Classical (and baroque, romantic, neoclassical, etc.) is an odd genre to a Christian. With modern music, we can easily filter out secular songs by their lyrics or artists. For some Christians, the extent of their musical tolerance is the year’s Wow Hits list. However, Christians should not and are not limited to expressly-purposed worship songs. David had no Christian lyrics to his music but was able to calm the demons in Saul. Psalm 150 describes giving praise to G-d with an entire orchestra of instruments.
Consider other forms of fine art. As a Christian, are you only allowed to admire the Last Supper as it paints an explicit Christian scene? The Great Wave off Kanagawa features no biblical vignette, but doesn’t the Bible use waves and the ocean as a direct example of G-d’s power?
Now, not every painting is an ode to G-d or His creation. Likewise, not every piece of music will be a pure representation of emotion using the rules that G-d created for auditory self-expression. For paintings, we judge the finished work. Lewd or disturbing paintings are not the best idea for a Christian. What do we do for classical music?
Music has the fantastic quality of reflecting the composer. We see this often nowadays. A singer experiences a tough break-up, and the next hit single reflects either the despair of a lost relationship or the elation of, “I’m free and I didn’t need you anyway!” We know from stories in the Bible that music can be used to affect the spiritual. When written by the pure of heart, it calms demons. It may be safe to assume that music written by those with evil intent could have an undesirable spiritual effect. This is why Christians tend not to listen to rock n’ roll or music by immoral artists.
Then the answer becomes clear. We can approve classical pieces by vetting the composers. Luckily, in those times, well-educated composers tended to grow up and work in Christian realms. Many pianists had their start as organists in churches. There are not many composers who did not meet the moral requirements of their time periods. Still, it is important to make sure that the composer believed in G-d and followed His teachings in their life.
I have compiled an incomplete list of composers that meet the cut and those to whom I do not listen. Feel free to do your own research and contest any judgment I have made here if a different reasoning presents itself.
Yes:
Bach
Beethoven
Brahms
Bruckner
Dvorak
Elgar
Handel
Haydn
Liszt
Mendelssohn
Messinaen
Mozart
Prokofiev
Rachmaninoff
Saint-Saëns
Salieri
Schubert
Schumann
Stravinsky
Telemann
Vivaldi
Weber
No:
Chopin
Debussy
Mahler
Rimsky-Korsakov
Shostakovich
Tchaikovsky
Wagner
Helpful resources:
Also, biographies of each composer.