I have a slightly different take on rating and slope relative to HCI. I'm not going to comment on course HC because that is a completely separate issue and quite frankly I know exactly the same as you, used in calculating CHC. Between the two numbers the course rating is the only number of interest to the scratch golfer and the slope rating is the primary number of interest for a less skilled golfer (strictly in terms of HCI). I'll slightly modify your example to illustrate this using some mathematics.

For example......what if I score 75 on 2 courses......

75CR 125 slope

75CR 145 slope.......

Will the differential be different?

No exactly the same regardless of what slope rating is chosen in fact. The closer one gets to scratch the progressively less important slope is to the calculation, culminating in the ultimate case of being exactly scratch.

(75 - 75) * 113/125 = 0

(75 - 75) * 113/145 = 0

Changing the example again to illustrate the less skilled case.

For example......what if I score 105 on 2 courses......

75CR 125 slope

75CR 145 slope.......

Will the differential be different?

(105 - 75) * 113/125 = 27.1

(105 - 75) * 113/145 = 23.4

Very definitely yes, where the scratch player saw zero difference (differential) between the score he shot and the result adjusted by the formula, this player sees 3.7 strokes difference between courses and 6.6 strokes total difference from the actual score shot in the 145 slope case. To illustrate that 6.6 stroke difference I need to introduce another special case, the standard difficulty course.

For a slope of 113 players of all handicaps will find the course plays to exactly the same difficulty.

For example on a course rated 75 CR and 113 SR what are the differentials for scores of 75 and 105?

(75 - 75) * 113/113 = 0

(105 - 75) * 113/113 = 30, or (105 -75) * 1 = 30, and finally just (105 - 75) = 30

And in fact it is clear that the slope rating can be left out entirely, the differential is purely based on the score shot relative to the course rating.

In summary the closer to scratch the less important SR, the further from scratch the progressively more important SR becomes (strictly in terms of HCI calculations anyway). Everything in between is a sliding scale. The official statement on SR describes the same thing but for whatever reason I don't find it particularly clear, but it might make the light shine for someone else.

Quote:

A "Slope Rating" is the USGA's mark that indicates the measurement of the relative difficulty of a course for players who are not scratch golfers compared to the USGA Course Rating (e.g., compared to the difficulty of a course for scratch golfers).