There are at least three different Gallifreyan alphabets. All of them are obviously sets of symbols used to write Gallifreyan languages, just as the Roman alphabet is a set of symbols used to write western European languages.
Old High GallifreyanEdit
The Old High Gallifreyan alphabet mostly looks like Greek (with some mathematical symbols and mirror-image characters thrown in). Probably the clearest image of Old High Gallifreyan text is River's 'Hello, Sweetie' message in "The Time of Angels" or Rassilon's rhyme in The Five Doctors.
It was used to write Ancient Gallifreyan and Old High Gallifreyan, apparently the languages spoken before and around the time of Rassilon. At the time of the classic series, few Time Lords knew either the language or the alphabet (similar to historical Earth scripts like Linear B or Phonecian). Since the Last Great Time War, apparently only the Doctor and River know it.
The Modern Gallifreyan alphabet looks like a connected-cursive script similar to Arabic in most images, but sometimes it looks more like a block script like Korean Hangul. The clearest image is probably in The Deadly Assassin.
Although nobody ever directly said so, it was obviously used to write Modern Gallifreyan, the language of the Doctor's time.
The Circular Gallifreyan alphabet doesn't appear to be an actual alphabet, but some other kind of writing system that's not analogous to anything humans use. It's made up of concentric and overlapping circles, and it only takes a few circles to convey quite a bit of information. This frequently appears on TARDIS console screens in the new series. Whether it's used to write the Modern Gallifreyan language isn't entirely clear; in one of the novels, it's implied that its meaning is independent of any language (maybe sort of the way the same Chinese character can have the same meaning in Mandarin, Cantonese, and even Japanese despite being pronounced completely differently, and even conjugated differently in Japanese).
Behind the ScenesEdit
None of these writing systems, or languages, has been actually constructed as a 'real language' the way that, say, Klingon or Tolkien's Elvish languages have been. Whenever the production team needs some Gallifreyan writing, they just make up something that looks cool.
However, many fans have invented their own Gallifreyan alphabets (generally as a direct character-for-character transliteration of English). You can find these systems on Google, or just invent your own.
- ↑ Since the TARDIS can't translate Gallifreyan languages (as confirmed in "A Good Man Goes to War"), it apparently looks like exactly what we're seeing. That probably explains the Doctor's college nickname "Theta Sigma", Omega's name (or, according to the audios, his college nickname), etc.
- ↑ This name was never used on-screen, but has been used in novels.
- ↑ This name was also never used on-screen, or in any novels, but is only a popular fan name. (People need some name in order to talk about it.)