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About diphthongs

A diphthong (or two-vowels) is a combination of two vowels to generate a one-syllable sound. Like consonant combinations, diphthongs change over time, some remain the same as in classical times, while others simplified.

At first, all diphthongs may have been pronounced seperately, then some combination became fix to generate one sound, so you'll see in the classical method, the sounds are more close to their 2-letter pronunciation.

A diphthong is always long.

Note that if a letter is marked with two dots at the top like this: Aloë, that means the two letters are not treated as a diphthong thus should pronounced in two syllables.

Diphthongs with same sounds in both methods

This group of diphthongs retain the 2-letter sound in both methods (strictly speaking they're not really diphthongs since they occupy two syllables):

OU AI
Retain their own sounds respectively. ou is as [ou] and ai as [ai]. English speakers should take care not to say ou like [əu] in no.
eg. prout, coutūntur, āit
AU EU
The two vowels form one syllable but both vowels must be distinctly heard like [a:u] [e:u]. The principle emphasis and interest belongs to the first which must be sounded purely.
eg. aurum, seu
EI
Apply the above rule to ei in the interjection: Hei, otherwise, this combination is not treated as diphthong and must be pronounced in two syllables.
This sound is rare in Latin. Do not pronounce as [ai] like German. eg. Hei! mei

Diphthongs that evolved

OE
Or written as Œ, in classical method, as [ɔi] in English boy. Simplified to [e:] in ecclesiastical method.
eg. coelicus (Classical, Ecclesiastical)
AE
Also written as Æ, in classical method, as [ai] in English bye. Also simplified to [e:] in ecclesiastical method. Thus in ecclesiastical, ae = oe = ē.
eg. Aegyptus (Classical, Ecclesiastical)
UI
In ecclesiastical method, always as [wi], but see previous page about qu cu and gu.
In classical method, first to determine if it's a diphthong. It's a diphthong if : 1. not combined with q to form the qu consonant combination. 2. U and I belongs to the same syllable.
As one syllable, it's pronouced as [ui], otherwise [wi].
eg. huic (Classical, Ecclesiastical)

Nota bene:

It's interesting to see there're different prereferences to simplifiy the diphthongs oe and ae into different languages. Italian pronounce it as [e:], while English speakers would say [i:]. That's a part of the reason why people in different countries "speaks" Latin differently.