Harmony

Sing! Day of song - Bobby McFerrin - Improvisation - A large crowd provides accompaniment for one of the finest vocal masters of popular music.
Jacob Collier Harmonises The Audience [Toronto] - Collier takes this idea further by harmonizing the crowd in 3 parts. Simple, but large scale and spontaneous.
Jacob Collier - Moon River - 4725 part harmony /hyp
Lizzy McAlpine: Tiny Desk Concert - The backing band here, Tiny Habits, has some of the tightest harmonies I've ever heard in popular music. Lots more available on Youtube etc.
Choral - Solo accordion, but lush chords and amazing gravitas. The ultimate track of a superb album.
Ryuichi Sakamoto - "andata" (from "async") - Another instrumental track, gravitas out the nose here.
MICROTONAL TETRIS - Magical harmonies and modulations that you may not have experienced before.
Vicentino: Madonna il poco dolce - Speaking of microtonal music, Vicentino's microtonal works may be of interest if you like Renaissance music.

Try these on for size, I'm curious which ones align with your request.

Following on from toshi's post, here's a little variety of things:

• Thomas Tallis - Spem in Alium -- from the Renaissance period, a choral piece written for 40 voices. This is just a single example; for my tastes, pretty much any Tallis is worth listening to.
• Giovanni Pierluigi da Palestrina - Missa Papae Marcelli -- As with Tallis, I think you can't go wrong with any Palestrina.
• Gregorio Allegri - Miserere mei, Deus -- a Baroque piece that's "stylistically conservative" so it has a lot in common with Renaissance styles. It's repetitive and not nearly as complex as the Tallis (for example), but I find it positively transcendental. I saw a live performance several years ago and would have been happy if it just repeated forever.
• Sergei Rachmaninoff - Divine Liturgy, Op. 31, Mvt. 18, "Let Our Mouths Be Filled -- There's a ton of beautiful Russian sacred choral music out there. This is just a sample that I'm partial to, as one of the many voices in this particular recording is mine.
• Robert Schumann - An Die Sterne -- Schumann is probably not best known for his choral music, but this is a gorgeous song for double choir, full of longing.
• Blind Boys of Alabama - Amazing Grace (House of the Rising Sun) -- an interesting arrangement of a traditional spiritual.
• Carbon Leaf - Gloryland -- made famous by Ralph Stanley.
• Great Big Dea - Old Brown's Daughter -- There are quite a few good examples from Great Big Sea that I could have picked, but this one is probably less known than some other others.

I'd love to know if you like any of these!

Some pop examples that come to mind just off the top of my head:

• Bon Iver's album 22, A Million is awash in vocal harmonies. "22 Over Soon" and "715 Creeks" are probably the most striking examples.
• Of Montreal's Gronlandic Edit uses harmony excellently as a dramatic device.
• Mother Falcon's cover of Radiohead's Lucky.

Most of the music I listen to features harmony, and there are quite a few choral tracks in there too. Rather than list a bunch of them, I’ll recommend one The Beginning and the End.

There is a certain degree of imperfection to the song that I have a hard time articulating; it’s almost as if some of the notes are off timing or pitch. But that is precisely the thing that makes the song so powerful. It’s just so incredibly human. It’s quite literally the only choral recording I’ve ever heard that does this. As a side effect if you are listening to it alongside all of these others you will probably not quite get it. But when you are able to listen to it on its own terms it will move you.

There's one song that immediately comes to mind when I hear the phrase "harmonies that create emotional response", Veil of Oblivion by Erang.

It's an incredibly simple track, 2 synths - only one at a time - and some sound effects. It's basically carried by the way the sound design and story interacts with the harmony (which is perhaps not what you're asking for, but I think it still counts). The harmony is almost a lullaby, mostly warm and comforting, but there's an atmosphere of longing as the progression... progresses. That longing is basically forced into the limelight by the sound design, a muted bass robs you of any warmth that the harmony could provide, and the remaining airiness takes over. The tone of the harmony is completely different when using the synth it does compared to if it had been played with a violin or some deeper instrument. The track isn't only cold, (to me) it embodies the feeling of becoming cold (I don't mean this in a metaphorical sense, I literally mean temperature-wise).

Then the two switch places. In the last minute of the song the second synth kicks in, basically the exact opposite of the first - bassy and muffled, the reverb further filling out the lows. It would be warm if it weren't for the harmony, which refuses to resolve, instead fading out without providing any relief. There's not a single moment of peace across the whole 4:05, despite it being almost within grasp.

The album art and title (Another World, Another Time) also paint a pretty good picture of the tone of the track, along with the context for the rest of the album. The first track in the album has an accompanying poem:

There are ages that left the memories of men


Like a timeworn stone under the rain.


And amongst them, one is lost forever: Far as the oldest stars, its light has gone out of our world…

Veil of Oblivion so perfectly captures the apocalyptic yearning that I'm basically forced to stop and listen with all my focus anytime it comes on.

This just turned into me rambling about one of my favorite songs, but it really was the immediate thing I thought of when I read your post, so I hope this has at least somewhat contributed to your curiosity :)

Here's a few that I go back to a lot that might fit the bill:

• The Big Ship, by Brian Eno
• The Bulgarian folk song Kaval Sviri
• Farewell, by Talisk
• Hide and Seek, by Imogen Heap.