Is there something above that would tell me if it leans gray. You are correct. Chroma tells you how close to a neutral white, gray or black (the grayscale) a color is. It's called the whiteness, grayness or blackness characteristic of a color. MT's Chroma is 1.48 and Jute is 1.42. So Jute is just a snidge more grayed than MT. Or you could reverse it and say MT is just a snidge more colorful than Jute. Sometimes when people perceive a characteristic of grayness they'll call it a "gray undertone". When they perceive a characteristic of whiteness, they'll sometimes call it "chalky". Whiteness, grayness, blackness is not only technically correct, I think it's a lot more specific and simpler. As far as warm and cool, every color in context of its color system can be categorized as inherently warm or cool. In the example above, once you get to the 5Y mark in the yellow hue family, the colors start to shift to the cool side. Every color in a color system has a warm side and a cool side. With color temperature, odds are the inherent color temperature of a color will hold true. But context matters, the light matters. In situ with other contents of a room, a color that's warm in context of its color system could take on a cool role - or vice versa a color that's inherently cool could take on a warm role. Here's the Colorography for Manchester Tan. Puts the color in context of its hue family and illustrates chroma; maybe makes the whiteness, grayness, blackness scale I'm talking about easier to see/understand.

Here's a link to Benjamin Moore that gives you some colors that go well with Manchester Tan. There's also a couple more suggestion that you can access here: Here is a link that might be useful: BM Manchester tan coordinating color

I agree Todd about them side to side. But the travertine is definitely a yellow tint (with New Venetian Gold granite) so probably calls for a yellow based color. The bathroom is its own room so I'm not sure if the paints in both areas have to flow or be the same/complimentary?

Manchester Tan under 3000K LED lights is the perfect greige color I spent years looking for. It is very neutral. It is a gray tan. It looks grayer in cool light, tanner in warm light. It never looks pink, lavender, peach, green, or yellow. (If the sunlight is bouncing of the grass or trees, yes it is greenish. Under incandescent light bulbs, it looks like Desert khaki, tan with a very slight olive brown to it.) Revere Pewter, Edgcomb Gray, and Elmira White looked lavender to the female half of the family, Once we said "Lavender" noone could stop seeing lavender. Gray is on-trend, but I keep seeing lavender or blue in most of the gray products out there. Neutral is hard because our perceptions can make the hue shift slightly. While Red always looks red, gray does not always look "gray". It is also known as Berber White. Spring in Aspen is a lighter version. Feather Down is lighter still.