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This question about notation is simple as dirt, but would be useful for me regardless, because of some work that I'm doing in music theory.

Basically, while there's a notation for subdividing the degree into arcminutes and arcseconds, so that "180 degrees, 30 minutes, 30 seconds" becomes 180º30'30", I can't find any notation for any unit which is then further subdivided into 360 degrees.

I thought there might be one for angles, so that rather than writing 1980º, I could write something like "5 cycles 180º" using some symbol for "cycles," but I couldn't find anything.

Is there any notation for something like this, or for any other unit of measure habitually subdivided into 360 parts, even if they aren't called "degrees?"

I would be interested both in any modern units of measure with some such notation, or in obsolete ones used only in antiquity.

Also, for those curious about how this question could possibly be
relevant to anything, it has to do with that the established
convention in music theory is to subdivide individual steps in the
Western 12-note musical scale into 100 "cents."

I'm writing something now about the presentational advantages of
instead subdividing the step by some highly composite number,
especially when the scale being subdivided is an arbitrary non-Western
scale. This can be useful for various specific music-theoretical
reasons that aren't relevant here, but which are related to the same
reason that highly composite numbers caught on when subdividing units
of measure in general.

A subdivision into 360 parts, specifically,
stands out for a few different mathematical reasons, also not relevant
here. But, while writing a proposal to this effect, I realized that I
knew of no established notation in which something is divided into 360
parts. So before I just make one up, so that two and a half steps
becomes 2;180º or something, I want to see if any such common or
historical notation is known to mathematicians.

A soft question, no doubt, but any insights would be much appreciated!

In general? I won't go so far as to rule it out completely, but I'll say with moderate confidence there probably isn't.

It's probably worth pointing out that while the Ancient Babylonians are credited with originating the 360 degree circle, this actually a half-truth that glosses over a point relevant here. They didn't actually have any concept of angle or arclength as numerical magnitudes. They just had length. But they could take some unit length, use it to describe a unit circle of radius 1, and simply construct an inscribed regular hexagon of perimeter 6 inside the circle, and see that the circumference of the circle must be slightly greater than 6. Since their numeral system was sexagesimal, they grouping and partitioning of units was by 60s, so if they wanted to refine the resolution of length measurement by an order of magnitude they would use 1unit = 60parts and get 6units = 6*60parts = 360parts.

Now in answer to your question, we might write 60º = 1', 60'= 1'' and so on. This superscript notation comes from the Romans. You'll note that the primes are in fact the Roman numerals for one and two, and the degree symbol is indeed a zero glyph which is functionally a decimal point.

You could call it a "decasecond" since $10'' cdot 360 = 3600'' = 60' =1^circ$ or $1$ hour.

Also it is almost the number of days in a year.

There does not appear to be an established notation for the unit circle, but there are pretty close things.

The notation of $^circ, ', ''$ derives from using roman numbers as column markers, as one might write 5h 3t 6 for five hundred, thirty, six. It is not specific to base 60, since the same scheme is used of feet, inches and lines (and downwards to points), on a duodecimal scale, for the french grade, using centisimal scale, and by the earliest decimalists, to denote dimes, cents and mills.

The measure at $^circ$ is a unit. When one writes $1^circ 30'$, one is writing 1.5 units. However, there are some interesting arguments that the circle, and not some fraction ought be the unit of measure.

When one measures angle, it is usually reckoned as a fraction of surface, measured in measures of the radius. The Sumerian systems suppose that $pi=3$, and has $2pi cdot 60$ degrees for circles in the sky, and $pi cdot 60$ ells of $24$ digits for real circles (like things you can walk around). This is in Sir Thomas Heath's 'history of greek mathematics'.

In the higher dimensions, one might want the same angle preserved when a cartesian product of full space is applied: that is, the angle between the planes is the same as the solid angle. This happens when all-space is taken as $1^circ$, and the primes, seconds, etc refer to fractions of it. So the angle where two square faces of a hexagonal prism meet is the same as the hexagon corner-angle, ie 0;40 = 1/3.

The measures are made in base 120, which greatly simplifies hand calculation, and because the first division gives the 12 hours of the clock, simplifies that too. Base 120 is historically attested in England, see eg 120 on the wikipedia for references.

Aslo, because a complete circle is shown with a circle-rune $^circ$, it some how makes some sense.

Why not simply re-use minutes and seconds?

It divides your unit in 3600 parts instead of 360 but just keeping seconds in tens should cure that.

Minutes and seconds were pars minuta prima and pars minuta secunda in Latin, ie "first small part" and "second small part". I believe that at one tine "thirds" were used as the subdivision after seconds, but the name is obviously confusing—you want it to mean $frac{1}{60^3}$, not $frac{1}{3}$. Presumably this is why it fell out of use.

It would make sense to indicate them with three prime marks though, just continuing the pattern after minutes and seconds. I think this notation has been used.

Archaeologists studying Babylonian mathematics use a different system for representing sexagesimal (ie base $60$) numbers which involves things like using a semicolon as the "sexagesimal point" and commas between the "digits".