diff --git a/doc/source/whatsnew/v2.2.0.rst b/doc/source/whatsnew/v2.2.0.rst
index 8cb4b3f24d435..85dfc3ebde873 100644
--- a/doc/source/whatsnew/v2.2.0.rst
+++ b/doc/source/whatsnew/v2.2.0.rst
@@ -195,6 +195,7 @@ Other enhancements
 - Allow passing ``read_only``, ``data_only`` and ``keep_links`` arguments to openpyxl using ``engine_kwargs`` of :func:`read_excel` (:issue:`55027`)
 - DataFrame.apply now allows the usage of numba (via ``engine="numba"``) to JIT compile the passed function, allowing for potential speedups (:issue:`54666`)
 - Implement masked algorithms for :meth:`Series.value_counts` (:issue:`54984`)
+- Improved error message that appears in :meth:`DatetimeIndex.to_period` with frequencies which are not supported as period frequencies, such as "BMS" (:issue:`56243`)
 - Improved error message when constructing :class:`Period` with invalid offsets such as "QS" (:issue:`55785`)
 
 .. ---------------------------------------------------------------------------
diff --git a/pandas/core/arrays/period.py b/pandas/core/arrays/period.py
index 5cdb30eeb2fed..a8c21cfbb6e2f 100644
--- a/pandas/core/arrays/period.py
+++ b/pandas/core/arrays/period.py
@@ -1179,7 +1179,7 @@ def dt64arr_to_periodarr(
     except (AttributeError, TypeError):
         # AttributeError: _period_dtype_code might not exist
         # TypeError: _period_dtype_code might intentionally raise
-        raise TypeError(f"{(type(freq).__name__)} is not supported as period frequency")
+        raise TypeError(f"{freq.name} is not supported as period frequency")
     return c_dt64arr_to_periodarr(data.view("i8"), base, tz, reso=reso), freq
 
 
diff --git a/pandas/tests/indexes/datetimes/methods/test_to_period.py b/pandas/tests/indexes/datetimes/methods/test_to_period.py
index aa217e895c30a..2c68ddd3d3d15 100644
--- a/pandas/tests/indexes/datetimes/methods/test_to_period.py
+++ b/pandas/tests/indexes/datetimes/methods/test_to_period.py
@@ -230,3 +230,11 @@ def test_to_period_nofreq(self):
         idx = DatetimeIndex(["2000-01-01", "2000-01-02", "2000-01-03"])
         assert idx.freqstr is None
         tm.assert_index_equal(idx.to_period(), expected)
+
+    @pytest.mark.parametrize("freq", ["2BMS", "1SME-15"])
+    def test_to_period_offsets_not_supported(self, freq):
+        # GH#56243
+        msg = f"{freq[1:]} is not supported as period frequency"
+        ts = date_range("1/1/2012", periods=4, freq=freq)
+        with pytest.raises(TypeError, match=msg):
+            ts.to_period()