Note that the principles of phonological theory have also been applied to the analysis of signed languages, with gestures and their relationships as the object of study.
In some languages the phonemes are directly linked to spelling, i.e. a phoneme is represented by a graphical symbol or a combination of them, a letter or a letter combination. However in English different phonemes can be spelled the same way ("good" and "food" have different vowel sounds), so one should use the International Phonetic Alphabet (IPA) to denote phonemes. To indicate that one means phonemes instead of phones the phoneme or sequence of phonemes is enclosed with '/'s (without the quotes or pluralization; see above examples).
Much of the phonological study of a language involves looking at data (phonetic transcriptions of the speech of native speakers) and trying to deduce what the underlying phonemes are and what the sound inventory of the language is.
Even though a language may make distinctions between a small number of phonemes, speakers actually produce many more phonetic sounds. Thus, the definition of a phoneme in a particular language is a set of phonetic sounds that all associated with the same phonemic sound in the brain.
Looking for minimal pairs forms part of the research to study the phoneme inventory of a language. However with this method it is often not possible to detect all phonemes so other approaches are used as well. A minimal pair is a pair of words, both from the same language, that differ by only a single phoneme, and that are recognized by speakers as being two different words.
When there is a minimal pair, then those two sounds constitute separate phonemes, otherwise they are called allophones of the same underlying phoneme. For instance, voiceless stops (p,t,k) can be aspirated. In English, word initial voiceless stops are aspirated, whereas non word-initial voiceless stops aren't aspirated (This can be seen by putting your fingers right in front of your lips and notice the difference in breathiness as you say 'pin' and 'spin'). There is no English word 'pin' that starts with an unaspirated p, therefore in English, aspirated [ph] (the h means aspirated) and unaspirated [p] are allophones of an underlying phoneme /p/. This is not true of all languages however - both Cantonese and Thai make the distinction between [p] and [ph], so in those languages, /p/ and /ph/ are separate phonemes.
Another example... in English, the glides, /l/ and /r/ are two separate phonemes (minimal pair 'lead', 'read'); however, in many Asian languages the two glides are allophones, and the general rule is that [r] comes before a vowel, and [l] doesn't (e.g. Seoul, Korea). A native speaker of Korean will tell you that the [l] in Seoul and the [r] in Korea are in fact the same letter. What happens is that a native Korean speaker's brain uses the underlying phoneme /l/, and depending on the phonetic context (before a vowel or not) this phoneme gets expressed as either the [r] sound or the [l] sound. Another Korean speaker will hear both sounds as the underlying phoneme and think of them as the same sound. This is how different languages can have varying numbers of sounds in their inventory, even though there are a constant number of distinct phonetic sounds that humans can make.
Noam Chomsky and Morris Halle[?] presented in The sound pattern of English a view of phonology where a phonological representation (surface form) is a sequence of units which have characteristic features. The features are from a universally fixed set and have the values + or -. The phonological representation reflects the underlying representation which is a concatenation of morphemes. Phonological rules govern how the underlying representation is transformed to the surface representation.
The particular sounds that a language decides to make distinctions between can change over time as new children learn the language. At one point, [f] and [v] were allophones in English, and these changed later into separate phonemes. This is one of the main factors of historical change of languages as described in historical linguistics (another being fast change resulting from influence by another language, e.g. French influence on English after 1066).
Stress and tone[?] are also part of phonology. In some languages, stress is non-phonological, e.g. in Finnish or in Germanic languages (to check. In contrast, most modern-day Germanic languages such as German or English, stress is indeed phonologically distinctive, although there are only few minimal pairs, e.g. /'august/ 'August (the name)' versus /au'gust/ 'August (the month)' in German, or /con'verse/ 'converse (to hold a conversation)' and /'converse/ 'converse (the opposite of something)' in English.
In 1976 John Goldsmith introduced autosegmental phonology. The phonological phenomena are no longer seen as one linear sequence of segments called phonemes ore feature combinations but rather as some parallel sequences of features which reside on multiple tiers.
John McCarthy, Alan Prince[?] and Paul Smolensky[?] developed Optimality Theory, where languages choose a pronunciation of a word that best satisfies a list of constraints[?] which is ordered by importance: it is better to not satisfy a less important constraint than a more important one. This is where most current research in phonology is done.