An illustrated note on diphthongs

Coverage of the diphthong and its relatives the triphthong and the tetraphthong is somewhat cursory in phonetics and phonology books I have access to, and ironically (or perhaps expectedly) the entries on the subject in Trask (1996) are the best I've found:

diphthong /'difthan/ n.
A single syllabic nucleus which begins with one vowel quality and changes more or less smoothly to a second quality, as in [ju] and [ai]. Usually one one of the two vocalic elements is more prominent than the others, this other consisting only of a preceding glide (an on-glide, as in [ju]), or a following glide (an off-glide, as in [ai]). The first type is called a crescendo (or rising) diphthong, the second a diminuendo (or falling) diphthong. Diphthongs may be further classified as wide or narrow, as closing or opening, or as backing, fronting or centring. Cf. monophthong, triphthong, tetraphthong.
NOTE the spelling and the pronounciation of this word. It is an error to use the term `diphthong' in the sense of digraph. Greek diphthonggos `double sound'.

Opening and closing diphthongs

Illustration: Opening diphthongs in Thai
opening diphthong n.
A diphthong whose second element is more open than the first, such as [ja]. Ant. closing diphthong.

It is not mentioned whether the diphthongs in the example are rising or falling, but they are all wide.

Illustration: Closing diphthongs in Am. English
closing diphthong n.
A diphthong whose second element is higher (closer) than its first, such as [ei] or [au]. Ant. opening diphthong.

It is not mentioned whether the diphthongs in the example are rising or falling, but they are all wide.

Fronting, centring and backing diphthongs

Illustration: Centring diphthongs in German
centring diphthong n. (also ingliding diphthong)
A diphthong during whose articulation the tongue moves toward the central vowel schwa, such as the diphthongs [ie], [ue] or [ee] in non-rhotic pronounciations of beer, poor and fair.
Illustration: Fronting and backing diphthongs in Cantonese
fronting diphthong n.
A diphthong in which the second element is farther front than the first, such as [oj]or [wi]. Ant. backing diphthong.
backing diphthong n.
A diphthong in which the second element is further back than the first, such as [ju]or [ew]. Ant. fronting diphthong.

An example of fronting and backing diphthongs in Cantonese. The fronting diphthongs are: ei ai ai åi oy ui. The backing diphthongs are: iu eu au au ou. Furthermore, all but ei and ou are wide.

Rising and falling diphthongs

These terms are found in Lass(1984) and Laver(1994) as well, 'crescendo' and 'diminuendo' diphthongs seems to be unique to Trask. A term Trask seems to have overlooked is 'level diphthong' as used in Laver(1994); neither rising nor falling.

rising diphthong n. (also crescendo diphthong)
A diphthong consisting of an on-glide followed by a more prominent second element, such as [ju] or [we]. Ant. falling diphthong.
falling diphthong n. (also diminuendo diphthong)
A diphthong consisting of a more prominent first element followed by a less prominent off-glide, such as [aj], [aw], [ee] or [oj]. Ant. rising diphthong.

There's no agreed on technique for marking the "weak" vowel (the second in a falling diphthong, the first in a rising diphthong), Lass (1984, chapter 11.2) uses a non-syllabicity diacritic [_^], as in [ai_^] and [a_^i], which is as good as any.

Narrow and wide diphthongs

narrow diphthong n.
A diphthong whose intial and final elements differ little in height, such as [ei] or [ow]. Ant. wide diphthong.
wide diphthong n.
A diphthong involving a large movement between its intial and final elements, such as [ai]. Ant. narrow diphthong.

Some triphthongs

Illustration: Triphthongs in RPIllustration: Some triphthongs

Whether triphthongs (and tetraphthongs) actually exist is also somewhat debated: the issue is, as with diphthongs, what status/difference is there between the glides (semi-vowels) /j/ and /w/ and "true" vowels. Is a sequence of three "vowels" a sequence of three vowels or are one or two of them glides?

The conveniently ignored bits...

The above does not cover the length of the elements of a diphthong; for the a in English {high} /hai/ and for instance Norwegian {hai} /hai/ is not the same.