User talk:SAKryukov/Keyboards based on the designs by Kite Giedraitis

Regular temperaments? Theoretical basics

Unfortunately, I can see a lot of mess and unclear statements on the theory on this site. To avoid miscommunication, I'll start with the formulation of the main points I'll need below in a brief thesis form. Before I do, I'll just list them in even shorter form. Comments are welcome.

• The scope is really Regular temperament, but I think the choice of the term is very unfortunate because it suggests tempering. In practice, on xen.wiki and other publications we rarely consider a combination of rational-number intervals with tempering in a single tonal system. Rather, we either work with pure rational-number interval systems, where the regular temperament is applicable and very useful, or EDOs, where the theory is applicable but way too trivial (rank-1 basis and perfect translational symmetry). From this point on this page, I'm going to discuss the pure rational-number interval systems only.

• Interval, set of intervals as a free Abelian group, each of three terms to be explained correctly and clearly

• Group as a module and group basis, how they are related

• Group actions, as related to musical intervals and tones

• Octave normalization as a part of group operation

• "Classical" 5-limit diatonic just intonation { 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 } as a rank-2 free Abelian group with the basis { 3/2, 5/3 }