For a short countdown of say 15 seconds, we could easily use music:
Conveying the number ten can be achieved by doing two sequences of 5: Do Re Mi Fa Sol. And then: Fa Sol La Si Do.
Or Do Re Mi Fa Sol. Then Sol Fa Mi Re Do.
There is an inherent rhythmic problem with the number 10, but
then again you can count to 9 with 3 trios and then reach the final tone by playing it to a count of 3 whole notes. So there you are.
In its simplest form, conveying a countdown or how much time is left, can be achieved by playing the final low note first, then playing a scale that will eventually reach it.
For example playing a note every interval:
Starting with C3 ... then C4, B4, A4, G3, F3, E3, D3, C3 C3 C3, with an ending part.
Or play the lower tone after each note:
Starting with C3 ... then C4, C3. B4, C3. A4, C3. G3, C3.
F3, C3. E3, C3. D3, C3. C3 C3! and an ending part.
Or only insert the low note every once in a while.
But why stay with single notes?
Each "note" in the sequence can be a song part or a tune played to its basic tone. That way the time left can be conveyed through a tune.
Of course, this is good only for short intervals like a countdown of up to 5 minutes for the oven or micro.
If you have to wait an hour and a half, you don't want the washing machine to play this unasked-for music. So you can set it to give you half-hour warnings, and then do a 10 or 30-second countdown tune.
Conveying an hour, 3/4, half and quarter hour, we may use a tune to four basic notes in increasing pitch, say a major chord, then the one that we are at is repeated several times.
So: to convey only a quarter of an hour passed and there is still 3/4 of an hour left, we could first play:
Do Me, Sol, Do4! to convey the full time that was expected, and then:
Do4 Sol, Sol, Sol and softly: Me Do,
to say that there is still 3/4 of an hour left to wait.