I have no clue what these even mean.
3rd tone? 6th tone? etc?
Thanks in advance.
The Byzantine tradition, like the Syrian, makes use of 8 Tones. Each of these represent a different musical scale around which the melodies for hymnody is based.
If you look at the Book of Eight Tones (Octoechos), which is used primarily at Matins and Vespers, each day of the week has a particular theme. Sunday, as you know, is the day of the Resurrection, on Monday we celebrate the Holy Angels, on Tuesday St. John the Baptist, on Wednesday and Friday the Cross, Thursday the Holy Apostles, and on Saturday the departed.
For each of those daily themes, the Octoechos contains eight different sets of hymns - one for each tone. Every week, a different tone is used. In order to accommodate these, many of the psalms and hymns that do not vary depending on the day of the week are sung according to the tone of the week. Lord, I have cried
at Vespers, for example, will therefore have 8 different distinct melodies in order to match what's being sung from the Octoechos.
Hymns from the Menaion, Triodion or Pentacostarion, which are only used on one particular day of the year, will only make use of a single melody. Many of these hymns will use the same melody (these model melodies are called prosomia), and so the tone will be indicated above the text in order for you to know in which tone to sing it, since it generally will not correspond to the tone of the week.
I hope I haven't confused you.