In Celtic, West African and electronic musical traditions, harmony is rarely conventional in a classical sense. The harmony is usually very static because it stays in the same place, often underpinned by a drone. This is the case in many world music genres.
Listen to "Hey Boy Hey Girl" by The Chemical Brothers and you will hear a drone on D. It is often accompanied by a riff, notated below, which just moves chromatically between a D and an E flat (listen from the drop at 0'49) and later by another idea that moves between an A and a G (from 1'53). These ideas don't contribute to any functional harmony but definitely add to the exciting, mesmerising atmosphere by remaining simple and static, allowing people on the dance floor to get really immersed in the music.
Just like the EDM example above, the harmony in Release is very simple and static. It is ultimately centred around C. Here are the main features:
A drone of C is heard. If you look at loop 11, 16, 21 and 22, they all have a sustained note of a C. The drone first fades in at about 0'19 and by 0'27 it is very audible. This then virtually underpins the entire song. This is particularly common practice in Celtic music and EDM. It helps provide a stable and calming basis for the song's harmonic fabric.
A C minor chord is heard at various times and is probably the most prominent triad in the song. For example, during the low whistle reel at 4'11 the accordion can be heard playing a C minor chord very rhythmically (loop 17). This is very harmonically static and this unobtrusive use of harmony allows the Celtic reels to come across clearly.
Confusingly, a C major chord is sometimes heard too! Looking at loop 11, you can see the synth pad sustaining an E natural. Listen to the male vocalist's verse at 2'54 and you will hear the synth bells being played (loop 4). At the start of every 4 bars is a high E natural that is very resonant. It is quite hard to make out but it is there! It gives the music more depth and harmonic complexity.
Other harmonies are created in the song but they can't really be seen/heard as triads in a conventional sense. In one slow moving synth strings loop (loop 10) there is a low G which moves up chromatically over 8 bars, playing A flat, A and B flat. Each of its notes creates a different harmonic interval with the C drone and adds to the profundity and perhaps spiritual atmosphere of the overall soundscape. This can be heard at 3'33 for the build up section. The bass drops out and this chromatic line comes out distinctly, giving an inspirational and uplifting feel. It can also be heard at 5'33, with the fiddle playing it with a double stopped rhythmic pattern (loop 23).
At 6'11 in the song (and at the very end) there are a number of electric piano loops heard simultaneously and this is perhaps the most harmonically complex section of the entire song because there are so many notes heard together, including C, D, E flat, F, G and B flat (loop 25). It almost has a hypnotic quality to it.
There are no cadences used at all. A cadence is a harmonic device that is used to establish a functioning key, so they do not work in this musical context.