The basic helix–loop–helix (bHLH) family of transcription factors recognizes DNA motifs known as E-boxes (CANNTG) and includes 108 members. Here we investigate how chromatinized E-boxes are engaged by two structurally diverse bHLH proteins: the proto-oncogene MYC-MAX and the circadian transcription factor CLOCK-BMAL1 (refs. ^,). Both transcription factors bind to E-boxes preferentially near the nucleosomal entry–exit sites. Structural studies with engineered or native nucleosome sequences show that MYC-MAX or CLOCK-BMAL1 triggers the release of DNA from histones to gain access. Atop the H2A–H2B acidic patch, the CLOCK-BMAL1 Per-Arnt-Sim (PAS) dimerization domains engage the histone octamer disc. Binding of tandem E-boxes^{, –} at endogenous DNA sequences occurs through direct interactions between two CLOCK-BMAL1 protomers and histones and is important for circadian cycling. At internal E-boxes, the MYC-MAX leucine zipper can also interact with histones H2B and H3, and its binding is indirectly enhanced by OCT4 elsewhere on the nucleosome. The nucleosomal E-box position and the type of bHLH dimerization domain jointly determine the histone contact, the affinity and the degree of competition and cooperativity with other nucleosome-bound factors.