Abstract
Reduction of unstressed /o/ and /a/ to [ɐ] or [ə] after non-palatalized consonants and to [ɪ] after palatalized ones in Contemporary Standard Russian (CSR) is systematic. But in certain inflectional suffixes [ə] occurs instead of the expected [ɪ] after palatalized consonants. In order to explain these apparent exceptions, I argue that vowel reduction after palatalized consonants is constrained by the morphology and that reduction to [ɪ] is blocked in certain cases of the paradigm by the interaction of Relativized Paradigm Uniformity (PU) and Paradigm Contrast (PCON) constraints (Rebrus and Törkenczy 2005; Steriade 2000). The main finding is that there is a critical contrast between the singular and plural within a given case, Number × Case, and [ɪ] is blocked when it would result in homophony with an /i/ [ɪ] suffix in the relevant situation. When the morphological contrast is implemented by some other means, then regular vowel reduction to [ɪ] takes place. The gen sg /-a/ suffix has special status due to type and token frequency of the [ə] variant. An analysis in terms of paradigm uniformity and contrast provides a more coherent account of the direction of language change in CSR vowel reduction than do appeals to stressed vowel faithfulness, spelling pronunciation, grammatical analogy or paradigm uniformity alone.
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Notes
Abbreviations used: sg = singular, pl = plural, nom = nominative, acc = accusative, gen = genitive, dat = dative, prep = prepositional, instr = instrumental, masc = masculine, fem = feminine, neut = neuter, adj = adjective, attr = attributive, poss = possessive adjective, C = non-palatalized consonant, Cj = palatalized consonant and /j/. Russian words are transliterated in the International Organization for Standardization system which uses diacritics (e.g., /Ʒ/ is ž, /ʦ/ is c) and italicized. References are given in the Library of Congress transliteration.
Because the [ət] is also sometimes found in Conjugation I verbs with liquid stems (Kuz’mina 1968b:142; Panov 1957, 2004:493; Shapiro 1968:18–19), loss of contrast between conjugation classes is not at issue.
After palatalized consonants the /u/ is somewhat fronted, and between two palatalized consonants it approaches [ʏ] (Avanesov 1972:50–52). The vowel /i/ is backed to [ɨ] after non-palatalized consonants and shibilants; the latter articulation is in large part due to velarization on the preceding consonant (Padgett 2001; Timberlake 2004:36–39). I use /i/ after any C below, with [ɨ] in phonetic transcription. Vowel reduction after [ʃ], [Ʒ], and [ʦ] is somewhat different for historical reasons (Avanesov 1972:64–65, 71–72, 186–187; Timberlake 2004:46–48) and is not discussed here. Some prefixes, compounds and borrowings are exempt from akan’e (Avanesov 1972:167–168, 175–176).
A slight contrast between unstressed non-high vowels (/e/, /o/, /a/) and unstressed /i/ is maintained in immediately pretonic position for some conservative speakers, with the former realized as a more open [ɪe] and the latter as [ɪ] before non-palatalized consonants (Avanesov 1956:105–133; 1972:45, 66–68; Padgett 2004; Padgett and Tabain 2005), but because most speakers of CSR do not make this distinction and have complete neutralization in immediately pretonic position after palatal(ized) consonants (Matusevich 1976:102–104, 109), ikan’e is represented by [ɪ] here. After palatalized consonants non-high vowels (including /a/) were neutralized to [e] in a process known as ekan’e sometime by the 14th century and then further reduced to [ɪ] in positions other than the immediately pretonic apparently by the end of the 14th century (Isačenko 1947:193–194). An /e/-pronunciation is found in the immediately pretonic syllable as late as the beginning of the 20th century in certain styles and among older speakers (Avanesov 1972:18–19; Košutić 1919:34–39), especially in St. Petersburg (Matusevich 1976:103, 107), in Russian emigre speech, and in many Russian dialects (Avanesov 1972:19, 27; Padgett 2004), but it is generally not found among younger CSR speakers (Matusevich 1976:107; Timberlake 2004:44). Jakobson (1929/1971:100), Trubetzkoy (1934), Halle (1959/1971:70–71), Lightner (1968) and others take ikan’e to be phonological in CSR—the complete neutralization of /a/, /e/, /o/, and /i/ after palatal(ized) consonants with recognized exceptions in certain morphosyntactic contexts. Thelin (1971: 68–81) argues against phonological ikan’e in favor of phonological ekan’e, and concludes that phonetic ikan’e is derived from ekan’e.
Compare the [ə] in [étə] from akan’e in /éto/ ‘this’, neut nom sg (mean F1 = 360, F2 = 1600) and /éta/ fem sg, (mean F1 = 360, F2 = 1640) in Thelin (1971:106). He took vowel midpoint measurements averaged over five male speakers of Moscow Russian who read the given word once in various contexts. Data were analyzed with a VoicePrint Spectrograph at a 20 mm/kcps frequency and a 300-cps filter (pp. 104–105). Bondarko and Verbitskaia (1973:44–45, 1975:185) find that post-tonic /a/ after Cj ranges from F1 = 490, F2 = 1000 to F1 = 300, F2 = 2400, depending on the speaker.
The morphemes neut nom/acc sg {-o}, masc/neut gen sg {-a}, prep sg {-e}, masc/neut instr sg {-om}, 1 pl verb nonpast{-om}, noun dat pl {-am} are also found under stress.
Avanesov (1972) represents the more conservative, prescriptive norm. In most cases the suffixal vowel can also appear under stress when the preceding stem requires accent on the inflectional ending. The glide [j] is not pronounced before /i/ or [ɪ] in CSR, especially after a vowel in unstressed syllables, e.g., /ij+e/ is commonly [ɪɪ], and the glide is a weak, non-syllabic [i] adjacent to other unstressed other vowels (see Timberlake 2004:63–64).
There are categories in which [ə] occurs after /j/ in CSR even when there is no corresponding stressed vowel in the given suffix (cf. the final vowel of the adjective neut and fem nom sg /-ojo, /-aja/).
Panov (1957, 2004) suggested that paradigmatic structure might play a partial role in ikan’e. He claimed that vowel reduction reflexes after palatalized stems are influenced by attraction to the stressed vowel and by the need to avoid lexical homophony, and he formulated a complex set of principles to account for the problem. Because his principles did not always make the right predictions, he resorted to a variety of extra-paradigmatic factors for additional explanations. In some circumstances vowels after /j/ were treated as special cases.
This is especially well documented for the word pole ‘field’ in Avanesov (1972:70), Jones and Ward (1969:196), and Thelin (1971), and in Leningrad/St. Petersburg speech the predominant pronunciation is with full ikan’e [póljɪ] (Kuz’mina and Glovinskaia 1974). Panov (1957, 2004:487), Kuz’mina and Glovinskaia (1974) insist on [ə] for Moscow speech, which most likely shows the effect of the preceding /l/ on the height of the vowel. There are only three neuter nouns with stem-final Cj other than /j/ (póle, móre ‘sea’ and góre ‘grief’), so the incidence of [ə] as a neut nom/acc sg suffix may be over-reported in the literature on the basis of the word póle [póljə]. In any event, both póle and móre have stress on the suffix in the plural /á/, thereby maintaining contrast with the singular. Most Cj-stem neuter nouns have stems in /j/ and fixed stem stress (Zalizniak 1967) thus more potential and actual homophony.
High vowels are phonetically fronted after palatalized consonants and more back after non-palatalized consonants under stress and in unstressed positions.
See Crosswhite (2001:57–117, 107–109) for a detailed analysis of Russian vowel reduction in OT. The shift of /o/ to [ɪ] poses a special problem for Crosswhite’s analysis because both /o/ and /u/ should surface as [i] given her constraint ranking. A solution in terms of constraint conjunction is explored and Max [round] selects [u] ([ʊ]) for /u/ but not for /o/.
The plural suffix is traditionally represented as /-ije/ in CSR, though the final vowels of long form suffixes are never stressed. Panov (1957, 2004:496–497) attributes the [ɪ] reflex in the final syllable to a type of phonetic assimilation after the loss of the glide following the high vowel (vs. retention of the glide after non-high vowels in /-ojo/ and /-aja/). He also points out that the plural paradigm as a whole is characterized by /i/ in suffixes: gen pl /-ix/, dat pl /-im/, prep pl /-ix/, and instr pl /-imji/, so /i/ generally marks plurals in adjectives. The final vowel in the nom pl is effectively /i/ [i]; morphology does not interfere with complete ikan’e in the nom pl. In the possessive adjectives, the nom pl is /i/ [ɪ], as opposed to the nom sg /o/ and /a/ which are [ə].
I am grateful to an anonymous reviewer for reminding me of these examples. Some of the masc nouns have newer forms with stem stress and a nom pl suffix in /i/, for example, pékari ‘bakers’ [pjékərjɪ], lékari ‘doctors, healers’ [ljékərjɪ], and the potential homophony with gen sg is not an issue. In any event, PU (Gen Sg-a) favors schwa in the gen sg.
The gen sg form is the same as the acc sg for masc animate nouns. Vakar (1966) is based on 10,000 words taken in 50-word samplings from 93 contemporary plays (1957–1960). Josselson (1953) counts 506,044 running words in 134 literary samples (1830–1948), mostly fiction and drama with some journalistic texts (20%) and literary criticism (14%). Štejnfel’dt’s (1963) dictionary of 400,000 words includes samples of Soviet children’s literature, adult fiction, plays, radio broadcasts and periodicals (350 texts).
Lönngren (1993) is based on 1,000,000 words from 600 texts of literary (1960–1988) and informative popular prose (25 subject areas for 1985–1989). Counts include word forms (tokens) and the most frequent 1,134 are given in his Appendix 6.
In the analysis above, the PU constraint was formulated in terms of the suffix /-a/ as PU (Gen Sg -a), but given that the gen sg /-i/ suffix also has a high frequency, the PU constraint could be a more general PU (Gen Sg). It would produce the same result for gen sg /i/-suffixes because they are generally faithful by Faith High V.
Reported experimental results do identify some differences that correlate with vowel type, though it is difficult to isolate this effect from the potential influence of orthography because the stimuli for instrumental measurements were for the most part read from written texts. Thelin (1971:122) finds that although there is a tendency to merge all non-high vowels to [ɪ]-like sounds (F1 = approximately 300 Hz, F2 = approximately 2,000 Hz) after palatalized consonants, “there are tendencies to a keeping apart of on the one hand /a/ and on the other /o, e/” and that “it is reasonable to postulate different phonetic target-values for /i/, /o, e/ and /a/ respectively in the position after soft consonants” only in certain post-tonic positions. In pretonic syllables “no significant differences between /a/ and /o, e/ have been observed in isolated pronunciation, though such differences have been observed in slow pronunciation” (p. 120). In phrase-final position /a/ is lower and more [æ]-like before a pause than the mid vowels, which are [ɪ] or [ə] (Matusevich 1976:103–104; Thelin 1971:92). Thelin (1971:104–105) used stimuli from 5 male speakers of Moscow CSR, aged 18–22, who read and recorded target words in isolation and in sentence context, where the target word form occurred within the phrase, at phrase boundary, and in absolute final position. Spelling pronunciation is most likely to be reflected in slow, careful speech. Bondarko and Verbitskaia (1973) and Bondarko (1981:164) find that post-tonic suffixes in /e/ and /i/ merge into /i/ but that post-tonic /a/ shows greater variation.
The effect of /l/ on posttonic CSR vowels is given in percents of [ə] tokens out of the total [ə] and [ɪ] tokens (Kuz’mina 1966): dýnjam ‘melons’, dat pl, 52%; détjam ‘children’, dat pl, 53%; but nedéljam ‘Sundays’, dat pl, 74%; and the difference between dýnjami ‘melons’, instr pl at 2% and nedéljami ‘Sundays’, instr pl at 49%. Thelin (1971:111–113) gives the following: náčat ‘begun’, past passive part with F1 = 340, F2 = 1770; náčal ‘he began’ with F1 = 390, F2 = 1360; zánjat ‘occupied’, past passive part, with F1 = 310, F2 = 1880; zánjal ‘he occupied’ with F1 = 280, F2 = 1490. As expected, this effect was also noted for the high vowel /i/, where /búdjit/ ‘he wakes’ had posttonic /i/ at F1 = 320, F2 = 1890 but /múʧjil/ ‘he tortured’ had /i/ at F1 = 370, F2 = 1290, and also for vowels within stems, e.g., /ʧjúʧjel/ ‘scarecrows’, gen pl, with F1 = 360, F2 = 1320, in sentence final position.
References
Avanesov, Ruben I. 1949. Ocherki russkoi dialektologii. Moscow: Gos. Ucheb. Ped. Izd.
Avanesov, Ruben I. 1956. Fonetika sovremennogo russkogo literaturnogo iazyka. Moscow: Izd. Moskovskogo Universiteta.
Avanesov, Ruben I. 1972. Russkoe literaturnoe proiznoshenie, 5th edn. Moscow: Prosveshchenie.
Barnes, Jonathan. 2006a. Strength and weakness at the interface: Positional neutralization in phonetics and phonology. Berlin: Mouton de Gruyter.
Barnes, Jonathan. 2006b. Phonetics and phonology in Russian unstressed vowel reduction: A study in hyperarticulation. Ms., Boston University.
Bat El, Outi. 2005. Competing principles of paradigm uniformity: Evidence from the Hebrew imperative paradigm. In Paradigms in phonology, eds. Downing, Laura, T. Alan Hall, and Renate Raffelsiefen, 44–64. Oxford: Oxford University Press.
Bondarko, Liya V. 1981. Foneticheskoe opisanie iazyka i fonologicheskoe opisanie rechi. Moscow: Gos. Ucheb. Ped. Izd.
Bondarko, Liya V., and Ludmila A. Verbitskaia. 1973. O foneticheskikh kharakteristikakh zaudarnykh fleksii v sovremennom russkom iazyke. Voprosy Iazykoznaniia 1: 37–49.
Bondarko, Liya V., and Ludmila A. Verbitskaya. 1975. Factors underlying phonemic interpretation of phonetically non-defined sounds. In Auditory analysis and perception of speech, eds. G. Fant, and M.A.A. Tatham, 177–190. London: Academic Press.
Bromlei, S.V. 1973. Protivopostavlenie I i II spriazhenii v russkikh govorakh i literaturnom iazyke. In Issledovaniia po russkoi dialektologii, eds. Iu.S. Azarkh, S.V. Bromlei, and L.N. Bulatova, 155–175. Moscow: Nauka.
Bukrinskaia, I.A. 1987. Forma imenitel’nogo chisla sushchestvitel’nykh srednego roda v russkikh govorakh. In Russkie dialekty: lingvogeograficheskii aspekt, eds. R.I. Avanesov, and O.N. Morakhovskaia, 124–129. Moscow: Prosveshchenie.
Burzio, Luigi. 1996. Surface constraints versus underlying representation. In Current trends in phonology: Models and methods, eds. Jacques Durand, and Bernard Laks, 123–141. Manchester: European Studies Research Institute, University of Salford.
Burzio, Luigi. 1998. Multiple correspondence. Lingua 103: 79–109.
Comrie, Bernard, Gerald Stone, and Maria Polinsky. 1996. The Russian language in the twentieth century, 2nd edn. Oxford: Clarendon Press.
Crosswhite, Katherine. 1999. Intra-paradigmatic homophony avoidance in two dialects of Slavic. In UCLA Working papers in linguistics 1: Papers in phonology 2, ed. Matthew Gordon, 48–67.
Crosswhite, Katherine. 2001. Vowel reduction in optimality theory. New York: Routledge.
Crosswhite, Katherine. 2004. Vowel reduction. In Phonetically based phonology, eds. Bruce Hayes, Robert Kirchner, and Donca Steriade, 191–233. Cambridge: Cambridge University Press.
Filin, F.P. 1972. Proiskhozhdenie russkogo, ukrainskogo i belorusskogo iazykov. Leningrad: Nauka.
Flier, Michael J. 2002. K voprosu o kriteriiakh foneticheskogo var’irovaniia i grammatikalizatsii v russkom iazyke. In Avanesovskii sbornik, ed. N.N. Pshenichnova, 246–251. Moscow: Nauka.
Gnanadesikan, Amalia E. 1997. Phonology with ternary scales. PhD diss., University of Massachusetts, Amherst.
Halle, Morris. 1959/1971. The sound pattern of Russian. 2nd edn. The Hague: Mouton.
Halle, Morris. 1965. Akan’e: The treatment of unstressed, non-diffuse vowels in southern Great Russian dialects. In Symbolae linguisticae in honorem Georgii Kurylowicz, ed. Witold Taszycki, 103–109. Cracow: PAN.
Isačenko, A.V. 1947. Fonetika spisovnej ruštiny. Bratislava: Slovenska Akad. Vied.
Jakobson, Roman. 1929/1971. Remarques sur l’évolution phonologique du russe comparée à celle des autres langues slaves. In Selected writings, vol. 1. phonological studies, 2nd edn. 7–116. The Hague: Mouton.
Jones, Daniel, and Dennis Ward. 1969. The phonetics of Russian. Cambridge: Cambridge University Press.
Josselson, Harry H. 1953. The Russian word count and frequency analysis of grammatical categories of standard literary Russian. Detroit: Wayne University Press.
Kenstowicz, Michael. 1996. Base-identity and uniform exponence: Alternatives to cyclicity. In Current trends in phonology: Models and methods, eds. Jacques Durand, and Bernard Laks, 363–393. Manchester: European Studies Research Institute, University of Salford.
Kenstowicz, Michael. 2002. Paradigmatic uniformity and contrast. MIT Working Papers in Linguistics 42: 141–163.
Kenstowicz, Michael. 2005. Paradigmatic uniformity and contrast. In Paradigms in phonology, eds. Laura Downing, T. Alan Hall, and Renate Raffelsiefen, 145–169. Oxford: Oxford University Press.
Kirchner, Robert. 1996. Synchronic chain shifts in optimality theory. Linguistic Inquiry 27: 341–350.
Korš, F.E. 1902. O russkom pravopisanii. Izvestiia Otdeleniia Russkogo Iazyka i Slovesnosti 7(1): 39–94.
Košutić, R. 1919. Gramatika ruskog jezika, 1. Petrograd: Russkaia Akademiia Nauk.
Kuz’mina, S.M. 1966. O fonetike zaudarnykh fleksii. In Razvitie fonetiki sovremennogo russkogo iazyka, eds. S.S. Vysotskii, M.V. Panov, and V.N. Sidorov, 5–24. Moscow: Nauka.
Kuz’mina, S.M. 1968a. Fonetizatsiia zaudarnogo vokalizma vo fleksiiakh. In Russkii iazyk i sovetskoe obshchestvo. Fonetika sovremennogo russkogo literaturnogo iazyka. Narodnye govory, ed. M.V. Panov, 42–56. Moscow: Nauka.
Kuz’mina, S.M. 1968b. Deistvie morfologicheskoi analogii. In Russkii iazyk i sovetskoe obshchestvo. Morfologiia i sintaksis sovremennogo russkogo literaturnogo iazyka. Narodnye govory, ed. M.V. Panov, 135–143. Moscow: Nauka.
Kuz’mina, S.M., and M.Ia. Glovinskaia. 1974. Fonetika. In Russkii iazyk po dannym massovogo obsledovaniia. Opyt sotsial’no-lingvisticheskogo izucheniia, eds. L.P. Krysin, and D.N. Shmelev, 38–155. Moscow: Nauka.
Kuznetsova, L.N. 1986. Glagol’nye okonchaniia -at, -jat v proiznoshenii akterov. In Russkoe stsenicheskoe proiznoshenie, ed. S.M. Kuz’mina, 51–57. Moscow: Nauka.
Lightner, Theodore. 1968. An analysis of akan’e and ikan’e in modern Russian using the notion of markedness. In Studies presented to professor Roman Jakobson by his students, ed. Charles E. Gribble, 188–200. Cambridge: Slavica Publishers.
Lönngren, Lennart. 1993. Chastotnyi slovar’ sovremennogo russkogo iazyka. Uppsala: University of Uppsala.
Matusevich, Margarita I. 1976. Sovremennyi russkii iazyk: Fonetika. Moscow: Prosveshchenie.
Nikolaev, V. 1960. Nekotorye dannye o chastotnosti upotrebleniia padezhnykh form v sovremennom russkom literaturnom iazyke. Russkii Iazyk v Natsional’noi Shkole 5: 19–26.
Padgett, Jaye. 2001. Contrast dispersion and Russian palatalization. In The role of speech perception in phonology, eds. Elizabeth Hume, and Keith Johnson, 187–218. San Diego: Academic Press.
Padgett, Jaye. 2003. Contrast and post-velar fronting. Natural Language & Linguistic Theory 21: 39–87.
Padgett, Jaye. 2004. Russian vowel reduction and dispersion theory. In Phonological Studies 7, 81–96. Tokyo: Kaitakusha.
Padgett, Jaye, and Marija Tabain. 2005. Adaptive dispersion theory and phonological vowel reduction in Russian. Phonetica 62: 14–54.
Panov, M.V. 1957/2004. O vliianii grammaticheskoi analogii na proiznositel’nye normy v sovremennom russkom literaturnom iazyke. In Trudy po obshchemu iazykoznaniiu i russkomu iazyku 1, eds. E.A. Zemskaia, and S.M. Kuz’mina, 479–510. Moscow: Iazyki slavianskoi kul’tury.
Panov, M.V. 1968. Umen’shenie srednei informatsii glasnykh. In Russkii iazyk i sovetskoe obshchestvo: Fonetika sovremennogo russkogo literaturnogo iazyka: Narodnye govory, ed. M.V. Panov, 22–31. Moscow: Nauka. Chapter 2.
Panov, M.V. 1990. Istoriia russkogo literaturnogo proiznosheniia XVIII–XX vv. Moscow: Nauka.
Rebrus, Peter, and Miklós Törkenczy. 2005. Uniformity and contrast in the Hungarian verbal paradigm. In Paradigms in phonology, eds. Laura Downing, T. Alan Hall, and Renate Raffelsiefen, 263–295. Oxford: Oxford University Press.
Russian National Corpus. 2010. www.ruscorpora.ru. Basic corpus. Accessed 12 September 2010.
Shakhmatov, Alekseı̌ Aleksandrovich. 1925. Ocherk sovremennogo russkogo literaturnogo iazyka. Leningrad: Gos. Izd.
Shapiro, Michael J. 1968. Russian phonetic variants and phonostylistics. Los Angeles: University of California Press.
Štejnfel’dt, E.A. 1963. Chastotnyi slovar’ sovremennogo literaturnogo iazyka: 2500 naibolee upotrebitel’nykh slov. Tallinn: Nauchno-issledovatel’skii Inst. Ped. Estonskoi SSR.
Stankiewicz, Edward. 1966. Slavic morphophonemics in its typological and diachronic aspects. In Current Trends in Linguistics, 495–520. The Hague: Mouton.
Steriade, Donca. 2000. Paradigm uniformity and the phonetics-phonology boundary. In Papers in laboratory phonology, V: Acquisition and the lexicon, eds. Michael Broe, and Janet Pierrehumbert, 313–334. Cambridge: Cambridge University Press.
Stoikov, Stoiko. 1963. Akane v govora na s. Trigrad, Devinsko. Bulgarski Ezik 13(1): 8–21.
Thelin, Nils B. 1971. On stress assignment and vowel reduction in contemporary standard Russian. Uppsala: Striv Service AB.
Timberlake, Alan. 2004. A reference grammar of Russian. Cambridge: Cambridge University Press.
Trubetzkoy, Nikolaj. 1934. Das morfologische System der russischen Sprache. In Travaux du Cercle linguistique de Prague 5.2. Leipzig: Harrassowitz.
Vakar, Nicholas P. 1966. A word count of spoken Russian: The Soviet usage. Columbus: The Ohio State University Press.
Zalizniak, Andrei A. 1967. Russkoe imennoe slovoizmenenie. Moscow: Nauka.
Zalizniak, Andrei A. 1977. Grammaticheskii slovar’ russkogo iazyka: Slovoizmenenie. Moscow: Russkii iazyk.
Acknowledgements
This research was supported by a National Endowment for the Humanities Fellowship (FA-52398-06). Any views, findings, conclusions, or recommendations expressed here do not necessarily reflect those of the National Endowment for the Humanities. I am grateful to Michael Kenstowicz, Ellen Broselow, the audience at FASL-19 held at Cornell University, and anonymous reviewers whose detailed and thoughtful comments significantly contributed to the improvement of the manuscript. Remaining shortcomings are mine.
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Bethin, C.Y. On paradigm uniformity and contrast in Russian vowel reduction. Nat Lang Linguist Theory 30, 425–463 (2012). https://doi.org/10.1007/s11049-012-9166-4
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DOI: https://doi.org/10.1007/s11049-012-9166-4