Suppose we choose to make the third valve alone give us a minor 3rd, while the second valve alone gives a semitone and the first
valve gives a whole step, and the fourth valve gives us a perfect fourth as usual:
If you tune your valve slides this way then the 2&3 valve combination will be sharp, and 1&3 will be very sharp.
A selection chart similar to the previous ones can be created for such an instrument.
A long half-step fifth valve can be set for the fourth semitone fingered 125. This gives you a good sixth semitone fingered 45, correcting the sharp 24 or the very sharp 123. The eighth semitone will be good with 234 but flat with 1245. The ninth semitone will be a bit sharp with 2345, and by luck we get a pretty good tenth semitone with 12345. The seventh semitone will not be good, and the eleventh is unavailable. Here is the fingering chart:
Short third valve, Standard fourth valve, Fifth valve = long semitone set for 125 = major third:
0 (open)....................open....................Bb.....C......Eb.....F
1 semitone...................2.......................A.......B......D.......E
2 semitones..................1......................Ab.....Bb....Db.....Eb
3 (minor 3rd)...............3.......................G.......A......C.......D
4 (major 3rd).............125.....................Gb.....Ab.....B......Db
5 (perfect 4th)..............4.......................F.......G......Bb.....C
6 (tritone)....................45......................E......Gb.....A.......B
7 (perfect 5th)............245+...................Eb.....F......Ab.....Bb
8 (augmented 5th).......234....................D.......E......G.......A
9 (sixth).....................2345+.................Db.....Eb....Gb.....Ab
10 (seventh).............12345...................C.......D.......F.......G
11 (major seventh).......NA....................B.......Db....E......Gb
A long whole-step fifth valve can be set for the fourth semitone fingered 15 instead of 23. This gives decent but redundant alternate fingerings for the third and fifth semitones. The seventh semitone is now 1235, the eighth is now 145 or 234, the ninth is now 345+, and the tenth is 1345. The only disadvantages seem to be the sharp sixth and ninth semitones (24+ and 245+, respectively) and the very sharp eleventh semitone. Here's the fingering chart:
Short third valve, Standard fourth valve, Fifth valve = long whole-step set for 15 = major third, slightly sharp so that 145 is only slightly flat and 1345 is perfect:
0 (open)....................open....................Bb.....C......Eb.....F
1 semitone...................2.......................A.......B......D.......E
2 semitones..................1......................Ab.....Bb....Db.....Eb
3 (minor 3rd)................3......................G.......A......C.......D
4 (major 3rd)...............15.....................Gb.....Ab.....B......Db
5 (perfect 4th)..............4.......................F.......G......Bb.....C
6 (tritone)....................24+....................E......Gb.....A.......B
7 (perfect 5th)...........1235-...................Eb.....F......Ab.....Bb
8 (augmented 5th).....234/145................D.......E......G.......A
9 (sixth)......................345+..................Db.....Eb....Gb.....Ab
10 (seventh)...............1345...................C.......D.......F.......G
11 (major seventh).....12345++..............B.......Db....E......Gb
A fifth valve can be tuned to a long minor third (like a conventional third valve) so that 25 is used instead of 23. This leaves you with the same sharp 24 that you had with a conventional 4-valve instrument, but gives you a slightly improved seventh semitone (135-), three possible fingerings fo the the eighth semitone (234, 1235+, or 245-), a slightly better ninth semitone (145+ instead of 134), a very good tenth semitone fingered 2345, and two choices (1345+ or 12345-) for the eleventh semitone:
Short 3rd valve, Standard 4th valve, Fifth valve = minor 3rd set for 25 = major third
0 (open)....................open....................Bb.....C......Eb.....F
1 semitone...................2.......................A.......B......D.......E
2 semitones..................1......................Ab.....Bb....Db.....Eb
3 (minor 3rd)................3......................G.......A......C.......D
4 (major 3rd)...............25.....................Gb.....Ab.....B......Db
5 (perfect 4th)..............4.......................F.......G......Bb.....C
6 (tritone)....................24+....................E......Gb.....A.......B
7 (imperfect 5th)........135(-)..................Eb.....F......Ab.....Bb
8 (augmented 5th)....234(-)....................D.......E......G.......A
9 (sixth)....................145+....................Db.....Eb....Gb.....Ab
10 (seventh...............2345....................C.......D.......F.......G
11 (major seventh)....1345+..................B.......Db....E......Gb
A fifth valve tuned to a major third can be used alone instead of 23:
Short third valve, Standard fourth valve, Fifth valve = major 3rd set 3 cents sharp:
0 (open)....................open....................Bb.....C......Eb.....F
1 semitone...................2.......................A.......B......D.......E
2 semitones..................1......................Ab.....Bb....Db.....Eb
3 (minor 3rd)................3......................G.......A......C.......D
4 (major 3rd)................5.....................Gb.....Ab.....B......Db
5 (perfect 4th)..............4.......................F.......G......Bb.....C
6 (tritone)....................24+....................E......Gb.....A.......B
7 (imperfect 5th)........235(-)..................Eb.....F......Ab.....Bb
8 (augmented 5th)........234....................D.......E......G.......A
9 (sixth)......................245+..................Db.....Eb....Gb.....Ab
10 (seventh.................345....................C.......D.......F.......G
11 (major seventh).....1345(-)................B.......Db....E......Gb
The diagonal line labeled "T4" on the chart suggests four ways to use a fifth valve to correct for the sharp 2&3 combination. Each one gives good fingerings for some notes in the extension register:
Interval...............Fingering..................Note Names
Interval...............Fingering..................Note Names
Interval...............Fingering..................Note Names
This gives us single-valve fingerings for the first six notes of a descending chromatic scale, which may well have been its original motivation. Unfortunately, it does not solve the sixth semitone (24+) problem. The seventh semitone fingered 235 will be some improvement over 124 but still a bit flat. The ninth semitone fingered 245+ will be a bit closer than 134+. The tenth semitone will be good with 345 or 1245. The eleventh will be either very sharp with 2345+ or a bit flat with 1345-. The twelvth semitone (alternative to the pedal note)can be played 12345+. In the fingering chart below I have set the 5th valve a bit sharp as a compromise between the fourth and seventh semitones:
Interval...............Fingering..................Note Names