See data and maps.
Melenchenko, M. (2023). “Numeral Bases”. In: Typological Atlas of the Languages of Daghestan (TALD). Ed. by M. Daniel, K. Filatov, T. Maisak, G. Moroz, T. Mukhin, C. Naccarato and S. Verhees. Moscow: Linguistic Convergence Laboratory, NRU HSE. DOI: 10.5281/zenodo.6807070. http://lingconlab.ru/dagatlas.
@incollection{melenchenko2023,
title = {Numeral Bases},
author = {Maksim Melenchenko},
year = {2023},
editor = {Michael Daniel and Konstantin Filatov and Timur Maisak and George Moroz and Timofey Mukhin and Chiara Naccarato and Samira Verhees},
publisher = {Linguistic Convergence Laboratory, NRU HSE},
address = {Moscow},
booktitle = {Typological Atlas of the Languages of Daghestan (TALD)},
url = {http://lingconlab.ru/dagatlas},
doi = {10.5281/zenodo.6807070},
}This chapter provides an overview of the distribution of bases of numeral systems in languages of the East Caucasian family and their neighbors. Here the base is defined as the lowest number which is multiplied in order to derive numeral expressions for higher numbers (Greenberg 1978: 288; Gvozdanović 1999: 3; Comrie 2013) (see a detailed discussion on the definition in (Mengden 2010: 32—35)). In practice, this definition means that the base is the multiplicand used to derive numerals from ‘20’ to ‘90’ (‘crowns’, in terms of (Calude, Verkerk 2016)). The clarification that the base is the lowest number is necessary to formally distinguish what is usually understood as the arithmetic base (e.g., ‘5’, ‘10’, ‘20’ in different languages) from higher multiplicands like ‘100’, ‘1 000’, etc.
There are many possible options of numeral bases in languages around the world. The most popular systems are decimal (base-‘10’) and vigesimal (base-‘20’) (and their mixtures), followed by quinary (‘5’) and body-part systems (Comrie 2013; Calude, Verkerk 2016: 3).
As noted in previous literature (Berg 2001: 165; Ganenkov, Maisak 2021: 107), East Caucasian numeral systems are either decimal, vigesimal, or mixed. In this chapter and on the corresponding maps, East Caucasian lects are coded with four values according to the numeral base they feature: decimal, vigesimal, both, and mixed. These categories are explained below.
An example of a language with a decimal (base-‘10’) system is Karata proper (Avar-Andic), which displays a regular pattern for the derivation of crowns from ‘20’ to ‘90’ by using the morph ‘10’:
| hac’a-da |
| ten-num |
| ‘10’ |
| k’e-jac’a-da |
| two-ten-num |
| ‘20’ |
| boʔ-ac’a-da |
| four-ten-num |
| ‘40’ |
Some languages use morphs of the root ‘10’ to derive only a subset of crowns. Very often, the number ‘20’ is expressed by a non-derived root, which is presumably cognate in all East Caucasian languages (Starostin, Starostin n.d.). For example, this is the case for Lak:
| ac’-wa |
| ten-num.iii |
| ‘10’ |
| q’u-wa |
| twenty-num.iii |
| ‘20’ |
| xːɵ-c’al-wa |
| five-ten-num.iii |
| ‘50’ |
Rarely, other irregularities in the derivation of crowns are attested. In several languages, the derivation of ‘30’ is rather opaque, as in Lak zuwi-wa (thirty-num.iii), while other crowns show a more transparent pattern (2). The same is true for ‘90’ in Upper Gakvari Chamalal: ač’un-na (ninety-num), which stands out among other crowns, cf. beƛ’a-c’a-da (eight-ten-num) ‘80’. The derivation patterns underlying such cases are not straightforward and require a separate discussion. However, these are just exceptions to an otherwise prevalent regular pattern of derivation. Moreover, these exceptions do not change the decimal pattern itself. They divide the numeric sequence in tens just like a regular decimal base, and the derivation of numerals in between the multiplicands of base does not differ: for example, in Lak ‘31’ is derived regularly as zuwi-lːij ca (thirty-lnk one).
In some languages, the regular derivation of crowns does not make use of a morph ‘10’. This is the case for Archi (Lezgic branch), Bezhta and Hunzib (Tsezic branch) (3). However, as for the exceptions discussed above, the inner structure of the numeric sequence is decimal, even though a different root is used to denote the meaning ‘10’.
| ac’ə.n |
| ten.num |
| ‘10’ |
| ɬa.na-rig |
| three.num-crown |
| ‘30’ |
Among non-East Caucasian languages, decimal systems are present in Armenian, Azerbaijani, Judeo-Tat, Kumyk, and Nogai.
Many East Caucasian languages use vigesimal numeral systems, i. e., their base is ‘20’. At the same time, ‘10’ is used as a base for addition in between the twenties, so that, for example, ‘70’ is derived as 3 ✕ 20 + 10.
This subtype of base-‘20’ systems is usually also called vigesimal, but sometimes other terms are used to point out the intricacy (for example, “vigesimal-decimal” (Edelman 1999: 223—224)). Such a system is found, for instance, in Tsova-Tush of the Nakh branch:
| t’q’a |
| twenty |
| ‘20’ |
| ša-uz-t’q’a-it’ː |
| two-mult-twenty-ten |
| ‘50’ |
Similarly to the decimal systems, there are some exceptions in the regular vigesimal patterns. Several crowns may be derived irregularly, with a single root instead of two roots (of the multiplicand and the base): ‘30’, as in Standard Avar ɬeber-go (thirty-num), or ‘40’, as in Lezgian jaχc’ur (forty). By the same reasoning, these systems are simply categorized as vigesimal.
In East Caucasian languages which use vigesimal numeral systems, ‘100’ is usually the next arithmetic base. It is expressed by an underived root that, in most cases, is presumably cognate for the family (Starostin, Starostin n.d.). This base is then multiplied to derive hundreds from ‘200’ to ‘900’. However, in Budukh and Kryz of the Lezgic branch, ‘100’ is clearly derived from a vigesimal pattern (5). This vigesimal ‘100’ is itself further used as a base in multiplication and addition; e.g., ‘200’ is derived as 2 ✕ (5 ✕ 20) (and not 10 ✕ 20). As the definition of base used here relies upon the lowest arithmetic base, both types of systems are considered vigesimal.
| q’ʷa-d | fə.qːa-r | firi |
| two-iv | five.twenty-i | man |
| ‘200 men’ |
Among non-East Caucasian languages presented in the data, a vigesimal system is found in Georgian.
For central Agul varieties, it is reported that speakers use two systems, one with base ‘10’ and another with base ‘20’. These varieties are spoken in the area between northern and southern dialects of Agul, which use decimal and vigesimal systems respectively (Magometov 1970: 94—97; Ganenkov, Maisak 2008: 7). It is likely that the simultaneous use of two systems is practiced in other parts of the region but it was not noticed by scholars.
Several lects are categorized as having “mixed” numeral systems, e.g., systems that use different bases for subsets of crowns. Unlike the cases discussed in 2.3, here different bases are distributed across a single numeric sequence, so that the patterns for the derivation of ‘40’ and ‘50’, for instance, may differ, but only one pattern is possible for each of them.
In Zaqatala and Kusur varieties of Avar, the original vigesimal Avar system remains for ‘20’, ‘30’, and ‘40’, whereas higher crowns (‘50’—‘90’) are replaced with decimal borrowings from Tsakhur (Saidova 2007: 107; Isakov 2020: 94).
In Inkhokvari and Kwantlada varieties of Khwarshi, a decimal system is used for ‘50’—‘90’, whereas for ‘30’ and ‘40’ a vigesimal system is used (‘20’ is atomic). The decimal crowns ‘50’—‘90’ are borrowed from adjacent Andic languages, presumably Tindi (Imnajšvili 1963: 85; Khalilova 2009: 169—170; Karimova 2013: 132). Similarly, in the neighboring Qidi and Sasitli dialects of Avar, the crowns ‘50’—‘90’ have been replaced by Andic decimal borrowings (Karimova 2013: 132).
In the village Kondik of the Tabasaran area, ‘60’ and ‘80’ are derived vigesimally, whereas ‘30’, ‘50’, ‘70’, and ‘90’ are derived decimally (Magometov 1965: 159—163). This is apparently due to the fact that Kondik is situated between the northern and southern dialects of Tabasaran, which use decimal and vigesimal systems respectively.
In the East Caucasian family, decimal and vigesimal bases are distributed according to patterns that do not straightforwardly correlate with genealogical or areal distribution. Several branches of the family, spoken in different parts of Northeastern Caucasus, include varieties with both types of systems. Moreover, in some cases, different dialects of a single language feature different numeral bases. Some examples of divergence in regards to numeral systems within a branch or a language can be easily attributed to language contact, whereas others are harder to explain.
Numeral systems in Avar-Andic languages are predominantly decimal. Within the branch, the distribution of bases is areal: Akhvakh, (Standard) Avar, and Tukita, spoken in the eastern part of the Avar-Andic area, are the only languages to have a vigesimal system. The distribution of bases is also areal within the Lezgic branch. Vigesimal systems are grouped in the southeastern part of the Lezgic area, while decimal systems are grouped in the northwest. The border between the two types is blurred for Agul and Tabasaran, which show corresponding dialectal variation: northern dialects of both languages use the base ‘10’, and southern dialects use the base ‘20’. In the center, in some cases both systems are used (see 2.3) or a mixed system has appeared (2.4).
The Tsezic branch is subdivided into two groups: West Tsezic languages use vigesimal or mixed (see 2.4) systems, East Tsezic languages use decimal systems. The Dargic branch and neighboring Lak all use decimal systems; the Nakh branch and Khinalug use vigesimal systems. Neighboring non-East Caucasian languages mostly use the base ‘10’ (Armenian, Judeo-Tat, and Turkic languages), the only exception being Georgian.
crown — suffix for crowns (numerals ‘30’–‘90’); i — class I; iii — class III; iv — class IV; lnk — linker; mult — multiplicative; num — numeral