Here is the full list of all 31 Symphony achievements. It takes around 20-25 hours to unlock all of the achievements on Windows.
Symphony No. 5 in C minor, Op. 67 (hint: 67 is the key)
"A note of shortened duration separated from the note that may follow by silence" - Find and equip the rare Sonic weapon
"work" or "labour"; a number generally assigned by composers to an individual composition or set of compositions on publication, to help identify their works - Finish the game
"A title of extreme respect given to a master musician" - Reach the 4-gold star rating on 10 songs
"A gradual increase in volume of a musical passage" - Find and equip the rare Charging weapon
"Bang the piano with the fist" - Get 100% completion and over 30,000 points at the Fortissimo difficulty
"Always" - Reach 30,000 points and finish without dying at the Fortissimo difficulty
"Performing with exceptional ability, technique, or artistry" - Perfect game (100% + no death) and over 30,000 points at the Fortissimo difficulty
Unlock Fortissimo difficulty
"Silent; do not play" - Get 0% completion
Unlock Forte difficulty
St Matthew Passion, BWV 244
"Warlike, bellicose, belligerent or combative" - Get 100% completion and over 30,000 points at the Mezzo-forte difficulty
"With fire / in a fiery manner" - Perfect game (100% + no death) and over 30,000 points at the Mezzo-forte difficulty
"Leads a musical ensemble by way of visible gestures" - Reach the 4-Silver star rating on 10 songs
Upgrade a weapon to level 5
"Obstinate, persistent; a short musical pattern that is repeated throughout an entire composition" - Reach 30,000 points and finish without dying at the Mezzo-forte difficulty
Unlock Mezzo-forte difficulty
"With spirit / vigor" - Get 100% completion and over 30,000 points at the Piano difficulty
"Ignited / on fire" - Perfect game (100% + no death) and over 30,000 points at the Piano difficulty
"Leader of the first violin section" - Reach the 4-bronze star rating on 10 songs
"A large instrumental ensemble that contains sections of string, brass, woodwind, and percussion instruments" - Equip 4 different weapons simultaneously
Unlock Mezzo-piano difficulty
Defeat the first boss
"Sustained, lengthened" - Reach 30,000 points and finish without dying at the Piano difficulty
Equip a new weapon
Reach a score target
"A group of people who perform instrumental or vocal music" - Perform a 10X chain
"A group containing five members" - Perform a 5X chain
Play 840 minutes
Symphony No. 104 in D major (Hob.I:104)