turntable motor speed

I understand that an AC turntable motor uses the incoming line frequency of 60hz to set the motor speed, but how does a DC motor determine speed and how do you vary the speed to adjust for changes in platter or pulley size. Is the voltage changed and therefore the speed is controlled by the voltage or is there some other method at work. For instance, if I am running a 24vdc motor, would increasing the voltage to 25vdc cause the motor to run faster? Is it important to ensure that the voltage is exactly 24vdc, or is anything close acceptable?
Increasing voltage will cause the DC motor run faster.
It's very important but nevertheless it's a large downside of DC motor that it depends on input voltage. The voltage stabilization circuits should be very precise.
There are motors that have quartz AC generators of a stable frequency.
All DC motors used for such a purpose have a tachometer and feedback regulation circuit that controls the motor voltage. It's most commonly built right into the motor itself (just like a cassette-deck capstan motor) but occasionally you'll find it as a circuit external to the motor.

Simply regulating the value of the input voltage will not keep a DC motor at an exact speed.
Ok, so what does the feedback regulation do to the motor to change its speed. Does it change the voltage, and if so, how? more for faster? I mean, its fine to say that a circuit regulates the motor, but my question is how does it do that. The motor only has two wires that feed it, so something in those wires is telling it to speed up or slow down. What is it? voltage?
Yes, the change in speed in a DC motor is directly proportional to its voltage input: more volts = more speed; less volts = less speed. A 24VDC motor typically exhibits its nominal operating characteristics (speed, torque) when operating at its rated voltage. Too much voltage and the motor will be destroyed; too little and the motor will stall.

The regulation can take on many different approaches: since most brushed DC motors have mediocre speed stability, constant compensation (closed-loop feedback) is a must. This compensation can take the form of a tachometer which measures some point of reference as it moves by (either on the motor itself, or the thing (like a platter or pulley) the motor is driving) and adjusts speed according to how far off the from the reference mark the resulting measurement is - more points of reference equates to higher levels of accuracy. Another form of feedback measures the average current draw of the motor, and compensates for any deviation from this ideal. There are several other types of feedback mechanisms as well.

Brushless DC motors behave much like multi-phase AC motors, and require substantially more complex drive and feedback circuitry to accurately 'time' each phase.

Given the complexities of DC motor control, one can understand why synchronous AC motors remain a mainstay of turntable manufacturers despite some of their drawbacks.
Most dc motor come with a controller which varies the voltage output to achieve the desired speed. The better the quality transformer and controller the better the speed stability (see Origin Live DC motors). A few of the higher end tables with a dc motor (Linn LP12 Radical) have a sensor that is installed under the platter and monitors the platter speed and feeds back results to the controller to either increase voltage or decrease voltage based on the RPM.
there are a number of fancy dc voltage converters that contain an input to run a wire back from the motor to the converter to ensure that the rated voltage is maintained. Would these converters be useful as a power supply for a turntable motor. I have used one as a Sota power supply, but have not attached the wire back to the supply. this is what I am talking about (Acopian B24G210 Power Supply). Some of these supplies are more than $500 new.
No. This is nothing more than a lab-grade highly regulated DC power supply. It's the motor itself that varies in speed, regardless of the accuracy or "purity" of the incoming voltage; you can put a $10K DC supply in front of the "best", most expensive DC motor and it will still drift in its speed stability - not a lot, but enough to be noticeable. Such is the nature of a dynamic load, like a turntable.

While the current-sensing option on the Acopian supply supply will work, my guess is that it will maintain the output voltage within a range of +/- 2 to 5%. Not bad for picking widgets off an assembly line, but insufficient for controlling a turntable's speed. You need something with an accuracy of around +/- 0.1%.

There aren't too many stand-alone DC controllers for turntables. The Origin Live box works OK, but once you pair the controller and the box together you're already looking at $600 or so. Oh, and Mark Kelly made a nice DIY kit controller as well...
So the problem is that it doesnt react fast enough to changes in load? I would be interested to see what actual fluctuations exist in load when turning a fairly heavy platter. I would think inertia alone on a 20lbs platter would smooth out any inconsitencies in load quicker and more thoroughly than any motor controller could. The motor itself wouldnt have enough torque to change a heavy platters speed that fast.
The above should have been 'once you pair the controller and the motor'.

No, the problem (for me) is long-term drift...such as the table running fast or slow up to 1 or 2% over the course of a side (we're talking simple voltage regulation-style control without a feedback loop). All those tiny errors add up. To me, the short-term, second to second errors of a DC motor are fairly benign, though certainly not insignificant when heard in the context of a good AC drive system, given its superiority of torque delivery, accuracy and so-called "PRaT" (and the spectral dividends that result from this approach). At least for me. YMMV and all that...

My experiences with various controller topologies for DC drive would imply a speed accuracy of around +/- 0.1% is achievable under ideal conditions.

Both systems (AC & DC) have inherent weaknesses and strengths; each listener has their own set of biases and expectations as well.