 226 are found in the Irian Jaya and Papua New Guinea region.  Of these totals, 42 (or less than ten percent) NAN language groups possess "pure" 2-cycle systems and 2 AN language groups also have them, that is less than one percent.  In none of these cases is there evidence of a secondary 5-cycle; also, in most cases, it is not possible to determine whether there is a larger secondary cycle, 10 for example, as the data are usually not sufficient.  Of the 42 NAN systems, 18 of these are associated with body-part tally systems which are discussed in Sections 2.3 and 2.4.

Several examples of this type of system, as it occurs on the New Guinea mainland, are given below.[17]

Table 3

"Pure" 2-Cycle Systems in Three NAN Languages of New Guinea

	Southern Kiwai	Giri	Gende
1	neis	ibabira	mapro
2	netewa	ppunini	oroi
3	netewa nao	ppuni kagine	oro gu mago
4	netewa netewa	ppunini ppuninin	oroi oroi
5	netewa netewa nao	ppunini ppunini kagine	oroi oroi mago

It may be observed that none of the systems shown is precisely "regular" in that "one" in combination is not identical to the numeral "one".  The pattern displayed here is quite common among those systems classified as Type (a).  In some cases the numeral 3 may be translated as "two and another"; in other cases copula may be used to link the compounding numerals so that  4 may be translated as "two and two" rather than just "two two".  Some systems have compound numerals which incorporate grammatical or syntactic elements.  Nomad, in the Western Province of PNG,[18] has:

                                               2       benau

                                               4       benau-ili benau-ili

                                               6       benau-ili benau-ili benau-ili

where "benau" is the numeral "two" and "-ili" is the dual form of the third person plural, that is "they-2", which appears only in compound numerals.

It is useful to comment at this point on the fact that two AN language groups are found to possess "pure" 2-cycle systems.  While it is not known whether specific counting system types were associated with the languages ancestral to the NAN languages now found in the New Guinea region, this is not the case for the Oceanic AN languages.  There is good evidence that the reconstructed Proto Oceanic (POC) numerals form a 10-cycle system and indeed the putative form of these numerals derives from the many 10-cycle systems belonging to contemporary daughter languages of POC.  However the data for the AN languages Sissano and Middle Watut  indicate that instead of there being at least ten distinct numerals we have in fact only two.[19]

Table 4

Examples of Two AN Languages Which Have Numeral Systems With "Pure" 2-Cycles

	Sissano	Middle Watut
1	puntanen	morots
2	eltin	serok
3	eltin puntanen	serok a morots
4	eltin eltin	serok a serok
5	eltin eltin puntanen	serok a serok a morots

It is apparent that, contrary to expectation, language groups which have a counting system with a given primary cycle may be influenced to change their system in the direction of a reduction  in the magnitude of the primary cycle, in this case from 10 to 2.  As we shall see, these are not two isolated cases and that throughout the New Guinea region a large number of AN languages have counting systems which have a primary cycle of less than 10.

2.2.2  Type (b):  With a (2, 5) or (2, 5, 20) Cyclic Pattern.

This is the most common type of counting system found among the NAN language groups.  Altogether 109 of these languages have systems which possess a primary 2-cycle and which have a secondary 5-cycle; for those languages for which sufficient data exist there is almost always a tertiary 20-cycle in operation as well.  Included in this group are variants, similar to those mentioned in the previous section.  One of these is such that the numeral 3, normally expressed as a simple combination of the numerals "two" and "one" (i.e  2+1 and never, incidentally, 1+2), is expressed as "two and another" and in which the numeral "one" does not appear explicitly.  Another variant which occurs in only one language group, the Fore  people of the Eastern Highlands of PNG, and this has 4 = 2+2 but 3 can be interpreted as having the construction "1+1+1".  Several instances of the Type (b) counting system, as found in three NAN languages, are given below.[20]

Table 5

Systems Which Have (2, 5, 20) Cyclic Patterns: Examples Taken From Three NAN Languages

	Sulka	Tairora	Fore
1	a tiang	vohaiqa	ka
2	a lomin (lo)	taaraqanta	tara
3	kor lo tige	taaraqanta vohaiqa	kakaga
4	kor lo lo	taaraqanta taaraqanta	tara-wa tara-wa-ki
5	a ktiek	kauqu-ru	naya:ka-mu
10	a lo ktiek	kauqa-tanta	naya:tara-mu
20	a mhelum	vohaiqa vaiinta	ka:kina

Each of these systems has a secondary 5-cycle and a tertiary 20-cycle.  This type of system is, in fact, a 2-cycle system augmented by a digit-tally.  The "number words" for 5 and 10 each contain "hand" morphemes and those for 20 contain a "man" (or "being") morpheme.  This type of system often contains grammatical or syntactic elements when compound numerals are formed; the compounds, too, may contain words which are tally prescriptions or directions.  The Asaro dialect of Gahuku-Asaro [21] illustrates both of these aspects:

Table 6

Translation of Tallying Terms of the Asaro Dialect of the Gahuku-Asaro Language

	Gahuku-Asaro	Translation
1	hamo've	one it is
2	sita've	two it is
3	sito-hamo've	two-one it is
4	sita've sita've	two it is two it is
5	ade hela osu'live	hand-our this at finished it is
6	ade hela osu'livo hamo've	hand-our this at finished being one it is
10	ade hela hela osu'live	hand-our this at this at finished it is
20	evene'hamo'gizene ana osu'live	person one foot-his hand-his finished it is

There are 18 AN language groups that possess this type of system and 13 of these belong to the Markham Family,[22]  all of which are located in the Markham Valley of the Morobe Province of PNG.[23]  This provides further evidence of AN languages having had a change in the cyclic nature of their counting systems from, originally, a (putative) 10-cycle to systems with (2, 5) or (2, 5, 20) cyclic patterns.  Several examples of these are given below.[24]

Table 7

Three Examples of AN Languages With Counting Systems Having (2, 5) Cyclic Patterns

	Tomoip	Wampar	Duwet
1	denan	orots	taginei, taine, ta
2	ro huru	serok	seik
3	horum detu	serok orots	seik mba ta
4	horumu horum	serok a serok	seik mba seik
5	ko liem	bangi-d ongan	lima-ngg alinan
10	liem	bangi-d serok	lima-ngg a seik
20	tamdil	ngaeng orots	lima-ngg a seik,
		a mbei-ngg seik

As was found in the case of the NAN systems, we have here a 2-cycle system augmented by a (5, 20) digit-tally system.  The "number words" for 5 and 10 in each of the examples above contain a "hand" morpheme.  For Tomoip  and Wampar, the word for 20 contains a "man" morpheme while that for Duwet  contains both a "hand" and a "foot" morpheme.

2.2.3  Type (c): With a (2', 5) or (2', 5, 20) Cyclic Pattern.

This type of system, which has a compound numeral "three" such that 3 = 2+1 but has a distinct numeral "four", is relatively uncommon.  No AN language group has this type of system while 27 NAN groups have it.  Six of those also have body-part tally systems as well [25] while a further 14 have systems with (2', 5, 20) cyclic patterns.  Members of the latter group are largely located in the Oro Province and the southern part of the Morobe Province.  It is possible that one language group, Biangai, may have a numeral 4 which has the meaning "one less than five" and of which we find several examples in the next type of system.  Several examples of the Type (c) system are given below.[26]

Table 8

Examples of the Type (C) 2-Cycle Numeral System

	Kol-Sui	Biangai	Au
1	'pusup	wame-nak	kiutip
2	te'tepe	na-yau	wiketeres
3	tetepe kosup	nayau keya nak	wikak
4	ke'a so	mango-bek-tau ono	tekyait
5	'a:meleng	mele-na-zik	his pinak
10	melem'be:ga	mele-yau	his wien

2.2.4  Type (d):  With a (2", 5) or (2", 5, 20 ) Cyclic Pattern

This system, with a distinct numeral "three" but with the numeral "four" compounded such that 4 = 2+2, is not uncommon.  Some 12 examples are found among the AN languages of PNG, five of these are members of the Buang Family,[27] located in the Morobe Province, and a further 4, perhaps 5, are located in the Milne Bay Province.[28]  Among the NAN languages there are 40 examples of this system; three of these may not have a secondary 5-cycle and are associated with body-part tallying methods.[29]   One AN and two NAN examples are given below:

Table 9

Examples of the Type (D) 2-Cycle Numeral System

	Mumeng (AN)	Kwanga (NAN)	Usarufa (NAN)
1	ti	findara	morama
2	yuu	frisi	kaayaqa
3	yon	lamor	kaomoma
4	yuu di yuu	frisi frisi	kaayaqte kaayaqteqa
5	vige vilu	tabanangki	mora tiyaapaqa

2.2.5  Type (e): With a (2', 4, 8) or (2', 4, 8, 10) Cyclic Pattern

There is only a single example of this type of system and this occurs in the Melpa dialect of the Hagen (NAN) language.  The structure of the system is:  1, 2, 2+1, 4, 4+1, 4+2, 4+2+1, 8, 8+1, (10), ...  There is an argument for including this system under the 4-cycle classification as the putative "2+1" construction for the numeral may not in fact be valid and there may be a distinct, uncompounded numeral 3.  There are two representations for the numeral 10: one has the meaning "hands one", that is "the hands of one man";  the other representation is as the compound "8+2".  The first ten number words are given below:

Table 10

Numerals of the Melpa Dialect of the Hagen Language

	Hagen (Melpa dialect)
1	tenta
2	ralg
3	raltika
4	timbakaka
5	timbakaka pamb ti
6	timbakaka pamb ralg
7	timbakakagul raltika
8	engaka
9	engaka pamb ti
10	engaka pamb ralg pip, or, ki tenta

2.3  SUMMARY OF 2-CYCLE DATA

If all the 2-cycle variants are combined we have a total of 218 NAN languages which have counting systems belonging to this category, just over half the total sample (430) of NAN languages.  The 218 languages are distributed mainly among the major phyla, the Sepik-Ramu, the Torricelli, and the Trans-New Guinea, as follows:

Table 11

The Distribution of 2-Cycle Variants Among Three of the Major Phyla of the NAN Languages

Thus more than half (58%) of the total sample of the Trans-New Guinea Phylum languages possess a 2-cycle system variant, as do 92% of the Torricelli Phylum sample and 36% of the Sepik-Ramu Phylum sample.

Altogether there are 32 AN languages which have a 2-cycle variant.  All of these are Oceanic and are found in PNG.  Twenty-four of these 32 languages belong to Ross's North New Guinea Cluster, 7 belong to the Papuan Tip Cluster, and 1 belongs to the Meso-Melanesian Cluster.  We should note that only two of the AN systems appear to be "pure" 2-cycle systems while the remaining 30 all have a secondary 5-cycle, and in many cases a tertiary 20-cycle as well.

Map 6 shows the distribution of the 2-cycle variants (a) to (d) in the New Guinea region for both the NAN and the AN languages.  The distribution is widespread throughout the region and is found in both the coastal and inland highland areas.

2.4  THE EXISTENCE OF BODY-PART TALLY METHODS

In his Native tribes of south-east Australia, published in 1904, Howitt reported two unusual methods of "enumeration of the days", of the stages of a journey, which tallied an ordered sequence of parts of the body.  One of these is as follows:

                               1        little finger

                               2        ring finger

                               3        middle finger

                               4        fore finger

                               5        thumb

                               6        hollow between radius and wrist

                               7        forearm

                               8        inside of elbow joint

                               9        upper arm

                             10        point of shoulder

                             11        side of neck

                             12        ear

                             13        point on the head above the ear

                             14        muscle above the temple

                             15        crown of the head

"from this place the count goes down the other side by corresponding places".[30]  If we take the crown of the head, being on the body's vertical axis of symmetry, as a unique mid-point, and with the 14 symmetrical counterparts of the ones given above, we have altogether 29 tally-points.  Howitt also notes, in passing, that "this method seems to do away with the oft-repeated statement that the Australian aborigines are unable to count beyond four or at most five."[31]  The second tally method had a further two symmetrical points at "the divergence of the radial muscles" giving a total of 31 tally-points altogether.  Howitt indicates that this system was probably widespread in southeastern Australia.

Some three years later, in 1907, Ray reported the existence of similar tally methods in the Torres Strait islands.[32] In both the western and eastern islands (the former are Saibai, Murulag, Boigu, Mabuiag, Badu, and Moa, and the latter are the Murray Islands, Erub, and Ugar), Ray indicates that 2-cycle counting systems are found.  He then describes a tally system used in the western islands with the following tally-points:

                               1        little finger

                               2        ring finger

                               3        middle finger

                               4        index finger

                               5        thumb

                               6        wrist

                               7        elbow joint

                               8        shoulder

                               9        nipple (left)

                             10        sternum

                             11        nipple (right)

and so on, continuing with the corresponding symmetrical counterparts in reverse order until the little finger is reached at the nineteenth tally-point.  For the eastern islands, Ray gives examples of two other tallies, one with 29 points and another with 25.

Ray also has data on a similar tally method used in the Papuan Gulf region of the New Guinea mainland by the Purari-speakers and which has a total of 23 tally-points.[33]  The Purari  also, according to Ray, have a 2-cycle numeral system.  Another group in the Gulf region, the Orokolo, are reported by Ray as having a tally method which utilizes 27 body parts.[34]  These reports, deriving from the turn of the century, of the coastal part of the mainland are supplemented by data collected subsequently in the interior highlands region.  In 1940, the Government Anthropologist for Papua, F. E. Williams, first reported the existence of similar body-part systems in use in the Southern Highlands Province of PNG by the Foe-speaking people of Lake Kutubu.  Williams indicates that their tally method utilizes 37 tally-points and says that it "is used with rather remarkable accuracy, especially as providing a timetable for feasts, etc."[35]  Further data on the body-part tallies of the Southern Highlands were forthcoming in 1962 in an article by two members of the SIL, the Franklins, who provided details of the "Kewa counting systems".[36]  The eastern and western dialects of Kewa both have 47-cycle body-part tally methods and these are used, as with the Foe, for calendrical purposes, and in particular for reckoning when a certain festival should occur within a cycle of festival dances lasting over many months.  The Franklins also note that the Kewa  have a 4-cycle counting system in addition to their tally methods and that it is this system which is used to express precise numerical quantities and not the tally method.[37]

The available literature, then, provides evidence of the existence of this most unusual tally method and that it is found in the widely separated geographical locations of southern Australia, the Torres Strait islands, and the highlands and southern coastal areas of PNG.  As far as I can ascertain, there is no record of this type of body-part tallying existing beyond the Australia-New Guinea region.  We will now summarize the extent to which it is found on the New Guinea mainland and the location of those groups which use it.

2.5  BODY-PART TALLY METHODS IN NEW GUINEA

This type of tally which uses the fingers and other tally-points which are situated on the upper part of the body are, on the New Guinea mainland, found only among certain NAN language groups and never among AN language groups.  It is thus unknown in Island Melanesia, Polynesia and Micronesia.  The data for New Guinea may be grouped into three categories:  (a) the case where the data are reasonably complete enough to determine the nature of the tally method, (b) the case where the data are sufficient to determine that a language group possesses a body-part tally method but are insufficient to determine the total number of tally points used, and (c) the data suggest that a language group has a body-part tally method but further data are required to confirm this.

In the last category we have 21 candidates which probably have a body-part tally method, 14 of which are language groups belonging to the Trans-New Guinea Phylum, the remainder belonging to the Arai Stock, the Sepik-Ramu Phylum, and the Amto-Musian Family.[38]  In category (b) we have 12 language groups which definitely have body-part tally methods but for which we are unable to determine the magnitude of the complete cycle.  Three of these language groups are members of the Sepik-Ramu Phylum and the remaining nine belong to the Trans-New Guinea Phylum.[39]  In category (a) we have 40 language groups which definitely possess body-part tally methods and for which we have sufficient information to determine the cycle of each tally.  Of these, four are members of the Sepik-Ramu Phylum, one (Yuri) is a Phylum-level isolate, and the remaining 35 are members of the Trans-New Guinea Phylum.  There is no evidence that members of the East Papuan  and Torricelli Phyla possessed this type of tally method.

The characteristics which define this type of body-part tally are as follows.  The tally usually begins on the fingers of the left hand (this is not universal and occasionally the tally will be started on the right hand).  The tally proceeds in order from the little finger through to the thumb, then to various points along the arm, usually beginning at the wrist (the 6th tally-point).  Other points, after those on the arm, may be the neck, ear, eye, and nose; occasionally the breast and sternum may be used.  In 31 out of 40 cases the tally has a unique midpoint which lies on the body's vertical axis of symmetry (and, unlike all other tally-points, does not have a symmetrical counterpart).  Once the points on one side of the body are tallied, the sequence continues on the opposite side of the body, tallying the symmetrical counterparts of the original tally-points in reverse order until the little finger of the hand is reached.  For those tallies which do have a unique midpoint, the complete tally cycle will, of course, be an odd number.  This type of tally does not employ the toes or indeed parts of the anatomy below the navel.

The mathematical term for the tally cycle is, properly, modulus  in that it is usual to start the tally from the beginning again once the full tally cycle has been completed.  As was noted above, the language groups which possess these tally methods use them for calendrical reasons and, in particular, for establishing when a given festival should occur within a cycle of festivals.  The magnitude of the tally cycles varies considerably between the different groups as the following table shows:

Table 12

The Frequency of Occurrence of the Cycles of Body-part Tallies

Cycle Size	18	19	22	23	25	26	27	28	29	30	31	35	36	37	47	68
Frequency	2	2	1	8	2	1	11	2	3	1	1	1	1	1	2	1

The modal cycle magnitudes are 27 and 23.  No data are available which suggest why these two tally cycles should predominate.  It is tempting to think that the 27-cycle may have some connection with the lunar month although this is, of course, 29.5 days and not 27.

One notable feature of these tally methods, which are used by widely separated language groups, is the similarity with which the first ten tally points occur.  We have, altogether, 24 groups with tally methods which follow the sequence:

                           1         little finger

                           2         ring finger

                           3         middle finger

                           4         index finger

                           5         thumb

                           6         wrist

                           7         forearm

                           8         elbow

                           9         upper arm

                         10         shoulder

We may note that the Aboriginal tally recorded in southeastern Australia, given earlier, also follows this sequence.  Some six other language groups omit the forearm and upper arm and thus have:

                           6         wrist

                           7         elbow

                           8         shoulder

This latter type also occurs in the western Torres Strait islands.  The language groups which deviate from the patterns given above are those which have adopted larger tally cycles, usually greater than 35, and these are all located in the Southern Highlands Province of PNG, the dialects of Kewa  being examples.

Several other methods of body-part tallying, not included in the data just discussed, exist.  The Ama  of the East Sepik Province (PNG)[40] first tally the fingers of one hand, as for the other body-part methods, however they then proceed:

                          6         navel

                          7         breast (1)

                          8         breast (2)

                          9         shoulder (1)

                        10         shoulder (2)

                        11         eye (1)

                        12         eye (2)

at which the tally terminates.  The Sakam  of the Morobe Province (PNG)[41] have what is essentially a digit-tally which is augmented by the two nostrils: tallying starts on the fingers of one hand and proceeds to the fingers of the other hand until a tally of 10 is reached.  The left thumb is placed on the left nostril for the 11th tally-point and then the right thumb is placed on the right nostril for the 12th tally-point.  After that the toes are tallied giving a complete cycle of 22.  A further method is reported by an SIL member in surveying the Nagatman  of the West Sepik Province (PNG).[42]  The unpublished report indicates that "tallying begins on the left side of the body, proceeding up the left arm and then down the left side.  The toes are tallied and upon reaching the big toe the tally stands at 36.  The side of the left foot is then tallied and then the symmetrical points on the right side of the body."  The complete cycle of this tally method is not given although it would have to be at least 74.  One other unusual tally method is that of the Abau,  also in the West Sepik Province.  This is essentially a digit-tally augmented by the navel, the breast, and the eye.  Tallying begins on the fingers of the left hand proceeding from the little finger to the thumb.  We then have:

                        6          "hand one navel with"

                        7          "hand one breast two"

                        8          "hand one breast two navel with"

                        9            one hand and the fingers of the other hand excluding the thumb

                      10          "hands two"

                      11          "hands two navel with"

                      12          "hands two breast two"

                      13          "hands two breast two navel with"

                      14          "hands two breast two navel eye one"

                      15          "hands two foot one"

                      20          "hands two feet two"

It is usually the case that a particular language group will possess a single tally method.  Two groups, however, are exceptions.  The Ketengban  of Irian Jaya have two tally methods.  According to Briley "the main one is based on twenty-five while the second is based on twenty-eight."[43]   The 25-cycle method is given below in full and this illustrates, in passing, the way in which left- and right-side symmetrical counterparts are distinguished verbally:

1	tekin	25	telepcap	(little finger)
2	betene	24	danabeteniap	(ring finger)
3	weniri	23	danaweniriap	(middle finger)
4	dombare	22	danadome	(index finger)
5	pambare	21	danapame	(thumb)
6	napbare	20	dananap	(wrist)
7	tabare	19	danata	(lower arm)
8	pinbare	18	danapin	(elbow)
9	topnebare	17	danatopne	(upper arm)
10	taubare	16	danatau	(shoulder)
11	kumbare	15	danakum	(neck)
12	amulbare	14	danamul	(ear)
13	mekbare			(crown of head)

The 28-cycle is obtained by augmenting the 25-cycle with the two eyes and the nose in the order: left eye (14th tally-point), nose (15th), right eye (16th), and then continuing with ear, neck, shoulder, etc., as given above.  Another language group which has two different tally methods is reported by Bruce.[44]   This is the Alamblak  of the East Sepik Province (PNG) which has a 29-cycle tally method used exclusively by men and another (similar) method used exclusively by women.

Those language groups which possess body-part tallies will, generally speaking, also possess a numeral system.  Of the 52 NAN language groups known to have body-part tallies, the data for 16, or less than a quarter, are such that we are uncertain whether there is an accompanying numeral system as well.  However, there are 33 groups which do have an accompanying numeral system: 28 of these have a "pure" 2-cycle system, or one of the 2-cycle variants, and a further 5 groups have 4-cycle numeral systems.  The latter groups, comprising Wiru and several Kewa  dialects, are all located in the Southern Highlands Province (PNG).

For the 28 groups which have both 2-cycle system variants and body-part tallies, some 15 are such that the first four finger names of the tally (usually "little finger", "ring finger", middle finger", and "index finger") are displaced by the first four numerals, that is a hybrid of the tally method and the numeral system is formed.  The remaining groups appear to not to form a hybrid and the tally and the numeral system are kept separate and distinct. 

2.6  THE DISTRIBUTION OF BODY-PART TALLIES

The distribution of the body-part tallies throughout the New Guinea mainland is shown in Map 7.  Generally speaking, they are located in the central interior region and in the southern coastal areas.  In PNG, the Provinces which have the majority of the tallies are the Southern Highlands, Western, West Sepik, and Gulf.  There are traces as well in the Enga and Madang Provinces.  In the northeast and eastern coast of PNG we have only a few examples: Giri, Murupi, and Baruga.  In Irian Jaya we have evidence that there is a predominance of tallies in the central border region; that there may be further tallies found to the west is likely as more data become available.  Except for the Southern Highlands Province, the body-part tallies are usually associated with variants of 2-cycle numeral systems.

As has been noted above, body-part tallies which are virtually identical to those found in New Guinea are also found in the Torres Strait islands and in southeastern Australia.  The close similarity of many of these quite complicated tallies which are located over a widespread geographical area suggests that the tally method has been diffused; in some cases local variations have occurred but there is nevertheless a marked uniformity between the various tally-methods, particularly with regard to the first 10 tally points.  The tallies are found in both NAN and Australian language groups.  Given that this type of tally is not found at all among the AN language groups, we will assume that its entry into the New Guinea region was with one of the migrations which occurred prior to the pre-POC  movement into northeast New Guinea.  The question then arises whether the body-part tally entered the Australia-New Guinea region with the earliest Australoid migrations or whether it came subsequently with one of the NAN migrations.  This issue will be taken up again in Chapter 8.

Map6:  Distribution of 2-cycle variants in New Guinea

Map 7:  Distribution of body-part tallies in New Guinea