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Those two bars catch my attention and I started analysing what is making it to sound so unique and pleasant. And I want to extract from it the universal bricks for the relations of music components. (It's a fragment from Dareka no Shinzou ni nareta nara)

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All those notes are within the key, so no surprises here. The tempo -Allegro- adds a feel of a gallop. e.g. enter image description here looks like a triad with the root on G# but with a third missing, but it still sounds "happy", and as I know, the third is defining the quality of the chord. Simply adding root note on another octave makes it sound "happy"?

For the bass line the progression is as follows: vi IV V iii, but I don't quite understand its relation to the melodic line, sometimes it covers same notes on different octaves, sometimes not but still it sounds good, "homogeneous". What techniques for analysing that kind of relationships are good for the piece? Checking the pitch distance between them?

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I would rather analyse the progression in G♯-minor, not B-major. Then it's

5 -  | sus9-sus4 - -/7add4

This is quite similar to - | - 7 which I'd consider a bit of a film-music-cliché progression. Omitting the minor tonic's third is common for getting a minor key's dramatic properties without bringing in the melancholy ones. The , which stubbornly stays on its suspensions instead of going major for the folky minor resolution - or to the relative major as F♯-B, has a similar effect.

The final chord is quite a bit odd, with the left hand clashing both the G♯ and the F♯ against the right hand's F×. Basically this chord is simultaneously minor (corresponding to the diatonic natural-minor scale) and major (harmonic-minor leading tone) and sus4 (which would normally come before de-suspending to the major third, which does indeed happen in the right hand but then the left echoes underneath that third). Adding a minor third, or ♯9 as it's often called, to a dominant seventh chord is common in jazz and blues, but it sticks out much more in the context of all those open-sounding fifth-only chords. In a fast tempo this may not actually be so noticeable though – essentially you only hear the minor chord from the harmony side, but the outer melody voices go in with the leading tones. So this is a bit of a mixture between modal and contrapuntal ideas.

The, for the most part, harmonic simplicity/openness evokes something of an archaic feeling, while the doubled octaves simply add power to the sound. All of it also gives quite a lot of room for the player to shape the outcome with performance details.

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