This paper introduces a new Tolerance Band Modulation (TBM) for the Modular Multilevel Matrix Converter (M3C) to improve voltage quality and efficiency. The TBM method calculates the optimal switching times by integrating the voltage error, which results in a dynamic switching frequency. A comparison with the state-of-the-art phase disposition Pulse Width Modulation (PD-PWM) demonstrates the TBM's capability to achieve superior voltage quality without increasing switching losses. In addition to the standard principles of TBM known for M2C, this paper introduces several features for TBM to ensure an even load distribution among all cells and prevent unwanted voltage overshoot across the cell capacitors. The proposed method is validated by both simulation and experimental results.