Fractions that you're not allowed to (or choose not to) reduce, like musical time signatures

[cmloegcmluin]

@Dave Keenan and I were looking at the 17n-EDO notations recently when he pointed out how a different sagittal was used for the 1/2 and 2/4 apotome here —  in 17- and 34- EDO, respectively — than the sagittal used for the 3/6 and 4/8 apotome —  in 51- and 68- EDO, respectively (see the Periodic Table). This is despite the fact that, of course, 1/2 = 2/4 = 3/6 = 4/8. And so we went off on a tangent trying to name these sorts of deliberately-unreduced fractions, and decided we were having too much fun so we'd like to invite the public in on the discussion. Here's what we've got so far:

• Dave
  
  I dunno what you call this, when you treat fractions that are not in lowest terms as distinct from their lowest terms partner and from each other. We do this with time signatures. 3/4 time is different from 6/8 time and no one would consider writing 1/1 instead of 4/4. You don't just want to know how much pie you're getting. You also want to know how many pieces it was cut into in the first place.
  
  That reminds me of a joke I learned from my friend Mike: "I only cut the pizza into 6 slices today because I'm not very hungry". The unstated assumptions are that he would otherwise have cut it into 8 pieces, and we're still going to eat half the pizza each.
  
  
  Douglas
  
  Curses, I really want to name this concept, but I have exhausted myself in the first pass.
  
  [Mike's joke] hurt my brain a bit because my first instinct was: fewer slices but eating a consistent count of slices means you'll eat more pizza than you would normally! But yes, I get the actual joke. As if the slice size could be constant and the whole might just dematerialize if you cut fewer of them.
  
  
  
  Dave
  
  I almost felt guilty about saying I didn't know what those things were called, because I knew you'd try to find or invent a name for them and I know you have better things to do with your time.
  
  Me:
  Is there a name for the mathematical objects that are similar to fractions but you're not allowed to reduce them, like the time signatures on sheet music?
  

  Yes, the mathematical objects you're describing are often referred to as "rational numbers" or "fractions" in general, but in the specific context of music, time signatures can be seen as "non-reducible fractions." In mathematics, these can also be viewed as "restricted fractions" or "irreducible fractions."
  
  In music, a time signature like 4/4 is understood in its conventional form, and while it can be expressed as other equivalent fractions (like 8/8), the standard notation maintains its form to convey specific rhythmic information. The term "canonical form" might also apply, as it refers to representing a mathematical object in a standard or simplified way without changing its essential properties.
  
  If you're looking for a term more specific to music theory, "time signature" is typically the accepted term, and it doesn't have a direct mathematical analogy beyond those mentioned.

  That's pretty garbled, dead wrong in some places and tautological in others. They are definitely not rational numbers and I've seen educational material saying "time signatures look like fractions but they are not fractions". It wrongly claims that 4/4 time can be expressed as 8/8 and canonical forms are exactly what we don't need.
  
  I already looked into "irreducible fractions" and found that this is synonymous with fractions that are already in lowest terms and so can't be reduced any further. It isn't that we can't reduce these objects, it's just that we're not allowed to or we choose not to. And I can't find any evidence that "non-reducible" (and lets throw in "unreducible" while we're at it) are anything other than synonyms for "irreducible", including when applied to fractions. I thought "anti-reducible" might be a possibility. I found it used in philosophy, of nature or the universe, when claiming that it is not able to be understood from the properties of its parts. I found it used in a couple of obscure math papers in which the only words I understood were words like "of" and "the", but they definitely weren't talking about fraction-like things.
  
  So the only remaining possibility here from ChatGPT is "restricted fractions". No. Turns out that's about restricting the domain, possibly of ratios themselves or possibly of the denominator and/or numerator, for example to ensure that the denominator is never zero.
  
  Nice try ChatGPT, but no Kewpie doll.
  
  I tried "fractionoids" and all I found was this old arcade learning game. https://www.mobygames.com/game/144730/fraction-oids/
  
  Oh wait.... I've got better things to do with my time.
  
  [Your take on Mike's joke] a perfectly valid first take, although not one that occurred to me. But then, I was there, and I know Mike's kind of jokes. I took it that Mike was pretending to be so dumb as to think that the "slice" was the relevant unit of measurement here, for the purpose of hunger reduction, and not realising the size of the slice mattered.
  
  The ol' brain just wouldn't leave it alone. How about "deliberately-unreduced fractions"? Or any other adverb that means "by choice". <looks up thesaurus> How about "designedly-unreduced", "intentionally-unreduced", "wilfully-unreduced", "wantonly-unreduced", "unreduced-with-malice-aforethought", "flagrantly-unrepentant", "fragrant-underwear"? fractions.
  
  OK. I got a bit carried away with those last few.
  
  
  Douglas
  
  I'm glad to hear that you only almost felt guilty, because I knew by engaging with it in that way I might send you on a guilt trip. It's alright, really.
  
  I'll spare you my own ChatGPT conversation breakdown from yesterday; suffice it to say that the outcome was remarkably similar to yours.
  
  So anti-reducible seems fine except it's already taken by higher maths.
  
  [You did get carried away at the end, but] the first few are alight, though.
  
  I thought about this a bit more while drifting off last night. I thought about it in a couple of different ways.
  
  One way was: are there perhaps separate notions of irreducible numerators and denominators?
  • With an irreducible denominator, we're essentially saying, we cut every whole pizza into 𝑑 slices, and slices simply can't be merged back together, so if you're working with six eighth slices, that's simply not the same thing as three fourth slices; this is the key idea, I think.
  • With an irreducible numerator, we're entering the portal to Mike's jocutopia, where the criterion of the blue AF EDO notations lives too, agnostic to the size of slice; and this maybe barely more than the way ordinary numbers work, with just a sheen of the implication that they are counts of slices.
  The other way I thought about it was: how might you represent the situation in math expressions by introducing a new symbol or special quantity, something inspired by 𝑖 = √-1 perhaps, a constant that makes a number into a different version of itself. In the case of 𝑖 we make an imaginary version of a number, and in our case we're trying to make an irreducible version of one. But we actually need to add a constant to accomplish this in this case, not multiply it, and not only that, but we need to add that constant to both the numerator and the denominator, e.g. if we have (6+𝑟)/(8+𝑟), then we can't reduce it to (3+𝑟)/(4+𝑟), when 𝑟 is basically treated as a sort of unknowable quantity. But that's way too cumbersome to be an effective notation. And worse, it doesn't even work in edge cases: we would want 3/3 to be able to be different than 1/1, but this approach doesn't work for that, because (3+𝑟)/(3+𝑟) = 1/1 no matter what = 𝑟.
  
  I thought of "absolute ratio" but people already use that where they haven't yet been exposed to our superior "undirected ratio".
  
  I thought about pizza slices, or perhaps just slices. Maybe there's a worthy power to this metaphor, the nature of slices' irrecombinability. These just aren't ratios. They're slice counts. There's no reason they can't be >1, like you can have 12 slices of pizzas that have been cut into 8ths. So maybe as I alluded to earlier, it's really just the irreducible denominator that's the main idea here. I dunno if that helps you think of good terminology or notation.
  
  Maybe we should transfer this convo to the for-fun subforum, and let others in on the fun.
  
  
  
  Dave
  
  Sure. But what will you name the thread? "Those mathematical objects that are like fractions but you're not allowed to (or choose not to) reduce them, like musical time signatures"?
  
  
  Douglas
  
  "Pizza slice problem"?
  
  
  
  Dave
  
  The thing is, if you give the thread a short name, you're effectively deciding the answer before you ask the question. e.g. You're implying they could be called "pizza slice numbers".
  
  As a separate matter, I don't think "pizza slice numbers" would be a good name, because in many cases, maybe even most cases, nobody cares whether they are getting 3/6 of a pizza or 4/8 or a pizza.
  
  
  Douglas
  
  Oh, yeah, well, I did think "pizza slice number" was not a good name for the concept too, and actually that's why I thought it might be a decent name for the topic, for the reason you give: because it wouldn't thereby imply that the problem was already solved. But I can see from your take that pizza slices wouldn't even be a good name for the topic. Your name would be an acceptable name for the topic, long as it is. In the meantime I will try to think of a well-known real-world example of something for which one would really care about the difference between receiving 3/6 of it vs 4/8 of it... the best example might just be time signatures. Which would be fitting for a microtonal music notation system's forum...
  
  
  
  Dave
  
  Another problem with the title "Pizza slice problem" can be seen by web-searching the phrase.
  
  There's more than one, but I just learned that if you cut a circular pizza into 8 slices, provided all 4 cuts go through the same point, if two people take alternate slices they are guaranteed to get the same area, no matter where that point is.
  
  
  Douglas
  
  Ah yes, I've seen a Numberphile video or something on https://en.wikipedia.org/wiki/Pizza_theorem.
  

And so I've taken Dave's suggested name, though I had to shorten it a bit, because it was longer than the maximum character count allowed for forum topic names.

And yeah, my current best suggestion is not "timesig numbers", because if these things were reducible to single numbers, then that would counteract their irreducibility. And I think "timesig ratio" also doesn't work, because "ratio" conveys proportionality and thus, again, reducibility. "Fraction" is better, I think, because we are still dealing with fractioning quantities here. So I suggest "timesig fraction".

Here's another interesting subtopic: what would a good notation be for these? I did already start to talk about it above. I still think it may be better to come up with a special sign, or symbol to add or multiply by, then it would be to come up with an alternative to the solidus or division bar.

Perhaps it could be related to the following, where it occurs to me to wonder how these timesig fractions would interact with other numbers multiplied or divided into them. I can see a few different cases, experimenting on the timesig fraction 6/24:

type	/3	×2
hard 𝑛	6/72	6/12
hard 𝑑	2/24	12/24
hard both	(6/24)/3	6/24)×2

The hard 𝑛 and hard 𝑑 concepts are continuations of the ideas I introduced earlier about irreducible numerators and irreducible denominators. So perhaps instead of a single constant like the 𝑟 I looked at above, it could be a different constant or prefix operator to lock the numerator from the one to lock the denominator. Or maybe all we need is a symbol that locks a number, making it irreducible, wherever it may find itself.

A musical time signature 𝑐/𝑣 (count/value) is a bit more of a hard 𝑑 situation, I think, where half a bar of 4/4 time is definitely 2/4 more than it is 4/8. So maybe "timesig fraction" is not a good name for the whole set of three types of hardnesses. (I'm not at all married to "hard", or "locked"... these are just the words that have occurred to me in the moment to describe these situations.) Maybe there's some other real-world application that's more of a hard 𝑛 case that could lend its name to those relations, and another one that's a hard both? I've said too much already. Stepping away now...

[Dave Keenan]

One way was: are there perhaps separate notions of irreducible numerators and denominators?

• With an irreducible denominator, we're essentially saying, we cut every whole pizza into 𝑑 slices, and slices simply can't be merged back together, so if you're working with six eighth slices, that's simply not the same thing as three fourth slices; this is the key idea, I think.
• With an irreducible numerator, we're entering the portal to Mike's jocutopia, where the criterion of the blue AF EDO notations lives too, agnostic to the size of slice; and this maybe barely more than the way ordinary numbers work, with just a sheen of the implication that they are counts of slices.

I can't see it. Fraction reduction always reduces both the denominator and the numerator. They are not independently reducible. A deliberately-unreduced fraction declines to reduce either of them. That's your first category above.

Your second category involves ignoring the denominator completely. So, as you suggest, it is simply an integer, which is why it can't be reduced. And yes, it just happens to be a slice-count. But the blue apotome-fraction (AF) EDO notation doesn't ignore the denominator. It is in your first category. Ups and Downs notation ignores the denominator and so is in your second category. Ups and Downs notation is exactly like Mike's pizza joke.

[cmloegcmluin]

Superb choice of example in Ups and Downs notation. Thanks for that!

So I agree with your criticisms of this material, which was from an earlier state in my development of these ideas. But I now believe that the core of this idea is not tied to fractions, or to reducibility, but more to a sort of enforced uncombinability. Please see the table later on in my post where I demonstrated some variations. Or just see this enhanced version of that table, using a makeshift notation, where ːnː is uncombinable/"locked".

	/3=	×2=
ː6ː/24	ː6ː/72	ː6ː/12
6/ː24ː	2/ː24ː	12/ː24ː
ː6ː/ː24ː	(ː6ː/ː24ː)/3	(ː6ː/ː24ː)×2

And here's some more illustrative examples:

ː6ː + ː2ː ≠ ː8ː
ː6ː × ː2ː ≠ ː12ː
ː6ː / ː2ː ≠ ː(6/2)ː = ː3ː
6/ː3ː + 2/ː3ː = 8/ː3ː
6/ː3ː × 2/ː3ː = 12/ː3ː

I briefly considered notating these by putting the numbers in the time signature font, but that's not ideal, because it would suggest that they are together in a secondary numerical universe, such that it might seem like ː6ː + ː2ː = ː8ː when it doesn't; better to suggest that each locked number is in its own separate uncombinable existence.

To be clear, I'm hoping to eventually find names for these sorts of things that aren't 10 syllables long (as "deliberately-unreduced fractions" is), such that people might actually one day use them.

[רועיסיני]

I think it's just a pair of numbers, or a (2d integer) vector. If you divide the numbers or take the slope of the vector it has the value of a fraction, which you can reduce, but you can still look at the vector which is not the same as another vector with the same slope.

Or just see this enhanced version of that table, using a makeshift notation, where ːnː is uncombinable/"locked".

	/3=	×2=
ː6ː/24	ː6ː/72	ː6ː/12
6/ː24ː	2/ː24ː	12/ː24ː
ː6ː/ː24ː	(ː6ː/ː24ː)/3	ː6ː/ː24ː)×2

And here's some more illustrative examples:

ː6ː + ː2ː ≠ ː8ː
ː6ː × ː2ː ≠ ː12ː
ː6ː / ː2ː ≠ ː(6/2)ː = ː3ː
6/ː3ː + 2/ː3ː = 8/ː3ː
6/ː3ː × 2/ː3ː = 12/ː3ː

This looks to me like just other names for variables, and if you make expressions with them you get formal rational functions (maybe "formal expressions" is better?), like the notion of a formal sum, although your last example looks like a mistake: I'd say that 6/:3: × 2/:3: = 12/:3:², like 6 meters × 2 meters = 12 square meters. Maybe you want to have the identity that :n:² = :n: for all n, but that means you can't substitute n in the place of :n: anymore, and I think dividing by it may cause weird stuff.

[רועיסיני]

@cmloegcmluin I don't know what you allow to do with your new symbols but I'm pretty sure I have a proof that if 6/ː3ː × 2/ː3ː = 12/ː3ː then ː3ː = 1:
First, multiply by ː3ː:
6/ː3ː × 2/ː3ː × ː3ː = 12/ː3ː × ː3ː
Then simplify:
6/ː3ː × 2 = 12
12/ː3ː = 12
Now multiply by ː3ː again:
12/ː3ː × ː3ː = 12 × ː3ː
and simplify:
12 = 12 × ː3ː
Now subtract 12:
12 - 12 = 12 × ː3ː - 12
Simplify and group the terms:
0 = 12 × (ː3ː - 1)
Finally, the last time I checked 12 is not 0, and so I think you'd agree that
ː3ː - 1 = 0
and we can add 1 to both sides to get that
ː3ː = 1
Did I accidentally do a thing I'm not allowed to do?

Also, regarding the name, I think the best term would be "formal ratio" (again, like formal sum), or at least it's the best one I have thought of so far: It's notated as a ratio formally but unlike a true ratio you can choose not to evaluate it and it has a different meaning in its non-reduced form.

[רועיסיני]

Goddamnit, now you got me interested in this concept too! I think that like rational numbers you can make a new term of formal ratios/fractions, notate them as say [a:b], and define new operations on them.
Obviously you have "regular" multiplication and division by numbers:
[a:b] * n = [an:b]
[a:b] / n = [a/n : b]
And you have multiplication and division that keep the numerator constant instead of the denominator. Maybe augmentation and diminution?
[a:b] (*) n = [a : b/n]
[a:b] (/) n = [a:bn]
Note that I don't want to allow these operations between two formal fractions, only between a formal fraction and a number
Now you can also add and subtract the fractions, and you have the rules that
[a:c] + [b:c] = [a+b : c]
[a:c] - [b:c] = [a-b : c]
But expressions like [1:2] + [1:3] can't be simplified, like if you take one step of 34 plus one step of 51 it's not the same as 5 steps of 102, because you can subtract back one step of 51 and get one step of 34, not 3 of 102.
Also it's possible to define inverted versions of these, but I don't think they would be very useful.
Another operation you can do, at least between single formal fractions (and not sums of them), is the mediant:
[a:b] (+) [c:d] = [a+c : b+d]
Which returns a value in the middle of the two, like if you know that 7\12 is a meantone fifth and 11\19 is also a meantone fifth then 18\31 would be a meantone fifth as well, and one that falls between them. You can also have the "inverse mediant":
[a:b] (-) [c:d] = [a-c : b-d]
Which gets you a number that is not between them, on the side of the one with the largest denominator (which is the first of you don't like negative denominators), like 18\31 and 11\19 are pretty good meantone fifths, but you get from them 7\12 which is on the shaper side, because 31's is shaper than 19's.
I'm not sure how I want to generalise these operations to the sums and differences of formal fractions, or if I even want to allow to perform both kinds of operations at the same time, the problem would be that I want them to be inverses of each other, i.e. that x (+) y (-) y = x, no matter what y is, but this becomes difficult when y is a sum or difference of multiple formal fractions (and what about 0?) anyway if something like that would work I probably want to have
([a:b] ± [c:d]) (+) [e:f] = [x : b+f] ± [y : d+f]
for some x and y.
You can also define the inversion of a formal fraction:
[a:b]^-1 = [b:a]

Now for the fun part: algebraic identities!
You have associativity of multiplication and division
([a:b] * n) * m = [an : b] * m = [anm : b] = [a:b] * nm
([a:b] / n) / m = [a/n : b] / m = [a/nm : b] = [a:b] / nm
And in fact they are different facets of the same operation, as
[a:b] / n = [a/n : b] = [a:b] * (1/n)
and so you can write all sorts of identities like
(x * n) / m = x * (n/m)
where x is a formal fraction.
The case with augmentation and diminution is also very similar:
([a:b] (*) n) (*) m = ([a : b/n]) (*) m = [a : b/nm] = [a:b] (*) nm
([a:b] (/) n) (/) m = ([a : bn]) (/) m = [a : bnm] = [a : b] (/) nm
and they are also different facets of the same operation, because you have
[a:b] (/) n = [a : bn] = [a : b/(1/n)] = [a:b] (*) 1/n
and so you can write other similar identities. These four operations also fully commute with each other:
(x * n) (*) m = (x (*) m) * n etc. (10 identities)
because operations of the same type can be thought of as regular operations of just the numerator or just the denominator and operations of different types work on different parts of the fraction.
Now, addition is commutative, associative and subtraction is the inverse of it by definition on fractions of different denominators, and with the same denominators this is also easy to show:
([a:c] + [b:c]) = [a+b : c] = [b+a : c] = [b:c] + [a:c]
([a:d] + [b:d]) + [c:d] = [a+b : d] + [c:d] = [a+b+c : d] = [a:d] + [b+c : d] = [a:d] + ([b:d] + [c:d])
([a:c] + [b:c]) - [b:c] = [a+b : c] - [b:c] = [a+b-b : c] = [a:c]
In fact, if you define negation by -[a:b] = [-a:b] then
[a:c] - [b:c] = [a-b : c] = [a:c] + [-b:c] = [a:c] + (-[b:c])
which means that also in general
(x + y) - z = x + y + (-z) = x + (-z) + y = (x - z) + y
and similarly
(x - y) - z = (x - z) - y
Multiplication (and hence division) also distributes over addition and negation, and therefore also subtraction. Again for different denominators I can just define it like that, and for the same denominator we get
([a:c] + [b:c]) * n = [a+b : c] * n = [(a+b)n : c] = [an+bn : c] = [an : c] + [bn : c] = [a:c] * n + [b:c] * n
(-[a:b]) * n = [-a:b] * n = [-an : b] = -[an : b] = -([a:b] * n)
(x - y) * n = (x + (-y)) * n = x * n + (-y) * n = x * n + (-(y * n)) = x * n - y * n
This also happens with augmentation and diminution:
([a:c] + [b:c]) (*) n = [a+b : c] (*) n = [a+b : c/n] = [a : c/n] + [b : c/n] = [a:c] (*) n + [b:c] (*) n
(-[a:b]) (*) n = [-a:b] (*) n = [-a : b/n] = -[a : b/n] = -([a:b] (*) n)
(the proof for subtraction is exactly the same)
Also, negation is multiplication by minus one, since
-[a:b] = [-a:b] = [(-1)a:b] = [a:b] * (-1)
and so also distributes over addition, which means that, for instance,
x - (y + z) = x + (-(y+z)) = x + (-y) + (-z) = x - y - z
In fact, the new kind of addition and subtraction is not the only one the new multiplication distributes over: regular addition also satisfies this:
[a:b] * (n+m) = [an+am : b] = [an : b] + [am : b] = [a:b] * n + [a:b] * m
but this doesn't happen with augmentation:
[1:6] (*) (1+2) = [1:6] + [1:3] ≠ [1:2] = [1:6] (*) 3
About mediants, they are also commutative and distributive:
[a:b] (+) [c:d] = [a+c : b+d] = [c+a : d+b] = [c:d] (+) [a:b]
([a:b] (+) [c:d]) (+) [e:f] = [a+c : b+d] (+) [e:f] = [a+c+e : b+d+f] = [a:b] (+) [c+e : d+f] = [a:b] (+) ([c:d] (+) [e:f])
and naturally the inverse mediant is the inverse of the mediant:
([a:b] (+) [c:d]) (-) [c:d] = [a+c : b+d] (-) [c:d] = [a+c-c : b+d-d] = [a:b]
and in fact, you can define (-)[a:b] = [-a:-b] and get that
[a:b] (-) [c:d] = [a-c : b-d] = [a:b] (+) [-c:-d] = [a:b] (+) ((-)[c:d])
and so have identities of the kind you have with addition and subtraction.
Another funny thing that happens is that the mediant has a component mixing identity:
[a:b] (+) [c:d] = [a+c : b+d] = [a+c : d+b] = [a:d] (+) [b:c]
I don't think it would be useful, but it's fun to note.
I couldn't find identities with both (inverse) mediants and additions or subtractions, but the multiplication and augmentation (and hence division and diminution) also distribute over it:
([a:b] (+) [c:d]) * n = [a+c : b+d] * n = [an+cn : b+d] = [an : b] (+) [cn : d] = [a:b] * n (+) [c:d] * n
([a:b] (+) [c:d]) (*) n = [a+c : b+d] (*) n = [a+c : b/n+d/n] = [a : b/n] (+) [c : d/n] = [a:b] (*) n (+) [c:d] (*) n
and the "mediant opposite" is both multiplication and augmentation (expansion?) by -1:
([a:b] * -1) (*) -1 = [-a:b] (*) -1 = [-a:-b] = (-)[a:b]
and hence you can also get the rest if the stuff I got for addition and subtraction for mediants and inverse mediants.
Finally, inversions! Apart from being an involution:
([a:b]^-1)^-1 = [b:a]^-1 = [a:b]
they also make multiplication into diminution and division into augmentation:
([a:b] * n)^-1 = [an : b]^-1 = [b : an] = [b:a] (/) n = [a:b]^-1 (/) n
(x / n)^-1 = (x * (1/n))^-1 = x^-1 (/) (1/n) = x^-1 (*) n
and of course in reverse too:
(x (*) n)^-1 = ((x^-1)^-1 (*) n)^-1 = ((x^-1 / n)^-1)^-1 = x^-1 / n
(x (/) n)^-1 = ((x^-1)^-1 (/) n)^-1 = ((x^-1 * n)^-1)^-1 = x^-1 * n
It also distributes over mediants and mediant opposites, and hence over inverse mediants:
([a:b] (+) [c:d])^-1 = [a+c : b+d]^-1 = [b+d : a+c] = [b:a] (+) [d:c] = [a:b]^-1 (+) [c:d]^-1
((-)[a:b])^-1 = [-a:-b]^-1 = [-b:-a] = (-)[b:a] = (-)([a:b]^-1)
(x (-) y)^-1 = (x (+) ((-)y))^-1 = x^-1 (+) ((-)y^-1) = x^-1 (-) y^-1

Also I just noticed it may also sometimes make sense to multiply or divide two formal fractions:
[a:b] * [c:d] = [ac : bd]
[a:b] / [c:d] = [a/c : b/d]
This in fact generalizes the notions of multiplication and division with numbers and also those of augmentation and diminution:
[a:b] * n = [an : b] = [a:b] * [n:1]
[a:b] / n = [a/n : b] = [a:b] * [1/n : 1] = [a:b] / [n:1]
[a:b] (*) n = [a : b/n] = [a:b] * [1 : 1/n] = [a:b] / [1:n]
[a:b] (/) n = [a : bn] = [a:b] * [1:n]
and in fact regular numbers behave the same as formal fractions with denominator 1:
[a:1] + [b:1] = [a+b : 1]
[a:1] * [b:1] = [ab : 1*1] = [ab : 1]
The new multiplication is commutative and associative:
[a:b] * [c:d] = [ac : bd] = [ca : db] = [c:d] * [a:b]
([a:b] * [c:d]) * [e:f] = [ac : bd] * [e:f] = [ace : bdf] = [a:b] * [ce : df] = [a:b] * ([c:d] * [e:f])
and it even distributes over addition:
([a:c] + [b:c]) * [d:e] = [a+b : c] * [d:e] = [ad+bd : ce] = [ad:ce] + [bd:ce] = [a:c] * [d:e] + [b:c] * [d:e]
and over the mediant:
([a:b] (+) [c:d]) * [e:f] = [a+c : b+d] * [e:f] = [ae+ce : bf+df] = [ae:bf] (+) [ce:df] = [a:b] * [e:f] (+) [c:d] * [e:f]
A funny stuff that happens is that the inversion actually doesn't switch multiplication with division and augmentation with diminution but multiplication with diminution and division with augmentation, if we define augmentation and diminution in the way that makes sense between two formal fractions:
[a:b] (*) [c:d] = [a/d : b/c]
[a:b] (/) [c:d] = [ad : bc]
[a:b] * [c:d]^-1 = [a:b] * [d:c] = [ad:bc] = [a:b] (/) [c:d]
[a:b] / [c:d]^-1 = [a:b] / [d:c] = [a/d : b/c] = [a:b] (*) [c:d]
I say this is "the way that makes sense" because it generalizes the notion of augmentation and diminution between formal fractions and numbers if we say we identify numbers with formal fractions of the form [n:1]:
[a:b] (*) [n:1] = [a/1 : b/n] = [a : b/n] = [a:b] (*) n
[a:b] (/) [n:1] = [a*1 : b*n] = [a : bn] = [a:b] (/) n
The operation that switches between multiplication and subtraction is 1/x (which is not the same as what I called inversion!):
[a:b] * (1/[c:d]) = [a:b] * [1/c : 1/d] = [a/c * b/d] = [a:b] / [c:d]
and this also switches augmentation and diminution:
[a:b] (*) (1/[c:d]) = [a:b] (*) [1/c : 1/d] = [ad * bc] = [a:b] (/) [c:d]
These operations commute with each other:
1 / [a:b]^-1 = 1 / [b:a] = [1/b : 1/a] = [1/a : 1/b]^-1 = (1 / [a:b])^-1
and both of them together switch between multiplication and augmentation and between division and diminution:
[a:b] * (1 / [c:d]^-1) = [a:b] * [1/d : 1/c] = [a/d : b/c] = [a:b] (*) [c:d]
[a:b] / (1 / [c:d]^-1) = [a:b] / [1/d : 1/c] = [ad : bc] = [a:b] (/) [c:d]
This also means that inversion can be expressed by using diminution:
x^-1 = 1 * x^-1 = 1 (/) x
(when 1 is a shorthand for [1:1])
This may actually be a better notation for inversion than the ^-1 one, because if we want to actually raise these things to the power of numbers we would want to have xⁿ * x^m = x^n+m.
All of this stuff probably provides an easier explanation for some of the identities I've written, but this is left as an exercise to the interested reader.

[רועיסיני]

Okay we've defined some operations, let's see what structure they give us!
So first, I noticed that I really don't want to allow both mediants and addition and subtraction to be done at the same time, at least if we're not limiting ourselves to add and subtract only formal fractions with the same "formal denominator". That's because you can get stuff like
[0:a] = [1-1 : a] = [1:a] - [1:a] = [1:a] + [1:b] - [1:b] - [1:a] = [1:b] + [1:a] - [1:a] - [1:b] = [1:b] - [1:b] = [1-1:b] = [0:b]
But with mediants they are different! For example [1:1] (+) [0:1] = [1:2] but [1:1] (+) [0:2] = [1:3]. However if you only allow yourself to add or subtract formal fractions with the same formal denominator this is fine because the expression [1:a] + [1:b] - [1:b] - [1:a] would be invalid and so you can't confuse the zeros of the different denominators.
So let's start with mediants, because they give us the simpler structures.
First, if you allow mediants and inverse mediants, multiplication, division, augmentation and diminution by numbers, and inversion, you get the lattice ℤ² of vectors of two integers (or the vector space ℚ² if you also allow usual fractions as formal numerators and denominators). All the operations are linear and in fact the opposite is also true: all linear operations can be defined using the new operations. In general, multiplication by the matrix

    (a b)
A = (   )
    (c d)

can be done by the operation
A(x) = x * a (/) d (+) 1 (/) x * b (/) c
because
[u:v] * a (/) d (+) 1 (/) [u:v] * b (/) c = [u:v] * a (/) d (+) [v:u] * b (/) c = [au : v] (/) d + [bv : u] (/) c = [au : dv] (+) [bv : cu] = [au+bv : cu+dv]
which agrees with matrix multiplication.
On the other hand, if you allow mediants and inverse mediants and general multiplication and division between formal fractions (when possible i.e. when you get a valid result out of it) but not inversion (so you also still allow multiplication, division, augmentation and diminution by numbers because they can be expressed by multiplication and division with formal fractions) you get the commutative ring ℤ² (or ℚ²). That's because the formal numerator and the formal denominator don't have any way to interact with each other.
So what if we give them a way? If we add inversion into the mix (which also adds general augmentation and diminution because they can be expressed by it) you get the algebra ℤ² (or the algebra ℚ²), which is basically a lattice (or a vector space) where you can also multiply elements with each other. Something fun you can have in the algebra ℚ² is define the function that takes a formal fraction and evaluates it:
eval(x) = x / (x * 0 (+) 1 (/) (x * 0))
which (when the denominator isn't 0) gives you
eval([u:v]) = [u:v] / ([u:v] * 0 (+) 1 (/) ([u:v] * 0)) = [u:v] / ([0:v] (+) 1 (/) [0:v]) = [u:v] / ([0:v] (+) [v:0]) = [u:v] / [v:v] = [u/v : 1]
and remember that if you allow addition and subtraction between formal fractions with the same denominator, formal fractions of the form [a:1] are just regular numbers .

Now, what if you don't allow mediants and instead allow addition and subtraction of formal fractions with different denominators? First if you just allow addition and subtraction with multiplication and division by numbers, you get the infinite-dimensional lattice ⨁_n∈ℤℤ (or the infinite-dimensional vector space ⨁_r∈ℚℚ) where every basis element is a unit formal fraction with a different denominator (like [1:3] or [1:4]). In this context (augmentation and) diminution by numbers can be thought of as a nice class of linear operations that switch the basis elements around, but we'll see a better view of it later.
If we also allow multiplication between formal fractions, something a bit interesting happens between fractions of negative denominators and the corresponding positive denominators:
([1:1] + [1:-1]) * ([1:1] - [1:-1]) = [1:1] * ([1:1] - [1:-1]) + [1:-1] * ([1:1] - [1:-1]) = [1:1] - [1:-1] + [1:-1] - [1:1] = 0
which means I really want [1:-1] be either [1:1] or -[1:1] = [-1:1]. I like formal fractions having a consistent numerical value, and so I choose [1:-1] = [-1:1], which means that in general
[a:-b] = [a:b] * [1:-1] = [a:b] * [-1:1] = [-a:b]
so a negative denominator is the same as a positive denominator with a negated numerator.
What about 0 denominator? It creates even more trouble:
([1:0] - [1:a]) * [1:0] = [1:0] - [1:0] = 0
which either means that [1:0] is equal to [1:a] for every a, that [1:0] is 0, that division goes out of the window or that having [a:0] is not allowed. I think it makes the most sense to just not allow zero as a formal denominator (which is not the case when we have mediants - there zero is a perfectly valid "denominator") so that what I'll do from now on.
Now, if we allow multiplication between formal fractions but only allow integer entries we get the infinite variable polynomial ring ℤ[x₂, x₃, x₅, ...] where every prime number has a variable x_p, which is interpreted as [1:p]. This way the number [a:b] is a∏_ix_pi^n_i where b=∏_ip_i^n_i. Here we can also see the better view of augmentation and diminution: We've already established that diminution by 2, for instance, is the same as multiplication by [1:2], which means it's multiplication by the monomial x₂, while augmentation by 2 is the same as division by it, and in general augmentation and diminution by numbers are just multiplication and division by monomials, inversely*. Another nice thing that can be seen here, is that if we only include, say, x₂, we can only have powers of 2 as formal denominators, which is exactly what happens in time signatures!
When you also allow rational formal numerators you get the polynomial ring ℚ[x₂, x₃, x₅, ...] and if you allow rational denominators you get ℚ[x₂, 1/x₂, x₃, 1/x₃, x₅, 1/x₅, ...], where you also allow functions such as x₂ + 3x₂/x₅, which means [1:2] + [3:2/5]. If you choose to also allow division of two elements you get the field ℚ(x₂, x₃, x₅, ...) of rational functions (notice the round brackets here) with infinite variables over the rationals, where you can also have expressions like (x₂+x₃)/(2x₂x₃-x₅), which means ([1:2]+[1:3])/([2:6]-[1:5]).
You can also add inversion at every step of the way, which, if the step included multiplication, would mean also adding augmentation and diminution by general elements, but I don't think this matches any known algebraic structure (where inversion can be mapped to a meaningful operation). I don't really like it because inverting sums of formal fractions with different denominators creates new expressions that can't be simplified to the expressions we already know. However with identities like 1 (/) x * 1 (/) y = 1 (/) (x * y) you can still simplify some expressions, like
1 (/) ([1:2] + [1:3]) + 1 (/) ([1:4] + [1:6]) = 1 (/) ([1:2] + [1:3]) + 1 (/) ([1:2] * ([1:2] + [1:3])) = 1 (/) ([1:2] + [1:3]) + 1 (/) [1:2] (/) ([1:2] + [1:3]) = 1 (/) ([1:2] + [1:3]) + [2:1] (/) ([1:2] + [1:3]) = (1 + 2) / ([1:2] + [1:3]) = 3 / ([1:2] + [1:3]) (which also equals 1 / ([1:6] + [1:9]).)
On the other hand it's cool that inversion of single formal fractions mixed the numerator (which is like the coefficient) with the denominator (which is like the variable), for instance [6:5] which is 6x₅ becomes [5:6] or 5x₂x₃ which is really not a usual operation.

Also, it's interesting to look at the evaluation operations (the ones that give expressions numerical values that make sense with giving formal fractions the values of the usual fractions they symbolize) in each case:
In the case of ℤ² (or ℚ²) it just takes the slope of the vector, and if it's an algebra I even defined it explicitly, but I don't think it has a name in the ring case. In the case of the infinite dimensional lattice/space it's just a linear functional that maps each basis term [1:n] to the value 1/n, and in the case of polynomials and rational functions it substitutes 1/p for the variable x_p. They are not fully defined where we may "divide by zero", which means all the cases where we can have mediants (because there is a zero formal denominator), in the infinite dimensional lattice/space (because the zero formal denominator can also exist there, although there is no problem with it not existing there, if you prefer), not in the polynomials case, but interestingly yes again in the rational functions case, because expressions like 1 / (x₂ - x₃ - x₂x₃) are allowed, but when you try to evaluate them you get 1 / (1/2 - 1/3 - 1/6) = 1/0.

*It really annoys me that English doesn't have a commonly understood way to express the "opposite of respectively", where in one listing the order is one way but in the other listing it's the other way. In Hebrew our word for respectively in this sense is "בהתאמה" (pronounced [behatʔaˈma]), which roughly means "in a matching way", and so we can say "בהתאמה הפוכה" ([behatʔaˈmahafuˈχa]), which roughly means "in a way that matches in reverse", but when I search how to say the same thing in English I can barely find a place where people even understood the question and when I finally find a stackexchange post where the question is asked clearly enough the top answer concludes with "just don't do it".

[Dave Keenan]

... formal fractions ...

I think "formal fractions" is an excellent name. Obvious now that you've given it. Captures the idea perfectly. And from a non-native English speaker. Well done.

I don't understand or find any use for Douglas's ideas of locking just the numerator, or just the denominator, and I have not tried to understand your analysis. The idea seems far too simple to me, to be worth so many words and formulae.

... and in general augmentation and diminution by numbers are just multiplication and division by monomials, inversely*.
...
*It really annoys me that English doesn't have a commonly understood way to express the "opposite of respectively", where in one listing the order is one way but in the other listing it's the other way. In Hebrew our word for respectively in this sense is "בהתאמה" (pronounced [behatʔaˈma]), which roughly means "in a matching way", and so we can say "בהתאמה הפוכה" ([behatʔaˈmahafuˈχa]), which roughly means "in a way that matches in reverse", but when I search how to say the same thing in English I can barely find a place where people even understood the question and when I finally find a stackexchange post where the question is asked clearly enough the top answer concludes with "just don't do it".

You would need to have a good reason not to simply reverse the second list.

So, "... and in general augmentation and diminution by numbers are just division and multiplication by monomials, respectively."

But assuming I did have a good reason, I would, in general, use "separately, in the reverse of the order given". I note that "respectively" means not only "in the order given", but also "separately". For example "John and Robert were awarded the Distinguished Conduct Medal and the Victoria Cross, in the order given", still allows that they each received both medals, but "John and Robert were awarded the Distinguished Conduct Medal and the Victoria Cross, respectively", does not.

But in your case, there is little danger that the reader would think that each of the former operations corresponds to both of the latter operations.

So, "... and in general augmentation and diminution by numbers are just division and multiplication by monomials, in the reverse of the order given."

"Inversely" or "reversely" may be OK in other contexts, but here, in the context of arithmetic operations, those words have other meanings that would make the sentence ambiguous, because inverse multiplication is division and reverse division of x by y is equal to y divided by x. "In the reverse of the order given" still suffers from that ambiguity to some degree.

If you really need a single word, I suggest most people would understand what you meant if you used "antirespectively", by analogy with "anticlockwise", and more specifically "antiparallel" (used as a relationship between vectors, molecules and electronic devices), where "anti-" means reverse. The other meanings of "anti-": against, hostile to, counteracting, neutralizing; don't make any sense when applied to "respectively", and so shouldn't cause any ambiguity.

So, "... and in general augmentation and diminution by numbers are just division and multiplication by monomials, antirespectively."

At least one responder agrees with me in each of these forum threads:
https://english.stackexchange.com/quest ... spectively
https://www.reddit.com/r/logophilia/com ... pectively/

Many other responders propose solutions that only work for lists of two items and would not work for 3 items or more in each list. Others seem not to have understood the question as they propose solutions that mean "in no particular order". "Antirespectively" does not have these problems. I suppose that, in general (but not in your case), a reader might suspect "antirespectively" meant merely "in no particular order", but in that case they should wonder why the author didn't just write "not respectively", "non-respectively" or "irrespectively".

[רועיסיני]

... formal fractions ...

I think "formal fractions" is an excellent name. Obvious now that you've given it. Captures the idea perfectly. And from a non-native English speaker. Well done.

Thanks! I'm glad you liked it.

... and in general augmentation and diminution by numbers are just multiplication and division by monomials, inversely*.
...
*It really annoys me that English doesn't have a commonly understood way to express the "opposite of respectively", where in one listing the order is one way but in the other listing it's the other way. In Hebrew our word for respectively in this sense is "בהתאמה" (pronounced [behatʔaˈma]), which roughly means "in a matching way", and so we can say "בהתאמה הפוכה" ([behatʔaˈmahafuˈχa]), which roughly means "in a way that matches in reverse", but when I search how to say the same thing in English I can barely find a place where people even understood the question and when I finally find a stackexchange post where the question is asked clearly enough the top answer concludes with "just don't do it".

You would need to have a good reason not to simply reverse the second list.

The reason is that "multiplication and division" is the usual order and "division and multiplication" just sounds bad to my ears.

If you really need a single word, I suggest most people would understand what you meant if you used "antirespectively", by analogy with "anticlockwise", and more specifically "antiparallel" (used as a relationship between vectors, molecules and electronic devices), where "anti-" means reverse. The other meanings of "anti-": against, hostile to, counteracting, neutralizing; don't make any sense when applied to "respectively", and so shouldn't cause any ambiguity.

That's an excellent idea! I must have missed that in the other threads, it was probably buried under too much nonsense like "disrespectively" and similar jarring suggestions.

[רועיסיני]

The idea seems far to simple to me, to be worth so many words and formulae.

I agree. As I said in the beginning of my first comment (and also as the first example in the fourth) it's most simply just a pair of numbers, or a vector, even simpler than a fraction because fractions you can also reduce. The rest is just me having fun.