Overcounting in Numerals
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Regarding query: http://linguistlist.org/issues/15/15-328.html#2
Finally, my summary on the occurrence of so-called overcounting (OC) in
numeral systems. Sorry for the undue delay, I hope a late and lengthy
summary is better than none!
Of course, I would be glad to hear about other cases of overcounting and
related numeral constructions.
2. a proposed subtype
3. other examples
4. comments to the examples of the query
Thanks to all respondents:
Stig W. Jørgensen,
Matthias Trautner Kromann,
This summary adds some results from an own typological survey (Hanke 2005).
1. a definition, cf. Comrie (1997: 53), Greenberg (2000: 774):
Overcounting expresses a value as part of a group with an upper limit.
In contrast to subtraction and division, overcounting is a variant of
serial addition, the addition of a sequence of numbers to a certain value,
called augend (Greenberg) or additive base (Comrie).
''Twenty-one'' to ''twenty-nine'' constitute a series with the augend 20.
In OC constructions, the next higher augend is mentioned, e.g. 40 in
Yucatec (Maya – Yasugi 1995: 307, original source from 1746):
32 ''lah-ca tu-y-ox-kal'' '10-2 OC-ligature-2-20'
Another example is the rarely used construction for 11+ in Tamabo
(Austronesian – Jauncey 2002: 613):
32 ''ngalai-tolu ngalai-vati-na arna'' '10-3 10-4-ordinal 2'.
Generally, the lower augend is left unexpressed, Tamabo being the only
exception without bodily expressions (see below). Using two augends, OC
constructions are always part of base patterns, in the examples with base
10 and 20.
The formulation in the query ''to include a variant of the following base''
was an attempt to get more, perhaps controversial examples.
2. a variant connected with bodily expressions for the augends 5, 10 and 15:
Some numeral systems express addition with items otherwise used for values
(Hanke 2005: 106-110). Such complex additive constructions have not
explicitly been described as OC before, an example from Moni (Trans-New
10 ''hamagi'' 'both:hands',
11 ''bado hago'' 'foot 1',
16 ''amo bado hago'' 'other foot 1'
''(amo) bado'' refers to the range of the series, cf.
15 '' bado idi'' 'foot 5'.
Rawa (Trans-New Guinea) includes the augend 10:
11 ''kande eraya ke-gidemboro gura-nangge'' 'hand 2 leg-plus 1-only'
'hand 2 leg' expresses 15, cf.
15 ''kande eraya ke-ngga'' 'hand 2 leg-DEF'.
Such constructions do not necessarily rely on bodily expressions, but are
not known without them:
*''ten one of five'' for 11.
3. the cases of overcounting found so far:
The lack of OC in regions like Africa and North America may be due to my
limited data. The evidence shows only limited areal effects.
The recognizable OC markers are mainly ordinal or ‘towards’ expressions
(Greenberg 2000: 778). OC is only rarely unmarked.
Søren Wichmann corrected the impression that overcounting is an areal
feature of Mesoamerica comparable to base 20. Rather, OC seems to be
limited to the Eastern branch of Maya and some Western languages sharing
certain items with them. Søren cites a proposed proto-Mayan ''*lajuunh
r-oox=k'aal ''10 towards (3 x 20)'' ''.
The attested ranges vary, e.g. Classical Quiché used it with both bases 20
and 400. Many languages have replaced OC by normal addition. For more
details, I refer to Yasugi (1995: 98f., 104f.). Besides Mayan languages,
OC is attested from two (historical) varieties of neighboring Zapotec
(Oto-Manguean - Yasugi 1995: 97) in the range of 20 to 60.
Gabriele Müller pointed to Estonian
11-19 ''1-9 teist (kümmend)'' '1-9 other_of_2 (10:partitive)',
with the same structure as their Finnish counterparts (Abondolo 1998: 168).
Mansi, a Finno-Ugric language of the Ob-Ugrian subgroup, builds complex
numerals either by unmarked addition or by two variants of '' 'adding' the
smaller number to the larger'' (Keresztes 1998: 412), examples being
21 ''waat-n akwa'' '30-lative 1'
31 '40 towards 1'.
Both variants seem to be limited to the decades.
Another case is older (Eastern) Turkic: ''In the runic inscriptions and
earlier Uygur manuscripts, cardinals from the second to the ninth decade
are formed with the digit of the lower decade from the lower plus the
higher decade'' (Erdal 1998: 144), e.g.
27 ''yeti otuz'' '7 30'.
An alternative is regular addition with or without ''artoqi'' 'its supplement'.
In the 19th century, Naga Ao (Tibeto-Burman - Mazaudon 2002: ) used OC, but
only for 4 numerals before an augend, e.g. 15 '10 5', 16 '20 maben 6'. This
system and the related one of Sema show constraints similar to those on
subtraction (cf. Greenberg 2000: 774).
The mentioned Tamabo is the only other case of overcounting I know of.
4. comments to the examples in the query:
Russian 90 ''devjanosto'' is a ''specifically Russian form, not shared in
general by other Slavonic languages. (Even Ukrainian has a more regular
nine-ty form, except under Russian influence.) Various etymologies have
been proposed, not all of which would involve overcounting (e.g. one is
'nonal hundred', assuming the contact of base 9 and 10 systems), though it
may be that synchronically the numeral is so analyzed, e.g. its morphology
parallels that of sto 100.'' (Bernard Comrie) My mentioning of Russian 80
was just a mistake.
The Danish constructions with 'half' and related constructions in other
languages (see below) are no instances of overcounting proper, but a
related use of division.
First, a list of the decades without typos:
10 ti, 20 tyve, 30 tredive, 40 fyrre(tyve),
50 halvtreds, 60 tres,
70 halvfjerds, 80 firs,
20 ''tyve'' derives from Proto-Norse 'two-ty-PL' with haplological loss of
the first syllable (Ross/Berns 1992: 606). Synchronically, this gives an
arithmetic mismatch between 20 and ''-dive/(-tyve)'' in 30 and 40. The
latter just correspond to other Germanic forms like thirty and fourty.
For 50 to 90, I quote Stig Jørgensen's translation of the FAQ of the Danish
Language Council (www.dsn.dk, probably by Anita Ågerup Jervelund):
''The numerals halvtreds, tres, halvfjerds, firs and halvfems are the
abbreviated forms of halvtredsindstyve, tresindstyve, halvfjerdsindstyve,
firsindstyve and halvfemsindstyve. They all end in -sindstyve, which is
made up of sinde, meaning “times”, and tyve (20). The strange numerals
halv- derive from a number of old numerals referring to “a basic number
minus a half”: halvanden (“half-second”), halvtredje (“half-third”),
halvfjerde (“half-fourth”), halvfemte (“half-fifth”), that is “one and a
half, two and a half, three and a half, four and a half”. Today only
halvanden survives as a word in itself.''
Stig added valuable comments:
''The abbreviated forms are the ones normally used in modern Danish. Saying
''halvtredsindstyve'', instead of ''halvtreds'', is quite old-fashioned,
but could perhaps be used for emphasis, at least among well-educated
people. However, the full forms are preserved in the ordinal numbers:
''halvtredsindtyvende'' = 50th. There is no abbreviated form for this.''
Typologically, Danish has Tibeto-Burman relatives: Matisoff (1997) gives
similar forms for 50, 70 and 90 in the ''super-vigesimal'' system of Chang.
Dzongkha (Mazaudon 2002: 100-105) uses even an expression for 'quarter to',
30 '20 1/2-da 2' 35 '20 3/4-da 2'.
These numerals do not function as augends: 31 '20 1 and 11'
Some other Tibeto-Burman cases are less completely described and/or no more
in use. For details, I refer to Mazaudon (2002). These cases confirm that
the use of fractions in numerals is limited to the division of bases by 2
or 4, with the apparaent restriction that 3/4 is the only quarter fraction.
Due to limited space, I just mention the use of similar constructions in
measuring, e.g. time, and the inspiring discussion of expressions like
''one and a half million'' in Russian and other European languages.
Abondolo, Daniel (1998): Finnish. In: Abondolo (ed.): The Uralic Languages.
London, New York, p.138-157.
Comrie, Bernard (1997): Some problems in the theory and typology of numeral
systems. In: Palek (ed.): Proceedings of LP‘96, Prague. Prague, p. 41-56.
Erdal, Marcel (1998): Old Turkic. In: Johanson / Csató (eds.): The Turkic
Languages. London, New York, p. 138-157.
Greenberg, Joseph H. (2000): 75. Numeral. In: Booij et al. (eds.):
Morphology. an international handbook on inflection and word-formation.
Berlin, New York, p. 770-783.
Hanke, Thomas (2005): Bildungsweisen von Numeralia. Eine typologische
Jauncey, Dorothy (2002): Tamabo. In: Lynch / Ross / Crowley (2002) (eds.):
The Oceanic Languages. Richmond, p. 608-625.
Keresztes, László (1998): Mansi. In: Abondolo (ed.): The Uralic Languages.
London, New York, p. 387-427.
Matisoff, James A. (1997): Sino-Tibetan numerals: prefixes, protoforms and
Mazaudon, Martine (2002): Les principes de construction du nombre dans les
langues tibéto-birmanes. In: Mémoires de la société de linguistique de Paris.
Nouvelle série; tome XII; La pluralité. Leuven, p. 91–119.
Ross, Alan / Berns, Jan (1992): Germanic. In: Gvozdanovic: Indo-European
Numerals. Berlin und New York, p. 555-715.
Yasugi, Yoshiho (1995): Native Middle American languages: an
areal-typological perspective. Osaka.
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