Question in title. Suppose my friend and I are discussing a song and I say it's written in 4/4 at 120bpm but my friend says it's written in 4/8 at 60bpm. Under what conditions would one answer be correct?
4/8 at 60 bpm and 4/4 at 120 bpm are two different things.
Each measure of music is related to a pattern of stresses. Since the measures of both options you presented have four beats, they would both have essentially the same stress patterns. At 60 bpm, each stress pattern lasts four seconds. At 120 bpm, each stress pattern lasts only two seconds. The denominators of the time signatures does not affect that.
There’s not always a single correct answer for how a song is or would be notated. If official sheet music for the song is published, then that is how it was written by the song writers or that is what has been approved for publication. But there’s often more than one way to notate the same music that would sound the same.
The notation is not the music, it’s just a guide for how to produce or reproduce the music.
(Edited) To expand a bit on the answer of @Toddwilcox and give an idea of their equivalents and how different they are:
4 quarter notes in 4/4 time at 120 bpm would sound the same as 4 sixteenth notes in 4/8 time at 60 bpm.
4 quarter notes in 4/4 time at 120 bpm would sound the same as 4 eighth notes in 4/8 time at 120 bpm.
4 quarter notes in 4/4 time at 120 bpm would sound the same as 4 quarter notes in 4/8 time at 240 bpm.
By”sound the same”, I mean take up the same amount of time, I’m not taking to account the natural accents of the different note values in their respective time signatures.
The question suggests listening to a recording and attempting to guess the time signature. In that case, 4/2, 4/4, 4/8, 4/16, etc., would be effectively indistinguishable. One could discern the upper "4" according to its canonical STRONG-weak-SemiStrong-weakest pattern of stresses, but the lower number serves as an indicator of notational convention. One could make some educated guesses about the lower number based on experience combined with the tempo and style of the piece, but it would be a guess.
In terms of relating time signatures via BPM, it depends on how one interprets "the beat". If "the beat" refers to the primary underlying pulse of the music, then 4/2, 4/4, 4/8, etc., would sound identical. At "pulse" = 120BPM, for example, there would be 30 measures per minute regardless the specific time signature. The time signature would only specify the type of note used to represent the basic pulse.
However, if "the beat" actually means "quarter note" (crotchet), that is, the BPM is tied to a specific notation rather than the pulse itself, then the differences suggested in the OP turn up. At quarter-note = 120BPM, then 4/2 = 15 measures/minute; 4/4 = 30 measures/minute; 4/8 = 60 measures/minute; .... That is, when holding the quarter-note steady 4/2 would be half the speed of 4/4, which in turn would be half the speed of 4/8.
No, they're not the same. Since the top number tells how many beats, and that is the same (4) for each, then that is the deciding factor. Whatever the lower number is - denoting what kind of beats those four are, it has no bearing on the tempo itself.
The bpm tells what speed those four go along at, regardless of what those four represent.
It's very easy to think 'a quaver is half a crotchet' - true, in the same piece, but that doesn't count when comparing one piece to another.
4/4 @120bpm means one whole bar takes two seconds to count.One beat is a crotchet.
4/8 @60bpm means one whole bar takes four seconds to count.One beat is a quaver.
Please bear in mind that just because one time sig. has quavers and the other crotchets, which are twice as long, that has absolutely no bearing on the tempo of any piece.
If you meant 4/8 @240bpm, then they would both take the same time, and be at the same speed, or tempo.
Are 4/8 in 60bpm and 4/4 in 120bpm the same?
They can be, to some extent (as in the last examples in this answer). The most natural way to make the change from 4/4 to 4/8, however, is to keep the same number of beats per minute, because the temporal unit associated with the beat in these two meters is not the same. In 4/4, it is the quarter note, while in 4/8 it is the eighth note.
When you write music out, there are several quantities that must be determined:
- The duration of a "beat"
- The absolute duration of each note of the melody
- The number of beats in each measure
I say "absolute" duration of each note because the relative duration is determined by the melody. That is, if you change the relative duration, you're changing the music. As an example, consider the first phrase of Aura Lee, perhaps better known to many as the tune of Love Me Tender. The seventh note has twice the duration of the first six, and the last note has twice the duration of the seventh:
X: 1 L: 1/4 K: G M: 4/4 DGFG|AEA2|GFEF|G4
We will put that melody into 4/4 and consider how we can write it out equivalently in 4/8. But first, a word about beats per minute.
A few months ago, I was disappointed to read a question on this site that concerned some software or other that uses "beats per minute" to mean "quarter notes per minute." This goes against two hundred years of practice in using the metronome to specify tempo, where "beat" is a variable that is set for any given metronome marking to some note value. It can be a quarter note, and it certainly often is a quarter note, but it can also be a half note, an eighth note. In compound meters, it is usually the dotted note representing the compound beat, whether a dotted quarter note or some other dotted note. This answer will nonetheless take this meaning of the term into account, calling it the "faulty understanding" of the term, in addition to the more traditional meaning.
Now, Aura Lee in 4/4:
X: 2 L: 1/4 K: G Q: 1/4=120 M: 4/4 DGFG|AEA2|GFEF|G4| w: As the black-bird, in the spring, 'neath the wil-low tree, DGFG|AEA2|GFEF|G4| w: Sat and piped, I heard him sing, sing-ing Aur-a Lee. BBB2|BBB2|BAGA|B4| w: Aur-a Lee, Aur-a Lee, maid with gol-den hair, BBcB|AEA3/2G/2|GFB3/2A/2|G4|] w: Sun-shine came a-long with thee, and swal-lows in the air.
The most straightforward 4/8 version of this is also in 120 beats per minute (although it is not 120 beats per minute under the faulty understanding, because there are 60 quarter notes per minute):
X: 3 L: 1/8 K: G Q: 1/8=120 M: 4/4 DGFG|AEA2|GFEF|G4| w: As the black-bird, in the spring, 'neath the wil-low tree, DGFG|AEA2|GFEF|G4| w: Sat and piped, I heard him sing, sing-ing Aur-a Lee. BBB2|BBB2|BAGA|B4| w: Aur-a Lee, Aur-a Lee, maid with gol-den hair, BBcB|AEA3/2G/2|GFB3/2A/2|G4|] w: Sun-shine came a-long with thee, and swal-lows in the air.
This makes the most sense, because it keeps the same relationship between the meter and the melody. There isn't really any way to say that a particular performance of this tune should be notated using one or the other of these first two examples.
If you change the relationship between the meter and the melody, however, there are other options. This example effectively puts the melody into 2/4 rather than 4/8, so it doesn't make a lot of sense unless there's another part that really is in 4/8, but it does put the melody at the same speed as the first example (and it is also 120 beats per minute under the faulty understanding):
X: 4 L: 1/4 K: G Q: 1/8=240 M: 4/8 DG|FG|AE|A2|GF|EF|G2-|G2| w: As the black-bird, in the spring, 'neath the wil-low tree, * DG|FG|AE|A2|GF|EF|G2-|G2| w: Sat and piped, I heard him sing, sing-ing Aur-a Lee. * BB|B2|BB|B2|BA|GA|B2-|B2| w: Aur-a Lee, Aur-a Lee, maid with gol-den hair, * BB|cB|AE|A3/2G/2|GF|B3/2A/2|G2-|G2|] w: Sun-shine came a-long with thee, and swal-lows in the air. *
Whether something is in four-beat meter or two-beat meter (with twice as many measures) is something you can argue about, although it remains a somewhat subjective question. For example, I began writing this answer planning to use Twinkle, Twinkle, Little Star as the example tune, but I decided not to because even though you can find it notated in 4/4, it works better in 2/4. For example, Mozart used 2/4 for his variations on the tune.
The previous example (nominally) changed the number of beats in each note of the melody. You can change that both ways: it's also possible to write this tune at the same speed in 4/8 with 60 beats per minute by putting twice as many notes in a measure:
X: 5 L: 1/16 K: G Q: 1/8=60 M: 4/8 DGFGAEA2|GFEFG4| w: As the black-bird, in the spring, 'neath the wil-low tree, DGFGAEA2|GFEFG4| w: Sat and piped, I heard him sing, sing-ing Aur-a Lee. BBB2BBB2|BAGAB4| w: Aur-a Lee, Aur-a Lee, maid with gol-den hair, BBcBAEA3/2G/2|GFB3/2A/2G4|] w: Sun-shine came a-long with thee, and swal-lows in the air.
X: 6 L: 1/16 K: G Q: 1/8=60 M: 4/8 DGFG|AEA2GFEF|G4DGFG| w: As the black-bird, in the spring, 'neath the wil-low tree, Sat and piped, I AEA2GFEF|G4BBB2| w: heard him sing, sing-ing Aur-a Lee. Aur-a Lee, BBB2BAGA|B4BBcB| w: Aur-a Lee, maid with gol-den hair, Sun-shine came a- AEA3/2G/2GFB3/2A/2|G4|] w: long with thee, and swal-lows in the air.
Once again, this is almost certainly a misuse of the meter. But it does give us the melody at the same speed as the first example.
So, in conclusion, the most natural way to write a melody in 4/4 and 4/8 equivalently is to halve the note values in 4/8 as compared to 4/4 and to keep the same number of beats per minute.