diff --git a/docs/changelog/1374.feature.rst b/docs/changelog/1374.feature.rst
new file mode 100644
index 0000000000..171911f34b
--- /dev/null
+++ b/docs/changelog/1374.feature.rst
@@ -0,0 +1 @@
+Add support for minor versions with multiple digits ``tox -e py310` works for ``python3.10`` - by :user:`asottile`
diff --git a/docs/changelog/1377.fix.rst b/docs/changelog/1377.fix.rst
new file mode 100644
index 0000000000..2834b85b0a
--- /dev/null
+++ b/docs/changelog/1377.fix.rst
@@ -0,0 +1 @@
+Fix regression failing to detect future and past ``py##`` factors  - by :user:`asottile`
diff --git a/src/tox/config/__init__.py b/src/tox/config/__init__.py
index 849b31f7e7..b5df358339 100644
--- a/src/tox/config/__init__.py
+++ b/src/tox/config/__init__.py
@@ -51,9 +51,6 @@
 Import hookimpl directly from tox instead.
 """
 
-default_factors = tox.PYTHON.DEFAULT_FACTORS
-"""DEPRECATED MOVE - please update to new location."""
-
 WITHIN_PROVISION = os.environ.get(str("TOX_PROVISION")) == "1"
 
 
@@ -549,11 +546,21 @@ def basepython_default(testenv_config, value):
         python conflict is set in which case the factor name implied version if forced
         """
         for factor in testenv_config.factors:
-            if factor in tox.PYTHON.DEFAULT_FACTORS:
-                implied_python = tox.PYTHON.DEFAULT_FACTORS[factor]
+            match = tox.PYTHON.PY_FACTORS_RE.match(factor)
+            if match:
+                base_exe = {"py": "python"}.get(match.group(1), match.group(1))
+                version_s = match.group(2)
+                if not version_s:
+                    version_info = ()
+                elif len(version_s) == 1:
+                    version_info = (version_s,)
+                else:
+                    version_info = (version_s[0], version_s[1:])
+                implied_version = ".".join(version_info)
+                implied_python = "{}{}".format(base_exe, implied_version)
                 break
         else:
-            implied_python, factor = None, None
+            implied_python, version_info, implied_version = None, (), ""
 
         if testenv_config.config.ignore_basepython_conflict and implied_python is not None:
             return implied_python
@@ -561,23 +568,20 @@ def basepython_default(testenv_config, value):
         proposed_python = (implied_python or sys.executable) if value is None else str(value)
         if implied_python is not None and implied_python != proposed_python:
             testenv_config.basepython = proposed_python
-            match = tox.PYTHON.PY_FACTORS_RE.match(factor)
-            implied_version = match.group(2) if match else None
-            if implied_version is not None:
-                python_info_for_proposed = testenv_config.python_info
-                if not isinstance(python_info_for_proposed, NoInterpreterInfo):
-                    proposed_version = "".join(
-                        str(i) for i in python_info_for_proposed.version_info[0:2]
-                    )
-                    # '27'.startswith('2') or '27'.startswith('27')
-                    if not proposed_version.startswith(implied_version):
-                        # TODO(stephenfin): Raise an exception here in tox 4.0
-                        warnings.warn(
-                            "conflicting basepython version (set {}, should be {}) for env '{}';"
-                            "resolve conflict or set ignore_basepython_conflict".format(
-                                proposed_version, implied_version, testenv_config.envname
-                            )
+            python_info_for_proposed = testenv_config.python_info
+            if not isinstance(python_info_for_proposed, NoInterpreterInfo):
+                proposed_version = ".".join(
+                    str(x) for x in python_info_for_proposed.version_info[: len(version_info)]
+                )
+                if proposed_version != implied_version:
+                    # TODO(stephenfin): Raise an exception here in tox 4.0
+                    warnings.warn(
+                        "conflicting basepython version (set {}, should be {}) for env '{}';"
+                        "resolve conflict or set ignore_basepython_conflict".format(
+                            proposed_version, implied_version, testenv_config.envname
                         )
+                    )
+
         return proposed_python
 
     parser.add_testenv_attribute(
diff --git a/src/tox/constants.py b/src/tox/constants.py
index 8dcc86bf8a..c31f2602fe 100644
--- a/src/tox/constants.py
+++ b/src/tox/constants.py
@@ -9,36 +9,11 @@
 _THIS_FILE = os.path.realpath(os.path.abspath(__file__))
 
 
-def _construct_default_factors(cpython_versions, pypy_versions, other_interpreters):
-    default_factors = {"py": sys.executable, "py2": "python2", "py3": "python3"}
-    default_factors.update(
-        {
-            "py{}{}".format(major, minor): "python{}.{}".format(major, minor)
-            for major, minor in cpython_versions
-        }
-    )
-    default_factors.update({exc: exc for exc in ["pypy", "pypy2", "pypy3"]})
-    default_factors.update(
-        {
-            "pypy{}{}".format(major, minor): "pypy{}.{}".format(major, minor)
-            for major, minor in pypy_versions
-        }
-    )
-    default_factors.update({interpreter: interpreter for interpreter in other_interpreters})
-    return default_factors
-
-
 class PYTHON:
-    PY_FACTORS_RE = re.compile("^(?!py$)(py|pypy|jython)([2-9][0-9]?)?$")
-    CPYTHON_VERSION_TUPLES = [(2, 7), (3, 4), (3, 5), (3, 6), (3, 7), (3, 8)]
-    PYPY_VERSION_TUPLES = [(2, 7), (3, 5)]
-    OTHER_PYTHON_INTERPRETERS = ["jython"]
-    DEFAULT_FACTORS = _construct_default_factors(
-        CPYTHON_VERSION_TUPLES, PYPY_VERSION_TUPLES, OTHER_PYTHON_INTERPRETERS
-    )
-    CURRENT_RELEASE_ENV = "py36"
+    PY_FACTORS_RE = re.compile("^(?!py$)(py|pypy|jython)([2-9][0-9]?[0-9]?)?$")
+    CURRENT_RELEASE_ENV = "py37"
     """Should hold currently released py -> for easy updating"""
-    QUICKSTART_PY_ENVS = ["py27", "py34", "py35", CURRENT_RELEASE_ENV, "pypy", "jython"]
+    QUICKSTART_PY_ENVS = ["py27", "py35", "py36", CURRENT_RELEASE_ENV, "pypy", "jython"]
     """For choices in tox-quickstart"""
 
 
diff --git a/tests/unit/config/test_config.py b/tests/unit/config/test_config.py
index 7893d012b7..d3b17970f1 100644
--- a/tests/unit/config/test_config.py
+++ b/tests/unit/config/test_config.py
@@ -2194,26 +2194,24 @@ def test_no_implicit_venv_from_cli_with_envlist(self, newconfig):
         assert "typo-factor" not in config.envconfigs
 
     def test_correct_basepython_chosen_from_default_factors(self, newconfig):
-        envlist = list(tox.PYTHON.DEFAULT_FACTORS.keys())
-        config = newconfig([], "[tox]\nenvlist={}".format(", ".join(envlist)))
-        assert config.envlist == envlist
+        envs = {
+            "py": sys.executable,
+            "py2": "python2",
+            "py3": "python3",
+            "py27": "python2.7",
+            "py36": "python3.6",
+            "py310": "python3.10",
+            "pypy": "pypy",
+            "pypy2": "pypy2",
+            "pypy3": "pypy3",
+            "pypy36": "pypy3.6",
+            "jython": "jython",
+        }
+        config = newconfig([], "[tox]\nenvlist={}".format(", ".join(envs)))
+        assert set(config.envlist) == set(envs)
         for name in config.envlist:
             basepython = config.envconfigs[name].basepython
-            if name == "jython":
-                assert basepython == "jython"
-            elif name in ("pypy2", "pypy3"):
-                assert basepython == "pypy" + name[-1]
-            elif name in ("py2", "py3"):
-                assert basepython == "python" + name[-1]
-            elif name == "pypy":
-                assert basepython == name
-            elif name == "py":
-                assert "python" in basepython or "pypy" in basepython
-            elif "pypy" in name:
-                assert basepython == "pypy{}.{}".format(name[-2], name[-1])
-            else:
-                assert name.startswith("py")
-                assert basepython == "python{}.{}".format(name[2], name[3])
+            assert basepython == envs[name]
 
     def test_envlist_expansion(self, newconfig):
         inisource = """
diff --git a/tests/unit/interpreters/test_interpreters.py b/tests/unit/interpreters/test_interpreters.py
index c713015fe7..31d67ac7ed 100644
--- a/tests/unit/interpreters/test_interpreters.py
+++ b/tests/unit/interpreters/test_interpreters.py
@@ -46,7 +46,7 @@ def assert_version_in_output(exe, version):
 
     p = tox_get_python_executable(envconfig)
     assert p == py.path.local(sys.executable)
-    for major, minor in tox.PYTHON.CPYTHON_VERSION_TUPLES:
+    for major, minor in [(2, 7), (3, 5), (3, 6), (3, 7), (3, 8)]:
         name = "python{}.{}".format(major, minor)
         if tox.INFO.IS_WIN:
             pydir = "python{}{}".format(major, minor)
diff --git a/tests/unit/test_z_cmdline.py b/tests/unit/test_z_cmdline.py
index 20e5d0ce54..f8681f74a6 100644
--- a/tests/unit/test_z_cmdline.py
+++ b/tests/unit/test_z_cmdline.py
@@ -164,26 +164,31 @@ def test_unknown_interpreter_and_env(cmd, initproj):
         "interp123-0.5",
         filedefs={
             "tests": {"test_hello.py": "def test_hello(): pass"},
-            "tox.ini": """
-            [testenv:python]
-            basepython=xyz_unknown_interpreter
-            [testenv]
-            changedir=tests
-            skip_install = true
-        """,
+            "tox.ini": """\
+                [testenv:python]
+                basepython=xyz_unknown_interpreter
+                [testenv]
+                changedir=tests
+                skip_install = true
+            """,
         },
     )
     result = cmd()
     result.assert_fail()
-    assert any(
-        "ERROR: InterpreterNotFound: xyz_unknown_interpreter" == l for l in result.outlines
-    ), result.outlines
+    assert "ERROR: InterpreterNotFound: xyz_unknown_interpreter" in result.outlines
 
     result = cmd("-exyz")
     result.assert_fail()
     assert result.out == "ERROR: unknown environment 'xyz'\n"
 
 
+def test_unknown_interpreter_factor(cmd, initproj):
+    initproj("py21", filedefs={"tox.ini": "[testenv]\nskip_install=true"})
+    result = cmd("-e", "py21")
+    result.assert_fail()
+    assert "ERROR: InterpreterNotFound: python2.1" in result.outlines
+
+
 def test_unknown_interpreter(cmd, initproj):
     initproj(
         "interp123-0.5",