Basis Vectors
All isomorphic keyboards derive their invariance from their relationship to rank-2 regular temperaments of Just Intonation.
A two-dimensional lattice is generated by two basis vectors, and so is a rank-2 regular temperament of Just Intonation. A keyboard lattice generated by two given musical intervals will be isomorphic in any rank-2 temperament that is also generated by those same two intervals. For example, an isomorphic keyboard generated by the octave and tempered perfect fifth will be isomorphic with both the syntonic and schismatic temperaments, which are both generated by those same two intervals.
Read more about this topic: Isomorphic Keyboard
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