That may seem a contrary point to make. Isn't the history of jazz told through its many "first ladies"? The names of Ella Fitzgerald, Billie Holiday, Nina Simone and Sarah Vaughan trip easily off the tongue. In the last few years, there has been the phenomenon of Diana Krall, and in Britain Clare Teal and Gwyneth Herbert have gained greater recognition than most jazz musicians dare imagine.
Look at the list again, though, and notice what they all have in common - they are singers. While they dominate the mass market end of jazz, very few singers have made a major contribution to the development of the music. And as so many of them lack the defining skill of a jazz performer - being able to improvise - the place of the singer in jazz is at best a marginal one.
The ongoing core of the tradition is provided by instrumentalists. Within that core, women have had a very different role. As the American activist Angela Davis put it at a debate at the Yerba Buena Center for the Arts in 2002, women "continue to be purged from the jazz imagination, except as singers or piano players". The piano is the one instrumental exception to the rule, and performers such as Geri Allen, Eliane Elias and Marian McPartland are rated no less highly than their male counterparts. Even here, though, they are still very much a minority. So much so, that at last year's Mary Lou Williams Women In Jazz Festival (named after another great female pianist) at the Kennedy Center in Washington, one observer noted that on the main stage male performers outnumbered women by two to one.
History offers many examples of women in jazz, right back to the early years of the 1920s and 1930s when all-female bands such as The Pollyanna Syncopators and The Gibson Navigators were popular. Like the band in Some Like It Hot that Tony Curtis and Jack Lemmon have to disguise themselves to join, however, many of these groups were essentially novelty acts. The bands that made more of a point of their members' femininity enjoyed greater opportunities to record, thus strengthening the stereotype of the female jazz musician as a more lightweight alternative, easier on the eye and ear than her male equivalent.
This stereotyping continues today. The British pianist Zoe Rahman, who was shortlisted in the Rising Star category in the 2001 BBC Jazz Awards, recalls several examples. "Quite often when I go into a club there's an assumption that I'm a singer," she says. "People don't look at a man and make that same assumption." Many have suggested to her that she should sing as well. "But I'm a piano player, not a singer," she says. Recently a booker asking for pianists was offered her name. "Oh, yes, Zoe Rahman," he said. "Pretty girl." Adds Rahman: "And that was that."
Another facet of the stereotyping is a sexist and unfounded, but still widespread, view that a female musician will lack sufficient aggression when necessary, or will have a more peaceful and harmonically conventional approach. Such basic prejudice is proved groundless by Rahman, Arriale, or Hiromi, a startlingly talented young Japanese pianist who can match any man for technique or forcefulness of attack. The prejudice remains, however. "Some promoters look at me and assume I'm going to play a certain kind of music," says Rahman. "They think I'm going to play some airy-fairy girlie tune. Then they're surprised when I sit down at the piano."
Examples to make the case that women have been accorded equal status in jazz can be plucked out of every generation. Count Basie's great tenorist, Lester Young, had a sister, Irma, who played the saxophone, the instrument also played by Marjorie Pettiford, the sister of the double-bassist Oscar Pettiford. Woody Herman hired a female trumpeter, Billie Rogers, while Lionel Hampton took on the saxophonist Elsie Smith. Today, the situation is probably better than it has ever been. In the UK, the guitarist Deirdre Cartwright and the trombonist Annie Whitehead are regarded as leading players in their fields, and bands such as the National Youth Jazz Orchestra contain several female members.
But as that authoritative tome The New Grove Dictionary of Jazz notes: "The appearance of a special entry on women in this dictionary is both testimony and corrective to this history." A corrective, too, to the rest of the dictionary, which is notable for the paucity of references to women - as are almost every such book on jazz and most television documentaries. One man involved in the BBC's (otherwise excellent) Jazz Britannia series even phoned a female jazz musician to apologise for the lack of women in the programmes.
Why is it that an art form whose foremost practitioners have always seen their mission as being to look forward, to innovate, should prove in practice to be guilty of the most old-fashioned male chauvinism? Guy Barker, Britain's foremost jazz trumpeter, concedes that jazz is male-dominated but pleads not guilty to the above charges. "If a female enters this world," he says, "I've observed that she will receive far more encouragement, and I say that as a positive thing. Musicians are the most open people I know. If someone's great, it doesn't matter if they're male, female, or from Mars."
Barker puts the dearth of women horn players down to personal choice. "It's nothing more mysterious than that," he says. "All it takes is for a woman to fall in love with the music of Clifford Brown or Miles Davis, pick up the trumpet and learn how to play well. But they don't do it. If you look at a classical orchestra, most of the women are in the strings, woodwind, on piano and on harp. In jazz, the instruments most used are the trumpet, saxophone and trombone. It's the instruments, not the music." According to Barker, jazz musicians listen with their ears and not their eyes. But Rahman argues that audiences often do the opposite. "The man standing in the corner with the saxophone - that's jazz," she says. "But when I was in the US I heard some young horn players talking about a female saxophonist, and they were saying `it's just inappropriate for her to have that instrument in her mouth'."
The image and mythology of jazz are rich in stereotypes, of the beautiful singer with a tragic past, or the rascally horn player getting high in low dives. Very often, of course, they're grounded in the truth. This is a world that has been characterised by such machismo that women may not always have wanted to join it. After Billie Holiday once applauded Lester Young as the world's greatest tenor saxophonist at a club, Coleman Hawkins, who had been standing at the bar, got on stage. Ordering the pianist to start playing at the fastest tempo he could, "Bean" then proceeded to demonstrate at interminable length why he was the greatest instead.
Such behaviour is really that of the cocky schoolboy, not the artistic, visionary man. He may have grown up a lot since then. But according to many women, he's got a long way to go. Jazz man is not yet new man.