1. Mozarabic Chant
All Medieval plainsong was originally written in neumes, but after Guido d'Arezzo developed the four-line staff in around 1026 CE, monks notated Gregorian and Ambrosian chants in the new system. For good reason--neumes were imprecise, so much so that without an unbroken performance tradition, we can't interpret them. Mozarabic chant developed in Spain around the same time as Gregorian chant, and from what we can tell, it had many unique characteristics. Yet, though we have thousands of Mozarabic chants, they're only written in neumes, so this wealth of music is lost to history.
2. Music by Johann Christoph Friedrich Bach
J.S. Bach's youngest son was a prolific composer and keyboard virtuoso. While he was overshadowed by his illustrious father and famous brothers, he still composed over twenty symphonies, significant keyboard works, and an extensive collection of vocal music. Tragically, much of his music was lost or destroyed during WWII, when the State Institute for Music Research in Berlin was bombed by the allies. While the Germans evacuated some of the music, the lack of security meant most of it disappeared, along with thousands of antique instruments.
3. Music Lost in Concentration Camps
During the Holocaust, Nazis ruthlessly suppressed modern music as well as music by Jewish composers. Music by composers like Viktor Ullmann, a Jewish composer who died in Auschwitz, was burned, suppressed, or simply lost in the chaos of war. The terrible loss of Jewish composers during the Shoah meant that many promising careers were cut violently cut short, leaving us bereft of their musical gifts forever.
4. Claudio Monteverdi's Lost Operas
Claudio Monteverdi composed one of the first operas, L'Orfeo, which is the earliest opera that's frequently performed today. Yet, though Monteverdi composed a total of eighteen operas, only three of them survive intact today. Many of his most beloved works from this time period, including his L'Arianna, are lost (apart from a few arias). In this case, it wasn't war or political suppression that destroyed Monteverdi's great works: it was a lack of regard. People at the time preferred new music to repeat performances, so once an opera's performance was over, the music was disregarded, with maddening results for music historians everywhere.
5. Shostakovich's Suite for Jazz Orchestra No. 2
This is another piece of music that was originally lost in WWII. Unlike most of the other pieces on this list, however, a piano score of the work was found in 1999, so at last some of the original music remains.