There are 495 ways to traverse the octave in 4 steps.

1. Ab A Bb B
2. C A Bb B
3. C Ab A B
4. C Ab A Bb
5. C Ab Bb B
6. C D A B
7. C D A Bb
8. C D Ab A
9. C D Ab B
10. C D Ab Bb
11. C D Bb B
12. C D E A
13. C D E Ab
14. C D E B
15. C D E Bb
16. C D E F
17. C D E G
18. C D E Gb
19. C D Eb A
20. C D Eb Ab
21. C D Eb B
22. C D Eb Bb
23. C D Eb E
24. C D Eb F
25. C D Eb G
26. C D Eb Gb
27. C D F A
28. C D F Ab
29. C D F B
30. C D F Bb
31. C D F G
32. C D F Gb
33. C D G A
34. C D G Ab
35. C D G B
36. C D G Bb
37. C D Gb A
38. C D Gb Ab
39. C D Gb B
40. C D Gb Bb
41. C D Gb G
42. C Db A B
43. C Db A Bb
44. C Db Ab A
45. C Db Ab B
46. C Db Ab Bb
47. C Db Bb B
48. C Db D A
49. C Db D Ab
50. C Db D B
51. C Db D Bb
52. C Db D E
53. C Db D Eb
54. C Db D F
55. C Db D G
56. C Db D Gb
57. C Db E A
58. C Db E Ab
59. C Db E B
60. C Db E Bb
61. C Db E F
62. C Db E G
63. C Db E Gb
64. C Db Eb A
65. C Db Eb Ab
66. C Db Eb B
67. C Db Eb Bb
68. C Db Eb E
69. C Db Eb F
70. C Db Eb G
71. C Db Eb Gb
72. C Db F A
73. C Db F Ab
74. C Db F B
75. C Db F Bb
76. C Db F G
77. C Db F Gb
78. C Db G A
79. C Db G Ab
80. C Db G B
81. C Db G Bb
82. C Db Gb A
83. C Db Gb Ab
84. C Db Gb B
85. C Db Gb Bb
86. C Db Gb G
87. C E A B
88. C E A Bb
89. C E Ab A
90. C E Ab B
91. C E Ab Bb
92. C E Bb B
93. C E F A
94. C E F Ab
95. C E F B
96. C E F Bb
97. C E F G
98. C E F Gb
99. C E G A
100. C E G Ab