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Are all three of these valid spellings and have I listed them in the chronological order they would have been used?
Would the C have been pronounced with a hard 'K', or a 'CH'? Does anyone happen to know the official International Phonetic Alphabet spelling for this word?
Yes, the spelling DVCITIS was absolutely valid in the Roman Republic and early Empire. The introduction of the letter U allowed people to mark the difference between the sound used as a consonant (sounding like our letter w) and the sound used as a vowel (sounding like our oo). I don't know when folks started using U, though.
Interestingly, there are different approaches to spelling Latin today, too. I prefer to alternate between v/u and j/i for consonantal/vocalic sounds. Church Latinists tend to do this as well. Most other folks seem to be to alternate v/u for consonantal/vocalic V but to use i for consonantal and vocalic I. I've encountered a number of texts printed in the last several years, however, that use only U and I for both consonantal and vocalic sounds; I'll admit to finding this a little annoying to read, but it could be that I'm used to alternating V/U.
In the time of Cicero, Cæsar, and their compatriots, yes, the C would have been pronounced like K. The change to a CH sound occurred some time between 300 and 700 CE; you can read more about it and the other phonological changes from classical to medieval Latin here.
The addition of the macron to the U in later years isn't really a spelling change; it's more a visual aide to let readers less familiar with the vocabulary know what vowels are long. So in classical Rome, you might for example find misspelled graffiti that read DVVCITIS, doubling the length of the vowel. No one who actually spoke Latin as a native tongue would require macrons.
The Romans did on occasion mark long vowels, but they did it differently, with what we think of today as the French accent aigu; you can read this meta post for a more detailed discussion. However, they usually did this only to mark vowels that, if left unmarked, might cause the words to mean different things (for example, the word anus means both "anus" and "old woman," so if there was any risk of confusion you might want to avoid it by marking the vowel). However, the use of apices isn't widespread these days, and if this is meant for a general audience I'd advise against it.
So I'd say you should go with either DVCITIS or DUCITIS.
All three are valid, and I would recommend 1 or 2.
The difference between 2 and 3 is the macron, the bar above the vowel U. The macron is used to indicate a long vowel, and it is an auxiliary mark. Such auxiliary symbols are often used in textbooks, dictionaries and such places where the reader is not assumed to be familiar with the word in advance and is expected to learn the correct length. However, if you want to write a more ceremonial text (like the title of an art project), I would consider it slightly bad style to include a macron. If you want to guide the reader — who is assumed to know the basics of pronouncing Latin — to read the title out loud, option 3 is good.
The letters U and V can be used to denote the same vowel. These two symbols were not distinguished in classical Latin. Therefore options 1 and 2 are essentially equivalent. At the time when U and V were the same thing, mainly upper case letters were used. Therefore I would prefer any of "DVCITIS", "DUCITIS" and "ducitis" to "dvcitis", but this is partly a matter of taste.
I would actually consider it ok to write a V with a macron as well, although it is less common.
In classical Latin the letter 'c' is indeed pronounced like 'k'. Maybe someone can provide you with a good guess of the classical pronunciation in IPA. I am not familiar enough with the alphabet to provide one. There is not going to be an official one, since Latin is pronounced differently in different countries and all ancient Roman officials are believed dead.
