A magnitude scale, M t, previously defined by the author, is unique among the many scales in seismology, because it is based on the instrumental amplitude of tsunami waves, its level being adjusted to agree with moment magnitude. Despite the simple definition, the M t scale reliably measures the seismic moment of an earthquake as well as the overall physical size of the source of the tsunami. In reality, the seismic moment estimated from M t for 39 events around Japan is found to be essentially equivalent to that estimated from seismic waves, with an uncertainty factor of two on average. The M t scale is useful for a quantitative discrimination of tsunami earthquakes that generate anomalously large tsunamis for their magnitude. Of 97 tsunamigenic events that occurred around Japan during the period 1894 to 1985, at least eight tsunami earthquakes are identified from the M t vs M s relation.