Is there a logical reason why music notes keep halving? In other words, why do notes go like :1, 1/2, 1/4, 1/8, 1/16… instead of this: 1, 1/2, 1/4, 1/6, 1/8, 1/10…
Ehr… because the divisions are that way. It’s base 2 division. Why do the usual Arabic numbers keep adding a new figure once the rest all go to 9, when counting up? Because we usually count in base 10. It makes a lot more sense to have the same base for every division, or for any number of figures, than to keep changing bases.
I’m not a musician, but I know that they have notes that don’t fall in the power-of-one-half pattern: dotted 1/8-notes and 1/4-note triplets are the equivalent of 1/6 notes. A dotted 1/4-note is a 1/3-note.
There are notations that allow you to divide a measure into any fractions you want (Google tuplet). But for the most part, you want the beat to occur in a predictable pattern, and for the most part that involves dividing a beat into two or (at most) three parts, even if you’re using a time signature that divides the measure into some weird pattern.
You can do any kind of fractional division of a beat that you want (in notation, this is typically done by bracketing together and labeling a group of n notes above the desired number of beats). But, if your basic system for counting doesn’t involve, um, evenly-fitting subdivisions (as Nava points out, constant halving is base 2 division), then you don’t have a ‘system’ at all.
“Say what you will about base 2 division, but at least it’s a system!”
We could (and regularly do) split our beats into 3s, though the written notation is less elegant than dividing into 2s.
But, really, you want to be able to add a whole-number amount of every main subdivision together to reach a longer, higher-level duration. Otherwise, you’d be trying to read and understand the relationship between, say a 1/6th beat and an 1/8th tied to a 1/10th, which is far from practical or useful.
Lol! Like I said, I’m not a musician, but I spent 10 minutes trying to figure out how you would possibly fit all that into a measure of music (let alone why!) and I couldn’t do it.
I’m gonna guess that not only do you not want to have to compute the least common denominator to figure out how fast to tap your feet while playing, but the difference between a 1/8 and 1/10 note would be pretty hard to detect. On the other hand, “roughly half as long” is probably a fairly intuitive calculation.
What time signature are we talking about?
Or possible.
A dotted 1/8-note is a 3/16 note, and a dotted 1/4 note is a 3/8 note. Neither is quite equivalent to a piece of a triplet.
There are also grace notes, which don’t have any particular set duration, but are basically just “really quick”. You only ever see them appended to the beginning of a longish note, which then correspondingly takes up slightly less time than it ordinarily would.
You’re absolutely right. Like I said, I’m not a musician, but I should have been able to figure that out myself. Not sure what I was thinking.
A 1/6 note would be one part of a quarter note triplet. A tenth note would be one part of an eighth-note quintuplet (which would be eighth notes usually marked with a 5 or 5:4 above them.)
Of course, then there’s irrational time signatures which can, in essence, have sixth notes, tenth notes and the like, but that’s a bit beyond the scope of this discussion, methinks.
There are also the compound time signatures, like 9/4, in which the first “division” is divided into three parts, with all subsequent “divisions” based on base 2. The trick, though, is that this is all made to work with the regular base 2 divisions.
Hey! I have a terrible sense of rhythm, but I tried my best to count out a beat like that time signature. I couldn’t do it very well, but I immediately thought, “That sounds like Blue Rondo a la Turk!” So I ask Siri what the time signature was for Blue Rondo a la Turk. She tells me it was 9/8.
Pretty cool!
You may not be impressed, but in my own mind I redeemed myself for that embarrassing error with the dotted notes. 
I like to play pieces in 6/8. Sets me feet ta dancin’.
Yeah, Blue Rondo a la Turk is 9/8, with subdivisions in 2+2+2+3 and 3+3+3. (Three bars of the first followed by one bar of the next if I’m playing it in my head correctly.) The 3+3+3 grouping is somewhat common, but the 2+2+2+3 (or any mix thereof) is much rarer in Western music. Bach’s “Jesu, Joy of Man’s Desiring” would be a typical 9/8 example where the beats are subdivided 3+3+3. Also, “Ride of the Valkyries,” but the Bach example is probably easier to hear and count.
Music doesn’t need to be written entirely in base 2. See Tuplet - Wikipedia.