There’s a very good reason why valved brass instruments can never be completely in tune without adjustments (trombone players like to hold this above the rest of the section). To lower a length of tubing by a half step, approximately 6% of additional total length is required. Assuming an initial length of 100” for ease of math, the second valve of a brass instrument would need to add 6 additional inches for a total of 106”. The next half step would need 6% of that length to lower it a half step (an additional 6.36’), and so on. Like trombone positions, the lower a valved instrument goes, the more tubing is necessary for each half step. Using the equation to find the necessary lengths of each of the valve tubings:
Table 1
| Tube | Half Steps Down | Equation | Final Length | Length of Additional Slide |
| Initial Tubing | — | — | 100 | — |
| Second Valve | 1 | 100 + (100 x 6%) | 106 | 6 |
| First Valve | 2 | 106 + (106 x 6%) | 112.36 | 12.36 |
| Third Valve | 3 | 112.36 + (112.36 x 6%) | 119.1 | 19.1 |
So far so good. The equations have identified the necessary lengths for the three valve slides. But what happens if we extend the equation to the other valve combinations?
Table 2
| Valve Combo | Half Steps Down | Equation | Final Length | Needed Slide Length |
Length Provided by Valve Combo (from Table 1) |
Discrepancy | Intonation |
| 1-2 | 3 | 112.36 + (112.36 x 6%) | 119.1 | 19.1 | 18.36 | Short by .74 | Sharp |
| 2-3 | 4 | 119.1 + (119.1 x 6%) | 126.25 | 26.25 | 25.1 | Short by 1.15 | Sharp |
| 1-3 | 5 | 126.25 + (126.25 x 6%) | 133.83 | 33.83 | 31.46 | Short by 2.37 | Sharp |
| 1-2-3 | 6 | 133.83 + (133.83 x 6%) | 141.86 | 41.86 | 37.46 | Short by 4.4 | Sharp |
Mathematically, while instrument makers could construct instruments with each valve in tune, combinations of valves would remain out of tune. Specifically, they become increasingly sharper as players add more tubing lengths. Manufacturers modify the length of each valve slide so that while each becomes slightly out of tune, combinations are more in tune. Even with this compromise, the 1-2-3 valve combination remains appreciably sharp and players must use additional means to adjust the intonation of their instruments.