GA /θ/ is a voiceless dental fricative: the tip of the tongue forms a light contact with the inner edge of the upper front teeth while resting on the cutting edge of the lower front teeth. There is a firmer contact between the rims of the tongue and the upper side teeth and gums. Since /θ/ is a slit fricative, the blade and tip of the tongue are flat and are shaped so as to leave a narrow passage with the upper front teeth through which the air escapes with weak friction. The soft palate is raised and the glottis is wide open.
GA /ð/ is a voiced dental approximant: the tip of the tongue makes a light contact with the inner edges of the upper and lower front teeth. The vocal cords are vibrating and the soft palate is raised. The dental contact is light and brief so that little or no friction occurs. The diacritic [ ̞] indicates a frictionless pronunciation: [ð̞].
In more formal and/or deliberate styles of speech and in singing, /ð/ may be realized as a voiced fricative. The tongue tip is held as for GA /θ/: It forms a narrowing with the upper front teeth while resting on the cutting edges of the lower front teeth, with the body of the tongue held relatively flat. The contact between the tip and the teeth will be weaker, though, than for [θ].
An interdental fricative is produced when the tip of the tongue is allowed to protrude between the teeth. This type of realization is not unusual in emphatic styles and in singing.
Interdental [θ] is usually the easiest realization to learn, so try pronouncing initial and final /θ/ in words like thin and myth by putting the tip of the tongue between the teeth, at the same time allowing the air from the lungs to escape gently between the tip and the front teeth. It is also a good idea to try holding this sound [θθθθ] to familiarize yourself with the feel of the dental articulation. Once you can pronounce interdental dental [θ] with confidence, you should try to pronounce the more common post-dental variety. Always make sure that the tongue is flat, to avoid [s]-like sounds, and not too tense, to avoid a complete obstruction of the airstream.
GA /ð/ is usually an approximant and should therefore be approached as such. Try to pronounce words like the, though by making a very light tap with the tip of the tongue against the insides of the upper and lower front teeth. The fricative realization should be used in relatively careful styles only. It is similar to that of /θ/, but the contact is even weaker. Note that in sequences like in the, at the, all the, alveolar /n,t,l/ will be dental and /ð/ will generally be very weak and pronounced without friction: In an attempt to pronounce a (post-)dental /θ/, non-native speakers may pronounce [f͜θ], a co-articulated labio-dental and dental fricative. Although the sound that is produced in this way is usually quite acceptable, the articulation is unnecessarily cumbersome. It also looks a lot like TH-fronting, which is socially marked because of its association with the speech of children and non-standard speakers, and therefore better avoided. Native speakers may use [f͜θ], as the realization of /fθ/ in fifth and twelfth, which are more frequently pronounced with elision of /f/: /fɪθ, twɛlθ/. If you find you do use [f͜θ], try to relax the lower lip to stop it touching the upper teeth, if necessary by holding it in place with a pencil.
Here are some words to practice the articulation of /θ/ and /ð/:
Initial /θ/ thin, thick, thumb, thorn
Final /θ/ myth, smith, math, both
Medial /θ/ method, author, ether, Cathy
Initial /[ð̞] this, these, though, then
Final [ð̞] seethe, breathe, loathe, smooth
Medial [ð̞] worthy, leather, either, breathing
Note that the approximant realization of (final) /ð/ is not subject to final devoicing. Compare the following pairs with a final voiceless dental fricative versus a final voiced dental approximant. As always, the burden of the opposition is carried by the duration of the vowel or diphthong preceding the final obstruent.
final [θ] final /ð/