A companion paper shows how a triple brass instrument can be designed to yield nearly perfect well tempered intonation. This cannot be applied to the larger low brass instruments because the inertia of triple valves is too great when made of brass. In May 2009 this author presented most of the known 3 and 4 valve systems including compensating ones. However well optimized they are not in tune within plus or minus 5 cents. To solve this problem an instrument with four double valves and a tritone switch valve is presented and optimized to minimize the RMS intonation error. The ratios of slide lengths to instrument lengths are given by 0.126, 0.0631, 0.198, 0.337, and 0.413. The compensating slide ratios are 0.0522, 0.0261, 0.172 and 0.139. The instrument is to be tuned 3 cents sharp on the open tones. Descending from the second open tone in semitones results in errors in cents of only 3.15, -2.97, -2.81, 2.65, 0, 0, 3.5, -2.97, -2.81, 2.65, 0 and 0. The RMS intonation error is 2.37 cents which is very acceptable in elite musical performance.