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 }

The tonal system and lattice

Let's consider the tonal system and lattice layout after by Kite Giedraitis in the form of this keyboard.

(Note the new name of my Microtonal project. Now it is migrated to the new fork under the new name "Microtonal Fabric".)

First of all, for future work, I need to know who invented what. Let me list what I can see.

• First of all, this is the choice of the tonal system. The tonal system itself is 7-limit. It can be represented by a free Abelian group of rank-4 with the basis { 3/2, 5/3, 7/4, 7/5 }. Of course, the basis could be chosen in several equivalent ways, but I prefer the choice with the minimal combination of numerator/denominator values. The tonal system is a superset of the "classical" 5-limit diatonic just intonation { 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 }.

• The presentation of this group in the form of the lattice has wonderful mapping: the basis element 3/2 moves the key location by ± one step in the horizontal direction, its inversion is 4/3. The basis elements 5/3, 7/4, 7/5, with inversions, move the key location to one of the neighboring rows.

• The lattice holds translational symmetry: the shift of the fingering in any direction preserves the intervals.

I understand that these properties of the tonal system, the lattice, and their mapping are not unique, and not sure that further generalization would bring something fundamentally new.