The Jewish perpetual calendar was arranged in a manner that ensures that the first day of Rosh Hashanah will never fall on Sunday, Wednesday or Friday. This guarantees that Yom Kippur will not fall on a Friday or Sunday (i.e. Saturday night), which would produce two consecutive days when preparing food and burying the dead is prohibited, and that Hoshanah Rabbah will not occur on Shabbat, which would interfere with the custom of taking the willows on this day.
The Rebbe explains that, kabbalistically speaking, the inner workings of Rosh Hashanah just don't jibe with the spiritual energies of these three days.
G‑d created the world using seven modalities. We call these sefirot, with the two most prominent being chesed (kindness) and gevurah (severity; discipline). The other sefirot derive from one or both of these.
Each of the seven days of the week corresponds to one of the sefirot. Sunday is the day when chesed is dominant. Wednesday relates to netzach, which stems from kindness. Friday is yesod, one of whose primary functions is the "sweetening" of gevurah. (Click here for more on the sefirot.)
So what's wrong with Rosh Hashanah falling on a day that's about kindness? Sounds like a pretty good idea...
Rosh Hashanah, however, is not really about undeserved kindness. It's about discipline and work on our part. On Rosh Hashanah we crown G‑d as our King. We say, "G‑d, we want You to rule over us," and that's what makes Him keep the world in existence for another year. G‑d sustains the world not out of pure kindness and generosity, but because we present ourselves as His subjects and sincerely accept upon ourselves his sovereignty, and so He consents to being our ruler. It all depends on our service. No Divine service on our part, no reason for G‑d to keep the world going. (For more on this, see The Kabbalistic Spin on Rosh Hashanah.)
Were Rosh Hashanah a day when G‑d says to Himself, "I feel like giving more to this world, just giving freely and openly, without reason or requiring anything in return," then Sunday, Wednesday or Friday would be a perfect match. But considering the real theme of the holiday, those days don't work.
Interestingly, the very first Rosh Hashanah, the day when Adam and Eve were created, was on a Friday. Though it may seem a bit strange that Rosh Hashanah can never reoccur on its original day of the week, it actually makes lots of sense considering the context.
That creation of Adam and Eve – which constituted the finishing touch of creation – on that first Rosh Hashanah, was on act of undeserved kindness on G‑d's part, considering that there was no one yet around to earn any kindness. Hence it is understandable why it fell on a kindness-oriented day. But that aspect of Rosh Hashanah didn't carry over to future years.
Let me know if this helps,
Malkie Janowski for Chabad.org