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The tuba in question (Eb) is the one on the left. The other two are BBb horns:

To answer JPNirschl's question... The valve section has always been an Eb and is original to this horn. The 4th piston was added by Sam Gnagy and is the #2 piston and casing from a like valve section. The 1-2-3 circuits were never altered.

I played the horn as my primary Dixieland horn for about five years and never worried much about a couple of 'quirky' notes since my Dixie playing style is a little 'ratty', anyway. I never played the horn in a section.

Playing most every big Eb tuba I've ever seen has always been somewhat of a compromise in the intonation department. I considered at one time making the 2nd valve slide 'pullable'.

Is it just me or does the 2nd valve tubing look the same length on both the BBb and the Eb?

Since I read the OP, this problem has murmurred to me. After all low Eb’s are very much part of my musical universe.

If I remember right the main problem boils down to inconsistencies in the tuning of the different partials played with the 2nd valve engaged. There may be two very obvious, but not necessarily very visible, reasons for this problem:

An ever so minor obstruction in the shape of dirt, corrosion, surplus solder, or an indenture in the inner knuckles of the 2nd piston or in the 2nd slide loop.

And/or an ever so small leak in the same area. Even rough edges of the valve or casing ports may create counterproductive turbulense.

Klaus