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The Mellocast for this week is on a topic not often discussed, that of ascending third valve horns. In short, this is a French system with its origins in the 19th century that is still seen in use today but not so often.
Over at one of my favorite sites, the Richard Martz Horn Collection, we read this introduction to the topic of the ascending third valve horn and find this first illustration. Note how the first and third valve slides look to be about the same length. That is because the first valve lowers the horn a step and the third valve raises the horn a step. As he explains there
This is an “ascending” horn. That is to say, when the third valve is pressed, the pitch of the horn is raised a whole step instead of being lowered by a step- and-a-half (minor third) as is the case on most three-valve instruments. This is accomplished by reversing the usual function of the third valve piston. The open horn air path includes the tubing of the third valve slide. When the valve is pressed the valve slide tubing is bypassed, shortening the length of the air path, thereby raising the pitch of the horn. For this reason a G-crook is used, since the rest of the horn corpus is the length of a standard F horn. (Horn math: the G crook plus the third valve slide equals the length of a standard F crook, therefore the open horn is pitched in F.)
These can and are also produced as double horns. They are not run into often but when they are I know they cause confusion as I have been contacted by people who ran into them a couple times. This horn in the second illustration is an Alexander Model 203 ST double horn. The full description is here but in German; in short, however, it works under the fingers very much like a regular double horn. A 23 combination still gives A-flat for example but acoustically it is arrived at differently. The photo is linked directly from the Alexander website; this is a horn you could purchase today.
They go into all of this in more detail in Episode 110 of The Mellocast with special guest Sandra Clark, Principal Horn of the Toledo Symphony. Check it out for more!