The top number in a time signature is easy to understand. Is the song in four? Count "one, two, three, four." Is it in three? Count "one, two, three." Is it in five? Count "one, two, three, four, five." That's all there is to it. However, the bottom number is another story. What is going on down there? I collected various examples of time signatures in this track I made, but I didn't understand why "Solsbury Hill" by Peter Gabriel is in 7/4 but "One More Night" by Can is in 7/8.
I'm not alone in finding this confusing. My students struggle with it too. They are right to! Every explanation I have ever seen of the bottom number is circular. Say we're talking about 4/4 time, so a beat is a quarter note long. Okay, so what is a quarter note? Well, four of them make a measure. And what's a measure? Uh... four quarter notes.
It only gets worse when the bottom number is some number other than four. Let's say we're talking about 6/8, so a beat is an eighth note long. And what is an eighth note? It's half a quarter note, so there should be eight of them in a measure, right? Well, no, because in 6/8 time, there are six eighth notes in a measure. So, what is the quarter note that a beat is half of? I feel like a crazy person just typing this. Imagine being a beginner trying to learn it.
It gets worse. How do you know if you're hearing 6/4 or 6/8? At some given tempo, you count 6/8 twice as fast as 6/4. But couldn't you just write 6/4 and double the tempo? You could, but 6/8 feels different. Feels different how? Well, it feels more like two slower beats divided into triplets. But couldn't 6/4 feel like that too? Logically, yes, it just... doesn't. But how as a learner are you supposed to know that?
I have read every explanation I could find of the bottom number in time signatures, and asked every music theory teacher I know. Some of them fall back on the same circular logic I described above. Some of them cheerfully admit that the bottom number has no logic to it, that it's a set of conventions and stylistic associations. You can only develop an intuition for these conventions after you play a lot of classical repertoire. Timothy Chenette says you should internalize them through conducting, even if you never actually conduct an ensemble. That's probably good advice, but it's a little late for me, and I don't know where I would fit conducting practice time into a one-semester theory or aural skills class.
One thing that has helped me is the idea that the bottom number historically conveyed tempo information. For a Baroque or Classical-era musician, the convention went like this:
If the bottom number is two, then the tempo is slow.
If the bottom number is four, then the tempo is medium.
If the bottom number is eight, then the tempo is fast.
To illustrate this idea to myself and my students, I made a song about it. The track is at a steady 120 beats per minute throughout, and it switches between 2/2, 2/4, 3/2, 3/4, 3/8, 4/2, 4/4, 6/4, 6/8, 9/4 and 9/8 time. The synth plays each beat on the corresponding scale degree from C Dorian mode, so beat one is C, beat two is D, beat three is E-flat, and so on. Here's an interactive Noteflight score.
My track doesn't give you the full picture. Baroque-era time signatures weren't just about metronomic tempo; the different numbers also spoke to the mood and feeling of the piece. A musician in Bach's time would know that 3/4 implied a minuet or waltz, whereas 3/8 implied a scherzo. Also, "smaller number means slower" is not always true. While 2/2 is supposedly half as fast as 2/4, sometimes composers use 2/2 for uptempo orchestral music. Why? Ask a conductor, I guess.
Ultimately, the best thing we can do is to listen to a lot of real-world examples. Ali Jamieson has a great list. Cadence Hira has a nice list drawn from game scores. And Wenatchee the Hatchet has many more classical examples. Thank you also to my Twitter friends for suggesting some of the examples below.
Examples in 2
2/2
Opera overtures, for example, The Magic Flute by Mozart.
The first half of the prelude to Bach's G minor Lute Suite (the slow part).
The first movement of Beethoven's Moonlight Sonata, apparently, even though the constant triplets make it feel like it's in three - I put a duple meter beat under it to help me hear it in 2.
2/4
Nursery rhymes like "Pop Goes The Weasel".
European folk and traditional music, like sea shanties, for example "Wellerman".
"Zorba's Dance" by Mikis Theodorakis.
Country music is supposedly in 2/4, but I have never in my life heard a country musician count it that way.
Examples in 3
3/2
"Short Ride in a Fast Machine" by John Adams.
"Widmung" by Robert Schumann.
Handel's Sarabande.
"Take Me to Church" by Hozier, apparently, though I would be very surprised if Hozier knew what 3/2 was.
3/4
Waltzes, like “The Blue Danube” by Johann Strauss II.
"My Favorite Things" from The Sound of Music.
3/8
Bach's two-part inventions in D major, D minor and E major; also, the second, faster part of the prelude to the Lute Suite in G minor.
"Hedwig's Theme" by John Williams.
3/16
The words "Mother Superior" in "Happiness is a Warm Gun" by the Beatles.
Examples in 4
4/2
Not widely used after the Baroque era; Franz Schubert used it for the Impromptu op. 90 no. 3 in G-Flat Major, but all the triplets make it feel like fast triple time.
4/4
Most Anglo-American popular music: the large majority of rock songs, almost all top 40 pop songs, just about every rap song, every techno and house track.
4/8
Exists mainly in arcane music theory discussions and tutorials; I can't find a single real-world piece of music written in it.
Examples in 5
5/4
"Take Five" by Paul Desmond (grouped 3+2).
"Anything Goes" as performed by Brad Mehldau.
The Mission Impossible theme by Lalo Schifrin (I see it described as 5/8 sometimes, I don't know how Schifrin wrote it originally).
"Four Sticks" by Led Zeppelin (the bridge is in 6/8).
5/8
The Bulgarian folk dance "Paidushko Horo".
Six Dances in Bulgarian Rhythm [3/6] by Béla Bartók.
Examples in 6
6/4
"Kashmir" by Led Zeppelin.
"Electric Feel" by MGMT.
6/8
"Norwegian Wood (This Bird Has Flown)" by the Beatles.
"Nothing Else Matters" by Metallica and many other power ballads.
"End of the Road" by Boyz II Men and many other R&B ballads.
Examples in 7
7/4
"Solsbury Hill" by Peter Gabriel.
"Estimated Prophet" by the Grateful Dead.
7/8
"One More Night" by Can.
"Theme from TRON" by Wendy Carlos (even though it's slow).
"Unsquare Dance" by Dave Brubeck.
Examples in 8
There is hardly any music that is notated in eight, aside from odd bars of modernist classical works that change meter a lot. But you could argue that most pop music that's supposedly in four is really in eight. Notice that dancers and choreographers often count in "five, six, seven, eight" rather than "one, two, three, four." There is plenty of music where everything is organized into groups of eight beats, but we express that idea with hypermeter, not meter. I have seen it argued that the tresillo rhythm should be considered 8/8, but everyone keeps their DAWs set to 4/4.
Examples in 9
9/4
"9/4 the Ladies" by Balkan Beat Box.
9/8
"Ride of the Valkyries" by Richard Wagner.
"Claire de Lune" by Claude Debussy.
"The Cool, Cool River" by Paul Simon.
"Blue Rondo a la Turk" by Dave Brubeck, though it's it's not grouped 3+3+3, it's grouped 2+2+2+3.
Examples in 10
10/4
"Playing in the Band" by the Grateful Dead (it's grouped 4+2+4).
"Unisphere" by Dave Brubeck.
10 /8
20 Caprices & Rhythmic Studies, Set 1: No. 1. Allegro giusto by Émile Jaques-Dalcroze.
Examples in 11
11/4
"Eleven Four" by Paul Desmond.
"Whipping Post" by the Allman Brothers, though really it's 12/4 with the last beat omitted.
11/8
Gankina Horo (sometimes written in 11/16).
Bachianas Brasileiras No. 9, second movement by Heitor Villa-Lobos.
Examples in twelve
12/4
12/4 - I can't find a single example; it would sound like 4/4 with three-bar hypermeter, like "It's About That Time" by Miles Davis.
12/8
Many blues and country songs.
"The Way You Make Me Feel" by Michael Jackson.
"Truckin'" by the Grateful Dead.
I am collecting these examples for the specific purpose of teaching music theory in a pop context. How many of these time signatures do pop musicians need to know? They certainly need to know about the top number, but I question the need for them to get familiar with the bottom number. DAW producers use tempo settings and finer grid subdivisions where composers of notated music would use time signatures. People who are playing and writing by ear tend to count inconsistently in four or eight for duple meter and three or six or twelve for triple meter, and aural communication is enough to keep everyone lined up.
I think you'd find Edwin Gordon's writings on this interesting, especially as he shares many of your complaints. I summarize some of it here: https://twochords.substack.com/p/meter
One of Gordon's more useful ideas is that of "enrhythmic" time signatures, so 2|4, 2|2, and 4|4 are different ways of notating the same thing.
I also like Orff-style time signatures where there is an actual note in the bottom half instead of a number, and it kills me that notation software that purports to be supportive to music education does not do this natively — it's only available as a workaround. (The vitriol toward this kind of notation from non-educators is also depressing.)
For performing musicians, it's important to know what it means to play in a 6/8 feel (even when notated in 3/4) or how to play a 4/4 piece in a 12/8 shuffle feel. These are perhaps practical issues of "style" more than theory, but if your students are going to write music and arrange it, they will encounter these.