The study of contemporaneous variations of the continuum flux and emission lines is of great importance to understand the different astrophysical processes at work in T Tauri stars. In this paper we present the results of a simultaneous $BVRI$ and H$alpha$ photometric monitoring, contemporaneous to medium-resolution spectroscopy of six T Tauri stars in the Taurus-Auriga star forming region. We have characterized the H$alpha$ photometric system using synthetic templates and the contemporaneous spectra of the targets. We show that we can achieve a precision corresponding to 2$-$3 AA in the H$alpha$ equivalent width, in typical observing conditions. The spectral analysis has allowed us to determine the basic stellar parameters and the values of quantities related to the accretion. In particular, we have measured a significant veiling only for the three targets with the strongest H$alpha$ emission (T Tau, FM Tau, and DG Tau). The broad-band photometric variations are found to be in the range 0.05$-$0.70 mag and are often paired to variations in the H$alpha$ intensity, which becomes stronger when the stellar continuum is weaker. In addition, we have mostly observed a redder $V-I$ and a bluer $B-V$ color as the stars become fainter. For most of the targets, the timescales of these variations seem to be longer than the rotation period. One exception is T Tau, for which the broad-band photometry varies with the rotation period. The most plausible interpretation of these photometric and H$alpha$ variations is that they are due to non-stationary mass accretion onto the stars, but rotational modulation can play a major role in some cases.
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