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Other than the fact that its relatively new compared to other orchestral instruments (e.g. the violin), why is the saxophone (any of them from the family, alto, baritone, tenor etc) not more prevalent in classical repertoire? Are there historical reasons? Is the timbre incompatible (and if so why?). I would prefer objective reasons for the latter if its true (and not subjective statements like "it just does not sound orchestral").

This question has been asked many times and various reason have been proposed:

The saxophone is too loud or doesn't blend

There are numerous recordings proving that the saxophone in the hands of a good player can play extremely quietly and blend anywhere.

There was a cartel of French instrument makers who wanted to protect their market and hindered the introduction of the saxophone.

This might have been true in France during Adolphe Sax's lifetime, but it doesn't explain why the saxophone didn't make its way into the orchestra later, or in other countries. Plus this boycott didn't stop the instrument finding its way into wind orchestras.

The saxophone had a reputation as a 'comedy' instrument.

This reputation didn't develop until the early 20th century in North America with ensembles like the "Six Brown Brothers".

I think the main reason is much more simple. The saxophone was fifty years too late. The symphony orchestra in its present form was complete by 1800. The fact that this combination still exists today, almost unchanged, shows how perfect it had become. There was no musical or acoustical reason to introduce a new instrument.

This r/classical comment by u/trambolino links to a 68-page thesis that answers your question.

Short version: 1. Most seminal symphonies were written before the invention of the saxophone. 2. When it was introduced, there weren't many skilled players. 3. The inventor Adolphe Sax manufactured all the parts of the saxophone himself, so part-makers and instrument makers rallied against his product (telling musicians to walk out of the orchestra if a saxophone would be included). 4. For the reasons mentioned above, there isn't much saxophone music from the past to inspire modern composers to include a saxophone in their symphonies.

Long version: THE MISSING SAXOPHONE:
Why the Saxophone Is Not a Permanent Member of the Orchestra
by
Mathew C. Ferraro. [Submitted to
The Dana School of Music
in Partial Fulfillment of the Requirements
for the Degree of
Master of Music
in
History and Literature.
YOUNGSTOWN STATE UNIVERSITY
May 2012]

I'll quote Ferraro's conclusion on p 54

      From its earliest days, the saxophone was well received by the musical
community. Approving reviews from outstanding composers and musicians who
supported Adolphe Sax quickly followed his introduction of the saxophone. Many
believed the saxophone had all the qualities to enhance the orchestra’s sound and thus
predicted its success and rapid implementation in the orchestra. Some composers wrote
parts for the saxophone in an attempt to implement the instrument in the orchestra. and
many stated they believed the saxophone had many qualities that could enhance and
expand the orchestra’s sound. Composers predicted its success and rapid implementation
in the orchestra.

      Sax’s initial intent was to create a bass instrument that was an alternative to the
bassoon and ophicleide, neither of which Sax was particularly fond. A gifted and selfassured
instrument maker, Sax eventually produced an entire family of saxophones that
many felt could bridge the gap in timbre between the brass and string sections of the
orchestra. Hector Berlioz, especially, hailed the superiority of Sax’s designs and felt the
instrument would make a fine addition to the orchestra. Other prominent composers of
the time, including Rossini and Meyerbeer, were also proponents of the newly invented
saxophone. All facts considered, it becomes clear that there were other mitigating
circumstances that negatively influenced the perception of the saxophone.

      Because of his many achievements and fierce defense of his skills, Sax created an
immediate rivalry with the established instrument manufacturers in Paris. His competitors
purposely sabotaged the introduction of the saxophone and its use in the orchestra.
Feeling threatened by Sax and the growing popularity of his instruments, his rivals

and p 55.

formed a unified front against him and were able to influence musicians and composers
to reject the instrument.

      The saxophone’s omission from the orchestra was also due to the lack of
professionally trained players. Although Adolphe Sax himself was appointed as instructor
of saxophone at the Paris Conservatory in the 1860s, he was dismissed after only a few
years and lessons were not offered again until 1942. Consequently the saxophone, on the
limited occasions it was required, was played or doubled by musicians who considered it
a secondary instrument. This lack of competent teachers and players resulted in
unsatisfactory performances that gave composers an inaccurate representation of the
possible applications of the saxophone.

      The saxophone continued to be overlooked through the remainder of the
nineteenth century, well after Sax’s patents had run out and his rivals had accomplished
their objective. The orchestra continued to include new and improved instruments with
the exception of the saxophone, which was making its way into other genres. It was
rapidly assimilated into military bands and by the turn of the century could be found in
vaudeville acts and other similar genres as a novelty instrument. All of these factors
combined suggest why the saxophone never became part of the orchestra. Whether these
factors were intentional, as was the case during the instrument’s introduction, or
consequential due to its popularity in mainstream music, it has left the saxophone with a
limited repertoire that begins nearly ninety years after its invention.

I think the reasons are economic rather than musical.

I don't really buy the argument that it was "invented too late", because it's just 11 years younger than the tuba (1846 vs 1835). And composers did start using it pretty early on - there are pieces by Bizet, Delibes, and others who included the sax.

But composers didn't really start considering the sax a serious instrument until the 1930s, when Sigurd Rascher started performing Borck's concerto (which was written for him). Then you start seeing it used by Shostakovich, Rachmaninoff, Bernstein, Prokofiev, Britten, Vaughan Williams, etc.

So there's a pretty decent repertoire out there today. But having spent 5 years on an orchestra's board, I can tell you that two things factor heavily into programming: ticket sales and royalty expenses. Pieces written after 1924 are likely to be more expensive to perform, and lesser known composers are far less likely to fill the seats. Major orchestras have to keep their budgets under control, and smaller ensembles simply don't have the money to perform much of the available music.

You beat me to it! In a nutshell Adolphe Sax had a battle in Paris between his invention, the saxophone, and French instrument makers who wanted to protect their market for other comparable instruments. That and his brusque personlaity saw the saxophone sidelined for policial reasons rather thna its virtues.