Commonly I've seen psychedelic rock starts with The Doors and others, but is there any other "psychedelic-something" (i.e. "psychedelic jazz") genre? What are the examples (of the psychedelic-something band's song)?
There is certainly a psychedelic funk genre, or so the creators of the band Funkadelic, a portmanteau of "funk" and "psychedelic", believed. They released three albums during their actual psychedelic funk phase, which was in the early 1970s (they later moved on to funk rock in the mid-1970s and straight funk by the late 1970s): Funkadelic, Free Your Mind And Your Ass Will Follow, and Maggot Brain. Their most well known songs from that period are:
- "Maggot Brain"
- "Can You Get To That", which you might recognize as being sampled in "Rill Rill" by Sleigh Bells", a song that was in an iPhone 5c commercial a couple of years ago.
- "I Bet You", which later covered by The Jackson 5
Some other songs of theirs from that era that trend heavily toward psychedelic funk:
- "Mommy What's A Funkadelic"
- "I Wanna Know If It's Good To You?"
- "Free Your Mind And Your Ass Will Follow"
- "Back In Our Minds"
As for psychedelic jazz in particular, that's regularly referred to as acid jazz. (Try searching for "psychedelic jazz" on Wikipedia.) I'm not too familiar with the genre, but The Brand New Heavies, Jazzanova, James Taylor Quartet, and Jamiroquai all seem to embody the sound. A song for each:
- "Brother Sister" by The Brand New Heavies
- "Another New Day" by Jazzanova
- "In The Hand Of The Inevitable" by James Taylor Quartet
- "Talullah" by Jamiroquai
For more genres and other info about Psychedelic music, see the Psychedelic Music Category page on Wikipedia.
There's also psychedelic folk, from Simon & Garfunkel and the Byrds in the 60's through to modern groups like Edward Sharpe and the Magnetic Zeros, and the Shins.
- Leaves That Are Geen - Simon & Garfunkel
- Mr Tambourine Man - The Byrds
- 40 Day Dream - Edward Sharpe & The Magnetic Zeros
- Saint Simon - Shins
Let's also not forget psychedelic soul (which overlaps with both psychedelic rock and psychedelic funk) --it was basically the black musicians' version of the psychedelic rock movement. Notable exponents of the sound include the (later) Temptations, Stevie Wonder, the Isley Brothers, the Chambers Brothers and the 5th Dimension. Modern singers such as Erykah Badu and Janelle Monae are arguably in the same general genre.
- Cloud Nine - Temptations
- I Believe - Stevie Wonder
- Voyage to Atlantis - Isley Brothers
- Time - Chambers Brothers
- Aquarius - 5th Dimension
- Bag Lady - Erykah Badu
- Prime Time - Janelle Monae
There is even psychedelic hip hop, which arguably reached its peak with the trippy rhymes of the 90's duo PM Dawn, but which has recently made a mainstream comeback in the music of Lupe Fiasco, Kid Cudi, Kendrick Lamar and others.
- Set Adrift on Memory Bliss - PM Dawn
- Gone Till November - Wyclef Jean
- B.O.B. - Outkast
- Provider - N.E.R.D.
- Sunny Day - Big Tymers
- Daydreamin - Lupe Fiasco
- Pursuit of Happiness - Kid Cudi
- i - Kendrick Lamar
Wikipedia also lists psychedelic pop (Beach Boys, Lemon Pipers) and of course there were the psychedelic revivals both in the 80/90's (XTC, Tears For Fears) and 2000's-present (Flaming Lips, Tame Impala), but those can all probably be considered as part of psychedelic rock.
Psy-chill (also known as psy-dub) is a psychedelic version of Chill-out Music (described by Wikipedia as "an umbrella term for several styles of electronic music characterized by their mellow style and mid-tempo beats", including "ambient, trip hop, nu jazz, ambient house and other subgenres of downtempo").
A psy-chill act like Shpongle would be the very definition of psychedelic music (at least for me). This is a project of Raja Ram and Simon Posford, both big names in the goa-trance scene and Shpongle does lean on that subculture, but is way more then just electronics. It is impossible to list one album to catch the magnificence of it, but this is a very good one nonetheless. It incorperates soundscapes, world music, dub, tribal influences to even salsa, hiphop and hardrock, all drenched in a fat layer of psychedelica.
This act, to answer your question, is a head figure in the psychedelic chill-out scene. And that scene has a very, very big psychedelic genre. Ranging from jazzy to trancy, to list a few: Koan, Ott, Kaya Project, The Mystery of the Yeti and so on. If you're into this, check out of the parties (like Ozora, Boom Festival, Psy-Fi, Lost Theory) - you're in for a treat.