We enhance constrained-based data quality with approximate band conditional order dependencies (abcODs). Band ODs model the semantics of attributes that are monotonically related with small variations without there being an intrinsic violation of semantics. The class of abcODs generalizes band ODs to make them more relevant to real-world applications by relaxing them to hold approximately (abODs) with some exceptions and conditionally (bcODs) on subsets of the data. We study the problem of automatic dependency discovery over a hierarchy of abcODs. First, we propose a more efficient algorithm to discover abODs than in recent prior work. The algorithm is based on a new optimization to compute a longest monotonic band (longest subsequence of tuples that satisfy a band OD) through dynamic programming by decreasing the runtime from O(n^2) to O(n \log n) time. We then illustrate that while the discovery of bcODs is relatively straightforward, there exist codependencies between approximation and conditioning that make the problem of abcOD discovery challenging. The naive solution is prohibitively expensive as it considers all possible segmentations of tuples resulting in exponential time complexity. To reduce the search space, we devise a dynamic programming algorithm for abcOD discovery that determines the optimal solution in O(n^3 \log n) complexity. To further optimize the performance, we adapt the algorithm to cheaply identify consecutive tuples that are guaranteed to belong to the same band--this improves the performance significantly in practice without losing optimality. While unidirectional abcODs are most common in practice, for generality we extend our algorithms with both ascending and descending orders to discover bidirectional abcODs. Finally, we perform a thorough experimental evaluation of our techniques over real-world and synthetic datasets.