Evolution, Physics and Usage of the Wagner Tuba
The evolution of the tuba in the nineteenth century begins with the ophicleide. (see fig 1) This instrument, inspired by a keyed bugle, was invented in 1817. (Baines 198) It was an improvement on the Serpent, a brass-type instrument from the end of the eighteenth century. Variants of the ophecliede remain in use today, such as the Russian bassoon. But this instrument is best remembered as an ancestor to the tuba.
fig 1 ophecliede

    After the invention of valves, brass was changed forever. Keyed instruments have weaker sounds when they keys are opened, but a valved instrument is strong on every note. After the invention and improvement in valves, the ophecliede was modified into the Bombardon (see fig2). As you can see from the picture, it still retains the ophecliede shape. Unlike the opheclied, keyed in B flat, the majority of bombardons "are in tuba pitch F" and are "up to 145 cm. tall with valve bore reaching 18 mm." (250) The instrument as a whole is very close in profile and shape to it's predecessor.
fig 2 a bombardon

    While the bombardons were still a relatively new invention, Moritz and Wieprect produced the first tuba. (see fig 3) This horn was also keyed in F, and had an unusual valve arrangement. (250) Most modern tubas follow the trumpet, where the first valve lowers by a whole step, the second a half step, and the third is equal to first two added together. On the first tuba, "the 1st and 2nd valves, for the left hand, lowered by a tone and a semi tone. The three for the right hand provided: a large tone, to make an exact two tones with the 1st valve; large semitone, to make an exact tone and a half with the 1st valve; and a perfect fourth." (250) Although some argue that this is a more useful valve arrangement, it soon was replaced with the one we are more familiar with.
The tuba, thus introduced, continues to evolve to this day. But in the nineteenth century, several people tried modifications to the tuba, with varying degrees of success, which no longer survive in regular uses. On of these people was Adolphe Sax. Sax did not primarily work on the tuba. His plan was to integrate the wildly divergent band instruments of the time into a single family of Saxhorns. Figure four shows some of the tenor and bass members of the family. These tuba-influenced instruments are narrower in bore than generally found on modern tuba (see fig 4). As his ideas were more of unification than modification, especially among the tenor and tuba size instruments, his changes did not significantly alter the sound or structure of the bass instrument. However, all of his changes were soon incorporated by other instrument makers into their own horns, so whatever improvements he made remain with us today, even if by a different name. (253)
Despite all this comparison between the bore of a particular horn and the "modern tuba," there was and still is a lot of variance in layout, size and bore. Figure five shows three Bass tubas of the same key and from approximately the same time period. Yet, even though the two horns oriented the same direction are called by the same name and in the same key, there will be a great difference between them in timbre. With this in mind, the various re-namings and deviants of tuba seem to be of less importance. They are all variations on a theme with pretty much the same sound. Yet one mutant tuba does stand out, the Wagner tuba.
The Wagner Tuba stands out not in its legacy to modern instruments, but in the wild divergence it possessed. The wagner tuba has several construction differences which greatly later its tone. The most immediately striking thing about the Wagner tuba (see fig 6) is the almost complete absence of a flared bell. In fact, the instrument is completely conical, including the valves. how does this change the sound? "The [sound] waves passing down a cone without a bell have their curvature suddenly changed at the open end." (Richardson 77) This makes the Wagner tuba less efficient at transmitting sound. It also affects the timbre. "If you give a tube a large flare you reduce the intensity of the upper partials in the note and render it more mellow." (78) the Wagner tuba would then have a brighter sound than either the french horn or a more standard tuba.
The next major difference between a wagner and a standard tuba is method of sound production. "They are intended to be played by french horn players using their own mouthpieces." (Morley-Pegge) (See fig 7) This certainly affects the tone of the instrument in a way different from other tuba-variants. "On mouthpieces of the horn type there is no flange and therefore nothing definite to form an edge-tone, hence the player is deprived of its help, with the result of a soft tone" (Richardson p 72) Therefore, we have simultaneously a bright sounding instrument with a soft tone.
While the shape of the mouthpiece is important, it is not as impotant as the dimensions. "The cup volume and the diameter of the consticted passage [of the mouthpiece] have significant effect upon the perfomance of a given mouthpiece, with the shape being a much less important variable." (Fletcher and Rossig 369-370) The depth of the french horn mouthpiece is 44 mm (Richardson 74). The higher Wagner tuba is in approximately the same range as the horn, so this is not striking. However, the lower Wagner tuba is in the same range as a tenor trombone, which has a cup depth of 64 mm (74). The varied cup volume and constriction of the french horn mouthpiece would also give a french horn sort of sound to the instrument.
    It is important not to underestimate the mouthpiece in sound generation and tone. Figure 8a shows the input impedance for a particular tube. Figure 8b shows the impedance for a particular mouthpice. The impedance of the horn resultant when they are joined togather and in figure 8c. As one can see, the mouthpiece has a profound affect on the impedance of the horn and the sound of the horn in general. The impedance, as modified by the mouthpiece, is also closely linked with the flare of the bell. (Fletcher and Rossig 372) The total sound of the Wagner tuba, therefore, will be unlike anything else in the orchestra, not like a tuba, and not like a french horn, but similar to both.
Despite these very unique characteristics, the Wagner tuba is really a hybrid between the saxhorn or tuba and the french horn. Aside from sharing mouthpieces and players with the french horn section, "in bore they are midway between the horn and the saxhorn." (Morely-Pegge) What lead Wagner to invent a hybrid instrument? In fact, "Wagner had already begun to compose Rheingold before he included the tubas." (Baines 263) So initially, at least, he did not even have them in mind. But in October of 1853, Wagner visited Sax. "(September 1865) A letter from the composer to King Ludwig refers to the 'extra instruments' which he had been scoring for in The Ring and which he had become acquainted with . . . in Paris 'at the maker Sax, whose, invention they were.'" (64) So at least by that time, Wagner had not thought of any innovations. Necessity, however, prompted him, as he was unable to find "those 'Sax'schen Instumente' or even possible substitutes for them in the military bands in Munich, or in Vienna either." (264) As he had already begun to compose for the Saxhorns at this point, he was faced with a dilemma.
    When he finally had the instruments built, he used them as a quartet, with two Bb instruments and two F. When they are used in The Ring, "they are played by the second quartet of horns, horns 5 and 7 playing the B flat instruments, and 6 and 8 those in F" (Morley-Pegge) Unsurprisingly, given Wagner's history of building his own opera house specifically for one opera, he only uses the mutant tubas in The Ring and nowhere else in his work.
    Bruckner and Strauss both made some use of the wagner tuba. Bruckner uses them in the adagio movement of his seventh symphony and also in his ninth. Strauss made use of the horns, but later revised them out. "The Tenor Tuba in Don Quixote was evidently first to have been a Wagner tuba: Strauss tells in Instrumentationslehre how several times he has written for a B flat tenor horn, and had found that as a melody instrument the ordinary Bariton (euphonium) was preferable to the 'harsh, awkward Wagner-tubas with their demoniac [sic] sound'"(Baines 265)
    Despite Strauss' criticism, it was Bruckner's seventh symphony which became his first commercial success, which must be attributed, at least in part, to the role of the tubas.
    Wagner failed to create a consistent transposition for his instruments. "In all the parts . . . of Rheingold, they sound . . . respectively a tone and a fifth lower than written." Yet, "In the other scores they appear in E flat and B flat, sounding a sixth and a ninth lower." (264) He apparently though the second version would be easier for the conductor. But in yet another "exception is in the Prelude to Gotterdammerung, where they are written in B flat and F in brass-band style . . . and sound a ninth and a twelfth lower." (264) This lack of consistency may indicate a lack of certainty on Wagner's part about the best way to notate his new instrument, or it may stem from a desire to irk french horn players.
    In Das Rheingold, Wagner uses D flat major for the tonality around Valhalla, as it is suggested "as the most obvious contrasting tonality for the framing scenes . . . before Valhalla . . ." (Bailey 54) Thus, the Valhalla theme is in D flat:
Wagner scores this for his tuba quartet. This theme, as is this key are paralleled in Gotterdamerung and the tubas are again used. As Bailey sates, Wagner uses many such parallels between the two, "The appearance of D flat, then, at the beginning and end of the main action of Das Rheingold serves to define the dramatic structure of the work, but at the same time Wagner reinforced the structural parallel of this opera with Gotterdammerung by concluding that opera in D flat also. The parallel uses of D flat are reinforced by the association of the Valhalla music with that tonality-music which is scored for the special sound of the so-called 'Wagner Tubas' in both operas. " (54) In fact, the first time the tubas get the melody in Das Rheingold (in the beginning of the second scene) it is a variant of the Valhalla theme in D flat major.
    Bruckner makes more diverse use of the Wagner tubas in his seventh symphony, which, as stated above, undoubtedly lead to its commercial success. The second movement opens with a wagner tuba quartet(see fig 9). Their part is marked hervortretend, which, if three years of high school German serves me right, means from the heart. Bruckner uses the wagnerian tubas in this way, primarily to carry emotion, and especially sorrow.
Bruckner dedicated that movement to Wagner because he died while Bruckner was writing it. The use of "the Master's" weird little tubas is undoubtedly done in tribute to the him. The use of sorrow in the tuba parts is then a reflection of Bruckner's sorrow over Wagner's death.
    When he makes use of the tubas, it is in the low register and often with only minimal support from the orchestra. Their solis introduce both the first and the second theme of the movement. His tubas are scored richly and tragically.
    Aside from its use by the composers mentioned above, the Wagner tuba has undergone a bit of a renaissance and is currently being used for film and television scores. Most of the other historic brass mentioned above is still here and there also. Nothing in music ever gets obsolete.