I think that most guitar players understand what the tuning ratio means for tuning machines. If the tuning ratio is 15:1 then you need to turn the tuning key 15 times for the string post to go around once. So, the higher the ratio, the finer the adjustment for each turn. So what about tuning ratio for Steinberger gearless tuners, LSR tuning machines or other tuning machines that don't use gears or string posts? If there aren't any gears and no gear ratio, than how on earth did Steinberger and others come up with 40:1 tuning ratios? In other words, 40 turns of the tuning key equals one turn of what? To answer this question, we have to figure out what gearless tuners do to tune the stings and compare that to the usual tuning machines. Tuning Ratio  What are we Trying to do? To try and compare tuning machines, let's talk about what these devices are trying to accomplish. To keep it real simple, the job of the tuning machines is to stretch the strings to the right tension so that we get the right pitch when we pluck the string. To stretch the string, we clamp both ends of the string at the bridge and at the tuning machine by tying or wrapping or locking or trapping the ends of the string somehow. We then move both ends further apart to increase tension. Let's have a look at how tuning machines do this. For a traditional tuning machine, we move the tuner end of the string away from bridge end by wrapping the string around a post. The amount of distance the moving end of the string moves away from the fixed end with each turn of a tuning key depends on a few things:
Let's assume that we did a really good job of clamping the ends of the string so there is no slippage at the anchor points but the string can slide freely on the post itself as it stretches and wraps around the post. Let's also assume that the diameter of the string is 0.017 inches (a common diameter for a regular gauge G string). Finally, let's assume that the diameter of the string post is 3/16 of an inch or 0.1875 inches. This is the diameter of the post on a generic electric guitar tuning machine. So how far does one end of the string move away from the other end in this situation? In other words, for one turn of the tuning key, how much further away does one end of the string get from the other? To do the calculation, we just need to make one more assumption and that's the position of the string before the string post starts turning. The following image shows the starting point. You can imagine that if the starting point was further clockwise, then the travel distance will not follow the circumference of the string post and could even be loosening the string! So, the distance that the string will travel with one turn of the string post is the circumference of the circle made by the center of the string.
The diameter at the center of the string is the diameter of the string post (0.1875 inch) plus the diameter of the string (0.017 inch) for a diameter of 0.2045 inches. Using the standard formula for circumference of a circle (circumference = diameter X pi) we get a circumference of 0.642 inches. For a 15:1 tuning machine, one turn of the tuning key will only turn the post 1/15th of a turn so we need to divide the circumference by 15 to get the string movement for one turn of the tuning key. So 0.642 inches divided by 15 is about 0.043 inches or 1.09 millimeters. In fraction inches, that's about 3/64 inches. So, the string pull of our generic 15:1 tuning machine using a 0.017 inch string is 0.043 inches per turn of the tuning key. Now, I know that this is a bit more complex than a gear or tuning ration but this can be applied to all tuners so that they can be compared apples to apples. The lower the number, the greater the tuning accuracy. In other words, you can make finer (smaller) adjustments to the amount the string is being pulled for each movement of the tuning key. How about the string pull of similar 12:1 or 21:1 tuners? That's easy now:
Tuning Ratio  How do Tuners with Gears Compare to Gearless Tuners? A 40:1 gearless tuner sounds really good right? With the example above, that would give us a string pull of only 0.016 inches per turn of the tuning key! Wow! Let's see how this stacks up to reality. For example, the RTrem used on Steinberger guitars such as the GTPRO moves the string on the bridge end away from a fixed string at the other end of the neck. Instead of using gears, a screw is used to move a claw that grips a ball end string. The screw used to move the claw is an M3 x 0.5. The "M3" means that this is a metric screw with a diameter of 3 millimeters. The "0.5" means that the pitch of the screw or the distance between each thread on the screw is 0.5 millimeters. This means that for each turn of the screw, a nut on the screw or the string claw in this case will move by 0.5 millimeters. In this case, string pull is really straight forward, it's 0.5 millimeters or 0.020 inches per turn. That's actually really close to the 40:1 calculation that we did (0.016 inches per turn), so Steinberger's number is very reasonable. The calculation is off by 0.004 inches which is about the thickness of a piece of paper. For that matter, any straight pull tuner that uses a M3 x 0.5 screw to pull the string can safely claim a 40:1 "ratio" even if we know that there is no gear ratio at all! There is no 40 turns to 1 something, there is only a string pull per turn. Congratulations if you read to the bottom of this post! It's pretty dry stuff but hopefully this explains the 40:1 claim if you ever wanted to know!
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