The main contribution of Regener [20] related to this thesis is the representation of diatonic pitches and intervals as two-dimensional vectors. The bases of the vector space are any two intervals whose linear combination can generate all other intervals. A particularly useful pair of bases is P5 and P8. The vector representation of pitch was discovered independently by the author of this thesis and later confirmed by his discovery of Regener's work, which, oddly, is not well-known in the tuning community.
Although fancy group theory formalisms seem popular among many mathematically inclined theorists [14][13], Regener's contribution was that he correctly identified the somewhat more pedestrian topic of linear algebra as having much to contribute to the study of pitches and intervals. Unfortunately, Regener arrives at the vector representation of pitches via an unwieldy intermediate representation in terms of diatone and quint, whereas it is possible to go directly from conventional pitch notation to vector representation, as we do in this thesis. Regener's work is confined to the study of regular tunings, although the vector representation can be used to study all diatonic tunings, as will be seen in this thesis. Regener's work is confined to a two-dimensional representation of pitch, whereas this thesis will show how a three-dimensional representation is a powerful theoretical tool for some types of tunings.