I’m going to explain the concept of irrational time signatures and how this seemingly complex subject can be utilised by music producers and beat makers on a digital audio workstation (DAW). Rather than on a score sheet where it is more commonly used in the case of more traditional music composers.
My favourite video, that is probably the most incisive on the subject of irrational time signatures and rhythms is this one by composer David Bruce, that I strongly encourage you to watch below;
To summarise, irrational time signatures (or non-dyadic time signatures) are essentially time signatures where the bottom number (i.e. the note value on each bar) isn’t divisible to the power of 2. The most common time signature in modern western music is 4/4 (4 quarter notes to one bar). Although other, albeit less common time signatures can include the likes of 3/4 (3 quarter notes to one bar), or 6/8 (6 eighth notes to one bar).
This piece in Study Bass explains the basics well in Time Signatures by Study Bass ;
“The bottom number tells you what kind of note to count. That is, whether to count the beats as quarter notes, eighth notes, or sixteenth notes. So the only numbers you will see as the bottom number (the denominator) will correspond to note values:
- 1 = whole note (you’ll never see this)
- 2 = half note
- 4 = quarter note
- 8 = eighth note
- 16 = sixteenth note
You could continue on with 32, 64, but you will hopefully never encounter them! After a while it gets a bit unwieldy. The most common bottom numbers are 4, 8 and 16.”
https://www.studybass.com/lessons/reading-music/time-signatures/
As I said, irrational time signatures are those where the denominator (bottom number) isn’t divisible to the power of 2. An example might be 4/6, or 3/5, or 3/7. Generally speaking this has more utility in sheet music as a way for performance musicians to better understand how the composer wants them to play a piece.
I’m a music producer more so than an actual musician, so there is less utility in this concept for me. However I was working on a very experimental song, called ‘REPHRASED YESTERDAYS’, where I spent a lot of time trying to play with different time signatures, (which is rare in hip hop). This prompted me to learn about irrational time signatures and I decided to experiment further and see how this can be utilised in a DAW.
The first hurdle I encountered was realising that I don’t think any DAW (at least not in Logic, which I use) actually allows you to use irrational time signatures. You’ll notice when changing the time signature in Logic, the bottom number can only be to the power of 2. I spent ages racking my brains trying to figure out a way round it. Then I realised from the David Bruce video, irrational time signatures only real utility is in conjunction with ‘rational ones’. From there I realised having that ability on a DAW isn’t really necessary, especially as there is quite a clear way round it. And that is by simply changing the song tempo! You may not be able to create a song in a 4/5 time signature in the time signature display itself. But you can essentially create the same feel by changing the tempo to a fraction of the original time signature.
There’s an equation you can use for figuring this out. Which is as follows;
(Old BPM/Old Numerator) x New Denominator = New BPM
So for example if I started off with a song that’s in 4/4 at 100BPM and wanted to create a 4/5 part of the song at 100BPM I can actually create this by changing the BPM rather than the time sig, using the formula above to calculate the difference between the note value of 4 and 5.
Therefore, I do this by dividing 100 (100bpm) in this case, by 4 (old denominator) which gives me 25.
I then take this figure and times it by the the new denominator, in this case 5;
25 x 5 = 125BPM
So if I started with a 4/4 time signature at 100BPM, then move to a 4/4 time signature at 125BPM, that essentially creates the same feel as would be the case if you were able to change the time signature to 4/5 at 100BPM. This is probably why DAW’s don’t seem to give you this option. If you can change tempo, it really isn’t necessary!
As another example, if you wanted to create a song that starts off at 4/4 100BPM but moves to a 3/3 time signature at 100BPM you would;
Divide 100 (100bpm) in this case, by 4 (old denominator) which again gives me 25.
Then 25 x 3 = 75BPM
So a part of a song at 3/4 time sig at 75BPM would give the same feel as 3/3 time sig at 100BPM if you were able to do so in the time sig display (which you can’t)….
It’s worth noting my reasons for wanting to learn about this and how to utilise it, is because I wanted to creatively experiment with a warped and wonky sense of time in my song ‘REPHRASED YESTERDAYS’.
In fact this song goes so warped and wonky with multiple changes in tempo to imply a change in time signature to different irrational time sigs, with some additional time sig changes thrown in. This is intentional to highlight a sense of absurdity I feel about our modern age, where it seems like we’re stuck on repeat in the world of art, music and culture with no sense of significant break or change in sight beyond comparatively small subtleties. ‘REPHRASED YESTERDAYS’ is basically my attempt at looking into the possibility of change, when we all seem culturally stuck in a rut amongst other things…
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