Charlie, I once read a claim by a musicologist that the Gregorian chants were taken from the tunes the Levites used in the Beit haMikdash, which are true Jewish music. In any case, Rav Moshe says that modern dress cannot be classified as Jewish or non-Jewish as they are made for everyone. Probably the same can be said for tunes. What matters is the words.
Shmil. even if the words are a translation of some verse in the Torah if there is a secular intent I would not classify it as Jewish. Similarly, someone who writes a name of Hashem with a secular intent does not write a shem kadosh. There is a joke about a boy and girl who meet fr a shidduch. The girl says “My name is Batka”. The boy says “Mine is Kelikaku”.