The period-luminosity sequences and the multiple periods of luminous red giant stars are examined using the OGLE III catalogue of long-period variables in the Large Magellanic Cloud. It is shown that the period ratios in individual multimode stars are systematically different from the ratios of the periods at a given luminosity of different period-luminosity sequences. This leads to the conclusion that the masses of stars at the same luminosity on the different period-luminosity sequences are different. An evolutionary scenario is used to show that the masses of stars on adjacent sequences differ by about 16-26% at a given luminosity, with the shorter period sequence being more massive. The mass is also shown to vary across each sequence by a similar percentage, with the mass increasing to shorter periods. On one sequence, sequence B, the mass distribution is shown to be bimodal. It is shown that the small amplitude variables on sequences A, A and B pulsate in radial and nonradial modes of angular degree l=0, 1 and 2, with the l=1 mode being the most common. The stars on sequences C and C are predominantly radial pulsators (l=0). Matching period ratios to pulsation models shows that the radial pulsation modes associated with sequences A, A, B, C and C are the 4th, 3rd, 2nd and 1st overtones and the fundamental mode, respectively.
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