We present a detailed analysis of the cosmic microwave background (CMB) structure in the Tenerife Dec.=+40 deg data. The effect of local atmospheric contributions on the derived fluctuation amplitude is considered, resulting in an improved separation of the intrinsic CMB signal from noise. Our analysis demonstrates the existence of common structure in independent data scans at 15 and 33GHz. For the case of fluctuations described by a Gaussian autocorrelation function, a likelihood analysis of our combined results at 15 and 33GHz implies an intrinsic rms fluctuation level of 48^+21_-15 muK on a coherence scale of 4 deg the equivalent analysis for a Harrison-Zel'dovich model gives a power spectrum normalization of Q_rms-ps=22^+10_-6 muK. The fluctuation amplitude is seen to be consistent at the 68 per cent confidence level with that reported for the COBE two-year data for primordial fluctuations described by a power-law model with a spectral index in the range 1.0<=n<=1.6. This limit favours the large-scale CMB anisotropy being dominated by scalar fluctuations rather than tensor modes from a gravitational wave background. The large-scale Tenerife and COBE results are considered in conjunction with observational results from medium-scale experiments in order to place improved limits on the fluctuation spectral index; we find n=1.10+/-0.10 assuming standard CDM with H_0=50kms^-1Mpc^-1.