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

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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 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 }.

I will clarify some of these items in detail if we discuss related topics being not exactly on the same page.

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, the 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.

The same can go with the round keys, 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.

The possibilities of the harmonic modulation

I think that we know from history, that historical tonal systems prior to some "well-tempered" (not necessarily EDOs) systems have been used without harmonic modulation. I think that this is commonplace to consider modulation as one of the major forces driving the implementation of the equivalence of tonalities. For the systems based on rational-number intervals, modulation was considered problematic.

Still, I don't think that even with the pure rational-number intervals the harmonic modulation is impossible. In any case, it seems to be at least a difficult problem for me.

I would like to hear some considerations or ideas on this topic.

I have some ideas of explanation of why I consider this possibility open. First of all, in contrast to traditional instruments, the tonal system of a software-based instrument doesn't have to be static. It can change its properties on the fly... How about that?

Invitation for the discussion

I have entered all essential initial content in a short period of time today. This part of the content can be recognized: all paragraphs come without my signature. From the time of writing, the page is open for discussion. — SASunday 2020 December 27, 22:26 UTC

Hey, SA, I don't know if this is a good place to post my comment, but I am trying to let you know that I wish to discuss some of my responses to this on Discord. --Aura (talk) 22:34, 27 December 2020 (UTC)
You can comment here, but this is my real intent: I want Kite Giedraitis to put his comments first and would ask other people to wait until we two basically shape the discussions on this page well. Kite promised to collaborate. What if we first have a preliminary discussion on Discord, which is much more convenient for intermittent things? I can start nearly immediately. When we have more mature content interesting to others, we can add it to How about that? — SASunday 2020 December 27, 22:47 UTC
Indeed, I think this would be most useful. Still, I think we can keep the comments that I've posted on this page seeing as this is what started the idea of the preliminary discussion on Discord. --Aura (talk) 22:49, 27 December 2020 (UTC)
Sure. — SASunday 2020 December 27, 23:26 UTC
Hi SA, just had a look. Some thoughts:
2nd section, "who invented what". The triangular 5-limit lattice goes way way back. I had the idea to extend the 5-limit triangles to 7-limit tetrahedrons, but it seems so obvious that I would be very surprised if I were the first one to do that. I imagine Erv Wilson probably did the same thing. We could call this a tetrahedronal lattice, but actually there are octahedrons in there too. I think of it as a 3-D lattice, but of course as a keyboard it has to be 2-D. The projection onto the 2-D page/screen/keyboard is nice and readable, but it lacks the ability to do ratios of 49-over or 49-under.
"The lattice holds translational symmetry". Only in 3-D. In 2-D, it doesn't. But I have another projection that I'm quite fond of, which does. It uses 2401/2400 to do an extremely slight microtempering to convert rank-4 7-limit JI to rank-3. 3rd pic on my user page User:TallKite. IIRC Graham Breed ( might have beaten me to publishing this lattice. The generators are {49/40, 7/4}, with two 49/40's adding up to 3/2. Of course you need a 3rd generator of 2/1 to make music with it. This is the big drawback to the colored triangular lattice keyboard you linked to. There is only one 2-D lattice that I know of that can do full 7-limit JI, that also has octaves of all the notes. It's rectangular not hexagonal, and the generators are one-fifth of 3/1 (very near 5/4) and one-twelfth of 3/2 (about 59¢). Furthermore 2 of the large generators equals 13 of the small ones. This puts the octave within easy reach at 3 large generators and 1 small one. Please do consider this layout.
3rd section, my color notation is mostly a spoken/written way to name things. I actually don't use literal colors all that much. Except in alt-tuner, where they do help keep track of ratios. But there are definitely limits, all the ones you mentioned, plus there's only so many shades one can distinguish. And also when the tonic is not on the white row, the whole thing sort of falls apart. (In alt-tuner, you can fix that by simply moving the notes around inside the lattice.) If you don't use the color words wa, yo, ru etc., then so far I don't think you're using much of anything original to me.
5th section, "change its properties on the fly' -- have you looked at alt-tuner? --TallKite (talk) 06:53, 30 December 2020 (UTC)
Kite, Thank you for your reply. I'll take a closer look, think, and write to you again. It looks like I was mistaken on the symmetry. Yes, I saw all-tuner, did not learn much yet. Happy New Year! — SAWednesday 2020 December 30, 16:01 UTC