Intro:
To start off, pull up the IPA for reference. We also need to get some terminology out of the way:
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"phoneme = contrastive/distinctive sound within a particular language (notation: /…/)."
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"allophone (or variant) = sound which counts as an alternative way of saying a phoneme in a particular language (notation: […])."
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minimal pair = a set of words that differ in only one sound.
In linguistics, solving a phonemic analysis problem entails determining whether or not two sounds are assignable to two different phonemes. If not, you must specify whether they are allophones in complimentary distribution, or free variation. When allophones are in complimentary distribution, they appear in different environments within the language. For example, between the sounds [ð] and [d] in Osage, the Siouan language of a Native American people who originally lived in what is now western Missouri, [d] is only found before the sound [a]. [ð] is found everywhere else a 'd-like' sound is used (forgive me for the informal term). On the otherhand, allophones in free variation are generally attributable to a regional accent. Gritchka points out the example of [k] sound at the end of 'pack'. It may be said with an aspiration, an explosion of air on that last sound in the word, notated like this [kh]. Or it may be said with a half-long unreleased [k`] sound. Despite their different sounds, the meaning of the word remains the same.
Algorithm:
What follows is an algorithm of sorts (in the vein of the old choose your own adventure books) for solving phonemic analysis problems:
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Beginning with two sounds from a language and a finite data set of words containing one or both of those sounds...
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Is there a minimal pair for the given sounds? If yes, go to 2. If not, go to 5.
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Do the words in the pair differ in meaning? If yes, go to 3. If not, go to 4.
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The sounds are contrastive, i.e. separate phonemes.
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The sounds are allophones in free variation.
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Describe the phonetic environment in which each sound appears; e.g. list what comes before and after each sound. Do the sounds occur in the same (or similar) environments, or are their environments complementary? If same/similar, go to 6. If complementary, go to 7.
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The sounds contrast so your best guess is that they're separate phonemes, and you'd expect to find minimal pairs with more data.
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The sounds represent allophones of a single phoneme.
Example Problem:
Czech is a West Slavic language of the Indo-European language family, spoken in the Czech Republic. In Czech, among the various stops (plosives) are two alveo-dental stops,[t] and [d], articulated by the tongue tip against the boundary between the upper incisors and the alveolar ridge behind them, and two palatal stops,[ty] and [dy]. To how many phonemes are these four sounds assignable? Consider the data below and support your conclusion.
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dej = 'give!'
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dyedyit = 'to inherit'
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dyej = 'action'
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dyelo = 'cannon'
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kotel = 'kettle'
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kotye = 'kitten'
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tedi = 'hence'
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tele = 'calf (animal)'
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tyelo = 'body'
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teta = 'aunt'
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tikat = 'to be on a first-name basis'
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titul = 'title'
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tyikat = 'to tick (clock)'
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vada = 'flaw'
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vana = 'bathtub'
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vata = 'absorbent cotton'
Solution:
Step 1 - List any minimal pairs.
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1 and 3
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4 and 9
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11 and 13
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14 and 16
Step 2 - Determine if the words in the pair differ in meaning.
The words in all of the pairs differ in meaning.
Conclusion -
All of the four Czech stops, [t, d, ty, dy], are separate phonemes, /t, d, ty, dy/, because they contrast.
Sources: Anthro/Ling 2040, http://www.spectrum.uni-bielefeld.de/~thies/HTHS/phonrules.html