A single t between vowels sounds like a d to me (or like an r in my language, Brazilian Portuguese).
May I say the tt spelling the same way, or does that only work for a single t?
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Yes, the spelling doesn’t matter, just the pronunciation.
In most North American and some Antipodean dialects of English, both Katie and kitty have an alveolar flap there, just like the one in the middle of Portuguese or Spanish cara. It is represented by [ɾ] in IPA.
In standard American English, the phoneme /t/ is pronounced, regardless of spelling:
As [tʰ] with a puff of air, at the beginning of a word (should be t superscript h, but I can't figure out how to make that on stackexchange).
As [t] without a puff of air, immediately before or after a consonant (e.g. in "wits" or "stretch" or "empty")
As the alveolar flap [ɾ] between vowels. This "between vowels" does not respect word boundaries: it is how the t in "put" and "it" are pronounced in Put it on the table.
As a glottal stop [ʔ] word-finally (unless the next word begins with a vowel.)
The phoneme /d/ merges with the phoneme /t/ between vowels (so both are realized as [ɾ]). Thus, for instance, "latter" and "ladder" become homophones. Because most American English speakers are literate, they will occasionally use the spelling pronunciation of "latter" to avoid the ambiguity in careful speech.
Similarly, the phoneme /t/ is not always realized when it occurs after /n/, so e.g. the words "winter" and "winner" become homophones (although this is not as universal as the phenomenon in the above paragraph). Again, speakers may consciously insert the [t] in "winter" to avoid the ambiguity.