> the defects of this tuning method became evident and the more flexible “well-tempered” tuning scheme was devised. This ensures that the ratio of pitch between every two adjacent notes is precisely the same.
The author seems to have confused well-temperament and equal temperament.
It is also odd that he calls just intonation a “simplification” of Pythagorean tuning.
How does it make sense to divide a vector by another? Can we at least multiply two vectors as well? (So that if the quaternion q is the vector a divided by the vector b, then q times b is a)
In a particular sense, division and subtraction are more general than their supposedly more primitive inverse operations.
For example, we can take the "difference" of two points to get a vector, i.e. the grocery store is in that direction, while it doesn't make much sense to add two points---this location plus the grocery store? Similarly with temperature (of a given scale), today can be 5 degrees hotter than yesterday, but saying "the total temperature between yesterday and today is 57 degrees" doesn't have a clear meaning.
So, in that vein, what operation transforms one 3D vector into another?
Well a combination of rotation and scaling, of course! There is some freedom in how you want to represent this scaling+rotation operation; we could use 3x3 matrices, meaning that the quotient of 3D vectors is reasonably described by a matrix. However, 3x3 matrices have 9 degrees of freedom, which is way overkill, since we just need 4 numbers: an axis (2 params), an angle, and a scale factor. For this reason, quaternions are a particularly natural representation of 3D vector division.
The general concept of division that we're getting at here has the terrible name "g-torsor". If you're not familiar with abstract algebra, though, the definition is not very enlightening. However, John Baez has a really accessible article[0] that comes to the rescue!
Side note: notice that in general, the results of division are different objects than the things doing the division. As such, it's somewhat unfortunate that we use the same units for both in many cases, e.g. temperature and temperature differences are really different things. Talking about units quickly turns into type theory, though, so I'll stop there.
It doesn't make sense in and of itself. We usually think
of division as a closed operation: we divide two things (like real numbers) and we get the same kind of thing out (another real number).
In Hamilton's original view of quaternions, he defined a "geometric quotient" of two 3d directed lines (one kind of object) as being a quaternion (another kind of object), and gave all sorts of complicated geometric formulas for how to calculate it.
> How does it make sense to divide a vector by another? Can we at least multiply two vectors as well?
Yes. That is the point of moving into the 4D complex space. This takes some higher level math, but I think it is essentially anyone can understand but just not taught in the standard coursework[0]. Here's a super quick crash course
This is all about how you abstractly define things like addition and multiplication. We'll call these actions an "operator". So you got these basic algebraic structures:
Group: a set (some numbers) and an operator (like + or *. I'll use ∘ to be general) that has associativity ( (a∘b)∘c = a∘(b∘c) ), an identity ( a∘=a ), and an inverse ( a∘ā=0 ). There's also closure which just means we don't create something that's outside the set ( ∀ a,b ∊ S, a∘b ∊ S )
Abelian Group: We also have commutativity ( a∘b=b∘a )
Ring: Think of as 2 groups. So we now have 2 operators ( ∘,▪ )[1]. This time ∘ is an Abelian Group and ▪ has closure, associativity ( (a▪b)▪c = a▪(b▪c) ), and is distributive with ∘ ( a∘(b▪c) = (a∘b)▪(a∘c) )
Field: Extend that ring so now ∘,▪ are both Abelian Groups (▪ got an identity element, an inverse, and commutativity), but we give an exception for the inverse on identity of ∘[2]
There's a lot more structures, but what we often really want is this Field thing. Our normal 1D numbers work work like that, the "every day" type of math. But think for a second, how do you do this same math in 2D? How do we make ab consistent? Well, you can't. At least not with our standard +,. So we introduce imaginary numbers. These numbers aren't imaginary so much as we're just changing the rule on +,* (∘,▪) to work differently. So we use imaginary numbers in 2D because it creates a field! You literally could do this with tuples as long as you remember the i^2 rule.
Now we extend this to 3D. Ops, we get a problem. Really no way to make ▪ () consistent here, especially around division. Bummer. We have a Ring.
But going up to 4D we can use a similar rule as to what we did in 2D and get a Field again! Now we can do consistent math. And this is actually why quaternions become useful for 3D rotation. But they can do a lot more, though come with some limitations as well.
This is super high level and just scratches the surface of all of this stuff but I hope it at least makes some sense here. Please ask questions and I'm also sure someone else will add some useful comments. But your question is completely normal and actually gets to the motivation of why kids these days are learning the difference between 39 and 93, despite having the same answer they do mean different things and understanding that earlier helps you when you get to abstract nonsense.
[0] It is taught later because this is when you first start getting into math as being about abstractions. You basically learn that you only ever learned one very specific kind of math and that all that breaks down. Your question is absolutely on point due to this. Like how you probably learned matrix multiplication and learned that AB ≠ BA
[1] Whatever ∘,▪ these will usually be referred to as "addition" and "multiplication" but just note that they aren't the same thing you're probably thinking of
[2] Let's return to less abstraction for a second. We have + and for (standard) addition and multiplication, right? 0 is the identity element for + and note that 0 doesn't have an inverse for multiplication. That is: we can't do a * (0^-1) = a/0. Note here that / is a shorthand for * with an inverse operator.
Also, "double" rotations, which have no analogue in 3D space. The extra degree of freedom in 4D lets you perform 2 orthogonal rotations. This means that you can rotate around in 4D space, leaving only the origin fixed. In 3D rotations always fix some axis.
Trying to think about this stuff can be mind-bending, since it's natural to try to visualize things. So to give a little intuition, consider the coordinates of a 4D vector (x,y,z,w) and notice that we can spin around in both the x-y and z-w planes at the same time without interfering with each other!
I took the liberty to replace my awkward wording with your "the result in column i and row j is the sum of product of elements in column i of the left cracovian and column j of the right cracovian". Hope you don't mind. Thanks!
One thing that comes up in the sort of code ML I like to write is a careful attention to memory layout. Cracovians, defined according to some sibling comment as (B^T)A, make that a little more natural, since B and A can now have the same layout. I haven't used them though, so I don't have a good sense of whether that's more or less painful than other approaches.
Thanks for the feedback, everyone. I pasted my Polish text into Gemini to translate it into English. Gemini hallucinated the translation of this example. Now it should be OK.
My map of the most frequent occupational surnames in Europe [1] also went viral in 2015. When I posted it on Reddit, newspapers from Ireland (they asked me for permission to reprint it) to Greece (nobody else asked) and countless people on Facebook republished it. Well, everybody has a surname, an occupation, and a country.
Late advice: when making a viral image, put your URL on it :-)
IMHO, Euclid's definition of a straight line in today's terms would be "a line that has the same direction on its entire length". His definition of a plane angle would be "a plane angle is the difference between the directions of two straight lines that have a common end in one point".
The problem I have with that version of Euclid's definition is that direction is not defined.
Playfair interprets Euclid as follows, I am using my own words here, a straight line is that figure which has the property that if it intersects its moved copy at 2 points it necessarily coincides with it everywhere. "Movement" is undefined, it has to be an isometry.
Hilbert's is more abstract and based upon sets. Line is a primitive (undefined name) that interacts with two other undefined names (points and planes) according defined relations (lies on, lies between and is_congruent).
How about a third unit, baud [1]? It looks no worse than hertz to me:
In telecommunication and electronics, baud (/bɔːd/; symbol: Bd) is a common unit of measurement of symbol rate, which is one of the components that determine the speed of communication over a data channel. It is the unit for symbol rate or modulation rate in symbols per second or pulses per second.
For a server, requests per seconds is the best SI unit you can do. In practice requests to a server are random so there is no periodicity and all periodic/frequency-related units do not apply.
