Wavelets with different integer scaling factor M can be combined in an orthonormal wavelet transform provided they share the same scaling function. These multiscale discrete wavelet transforms are not restricted to dyadic or M-scale signal sizes. Wavelets of this kind have recently been applied in meteorology to describe the background error covariances in the numerical weather prediction model ALADIN. In this note, we give mathematical details of the structure of these wavelets. We explicitely construct 2 multiscale wavelet families, based on M-scale generalisations of the Haar and Meyer wavelets.