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.

Not considering colors or notation

I don't want to consider colors here. I have a long story considering them and can explain it:

• Yes, synesthesia, the color-music association has a long history. However, not all authors have the specific background the in color sensory system I have. From my background, I know something which makes color associations questionable.

• Many reported that color associations help to understand or memorize musical structures. It may prove the merit of the color system but it does not prove any natural mapping between colors and tones or intervals. This is a mere color-coded mnemonics. Any mnemonical system would work if it uses nearly any sets of well-recognizable colors.

• Color perception is unique in different people. The properties of individual color perceptions are measurable. It's not very typical that a random person's perception fits some common "standard". Some minor but distinctly measurable perception peculiarities are probably most typical.

• Colors are majorly conceptual and imaginary. Many concepts understood by people as "colors" do not objectively exist in nature, but are the combined products of perception, personal experience, and human culture. In other words, people see not what they see, but, to a certain extent, what they expect to see. For example, the same scene at a different time of day has radically different color distributions. Each color taken separately can be perceived as different, but when we see the big picture, we barely notice the difference. We expect what we approximately should see and perceive according to the expectation. Moreover, some color concepts do not objectively exist at all. For example, there is no such thing as "brown color". We consider an object as brown not only if it fits in a certain color space area, but also only if it is put in a certain visual context. In a different context, we won't perceive the same color as "brown".

Interestingly, in musical perception, people have a lot more in common. It may sound like an absurd or a paradox, this is not at all obvious but this is true. Not too surprising, musical perception is simpler!

I do not argue against the color notation after Kite Giedraitis. I simply suggest that the subject is much more complicated than it seems to be and that it should not be in the scope. In the present context, I want to discuss only the tonal systems, layouts, lattices, and practical aspects of composition and playing music.

As to the problem of microtonal notation: I don't think that a microtonal notation is a fruitful subject for the development unless we fix the entire idea of notation. Traditional notation is wrong not because it is bad, but because the entire idea of having mapping between music and graphics is wrong. Notation should not be graphical in principle, but graphical forms could present a separate and polymorphic layer. If we don't accept it, notation work will be a waste of time. Yes, we badly need notation...

Different geometry

Let's take a look at the keyboard again. Do the keys have to be rounded? How about hexagons? I don't think so.

First of all, layout of a lattice and geometry are subtly different things. In my article, I criticized the Wicki–Hayden note layout.

Hexagonal image may seem very appealing, but the problem I can see is: it suggests higher symmetry than the actual tonal system layout has. I suggested rectangular or square keys: it lowers the symmetry and puts everything just right. It does not change the layout itself.

Same can go with the round key, but it would also fix one more problem: with round keys, glissando behavior is different, because at times a touch gets "between keys". This is not good, because glissando is very important and has to be smooth. I also see some reason to rotate the layout by 90°, but it would not make any fundamental difference.