In Brave New World, sky-signs feature the name of a band (?) called The Sixteen Sexophonists. Is the word Sexophonists a play on words by the author, and if so, what is the joke?
Possibly the more important word play is the part of the band name which you missed out. The full name is 'Calvin Stopes and his Sixteen Sexophonists'.
In 1918 and 1919 Marie Stopes published two books: 'Married Love', which included a chapter on Birth Control and 'Wise Parenthood' which advocated Birth Control to married women. She also published a pamphlet aimed at poor women, which was a digested version of 'Wise Parenthood' which ran to (a possibly significant) sixteen pages.
The following year, Stopes published A Letter to Working Mothers on how to have healthy children and avoid weakening pregnancies, a condensed version of Wise Parenthood aimed at the poor. It was a 16-page pamphlet and was to be distributed free of charge. Stopes's intended audience had—until this work—been the middle classes. She had shown little interest in, or respect for, the working classes; the Letter was aimed at redressing her bias.
At around the same time Stopes founded the Society for Constructive Birth Control and Racial Progress
Stopes explained that the object of the Society was:
"...to counteract the steady evil which has been growing for a good many years of the reduction of the birth rate just on the part of the thrifty, wise, well-contented, and the generally sound members of our community, and the reckless breeding from the C.3 end, and the semi-feebleminded, the careless, who are proportionately increasing in our community because of the slowing of the birth rate at the other end of the social scale. Statistics show that every year the birth rate from the worst end of our community is increasing in proportion to the birth rate at the better end, and it was in order to try to right that grave social danger that I embarked upon this work."
The objects of the Society For Constructive Birth Control and Racial Progress showed eugenic aims, summarised in point 16*: "In short, we are profoundly and fundamentally a pro-baby organisation, in favour of producing the largest possible number of healthy, happy children without detriment to the mother, and with the minimum wastage of infants by premature deaths. In this connection our motto has been 'Babies in the right place,' and it is just as much the aim of Constructive Birth Control to secure conception to those married people who are healthy, childless, and desire children, as it is to furnish security from conception to those who are racially diseased, already overburdened with children, or in any specific way unfitted for parenthood." Stopes advocated the compulsory sterilisation of those considered unfit for parenthood in 1918. and in 1920.
*oh look! another 16...
These seem to be relevant themes for the book, and to have been recent/current events as it was being written.
I note that the frontman's first name is Calvin and there may be some relevance to the theologian of that name, but I'm afraid that's rather beyond my reach, possibly something about predestination?
Edit: Quickly refamiliarising myself with the book via Gutenberg, I see that 'Social Predestination' is a significant stage in the cloning process, where embryos are subject to differing hormone levels and the addition of alcohol to their fetal environment to ensure that they are only suited for the category to which they have been assigned and can have no hope of self determination.
John Calvin, theologist 1509-1564, emphasized the sovereignty of the scriptures and divine predestination—a doctrine holding that God chooses those who will enter Heaven based His omnipotence and grace. The Director of Social Predestination in Brave new World is then, in some regard, set in the place of God.
So whatever the relevance of the Sixteen Sexophonists, their frontman brings together the concepts of predestination, eugenics and artificiality in the production of human beings through the linking of the names Calvin and Stopes.
I am not aware that there is any more to the word ‘Sexophonist’ than the obvious pun on ‘saxophonist’, and with the musicians playing music inspired by, evocative of and designed to inspire desire of- sex.