Add list of bright Rahn normal forms

setclass/setclass.py

@@ -290,3 +290,18 @@ class SetClass(list):
         minute on an Intel i7-13700H CPU).
         """
         return set(i.rahn_normal_form for i in SetClass.all_of_cardinality(cardinality, tonality))
+
+
+def bright_rahn_normal_forms():
+    """
+    John Rahn's normal form from _Basic Atonal Theory_ (1980) is an algorithm to produce a unique
+    form for each set class. Most of the time it is also the "darkest" (smallest brightness value),
+    except for all the times when that is not the case; this function returns that list, as a set of
+    (cardinality, darkest, rahn) tuples.
+    """
+    cases = set()
+    for C in range(13):  # 0 to 12
+        for sc in SetClass.all_of_cardinality(C):
+            if sc.darkest_form.brightness < sc.rahn_normal_form.brightness:
+                cases.add((C, sc.darkest_form, sc.rahn_normal_form))
+    return cases

setclass/tests/test_exploration.py

@@ -1,4 +1,4 @@
-from setclass.setclass import SetClass
+from setclass.setclass import SetClass, bright_rahn_normal_forms
 
 
 def test_brightness_conjecture():
@@ -19,3 +19,43 @@ def test_brightness_conjecture():
                 test = sc.complement.inversion.darkest_form.brightness - sc.brightness
                 ΔB = int((T - 1) * (T / 2 - C))
                 assert test == ΔB, f"T={T}, C={C}, ΔB={ΔB}, diff={test}: {sc.darkest_form}, {sc.complement.inversion.darkest_form}"
+
+
+def test_bright_rahn():
+    """
+    There are 63 instances where the Rahn normal form is not also the "darkest" (smallest sum of
+    pitch class values).
+    """
+    R = bright_rahn_normal_forms()
+    assert len(R) == 63
+    counts = {
+        0: 0,
+        1: 0,
+        2: 0,
+        3: 1,
+        4: 3,
+        5: 17,
+        6: 7,
+        7: 23,
+        8: 9,
+        9: 3,
+        10: 0,
+        11: 0,
+        12: 0
+    }
+    for C in range(13):  # 0 to 12
+        bright_rahns = set(i for i in R if i[0] == C)
+        assert len(bright_rahns) == counts[C]
+
+
+def print_bright_rahn():
+    """
+    Convenience function, to print the list of set classes where John Rahn's normal form is not the
+    "darkest" form (that with the smallest brightness value), as pairs of "darkest" and Rahn normal
+    form pairs, using Leonard notation, sorted by cardinality then brightness (sum of pitch class
+    values).
+    """
+    print("Smallest brightness, Rahn normal form")
+    set_classes = sorted(bright_rahn_normal_forms(), key=lambda x: x[0] * 1000 + x[1].brightness)
+    for (C, d, r) in set_classes:
+        print(f"{d.leonard_notation}, {r.leonard_notation}")