"It's all about the phrasing; almost like being an actor when you're singing. You're living the songs as you sing them, not just simply singing words that are on a piece of paper.
"It's all about making the songs conversational and Sinatra was so good at that.
"He made it so that it sounded like he was telling you a story. By the end of the song, you understood what he was singing about."
"Sinatra is at the top of the heap as far as American popular music singers goes."
Anthony got his first big break in 1997 when he went on tour with the Glenn Miller Orchestra, later joining the Tommy Dorsey Orchestra. Widely considered to be one of America's best big bands, the orchestra was led by trombonist Dorsey in the 1930s and 40s. It had multiple and introduced the world to Frank Sinatra in the early 40s.
Singing with the Tommy Dorsey Orchestra is one of the great thrills, says Anthony, "because of all the great vocalists who have been with the band in the past".
"Sinatra of course is the big one, but you had Dick Haymes, you had Jack Leonard, who also had lots of hit records in the early days with the band. It's an honour to be able to sing this American popular music and to just keep the tradition going."
Twenty-five musicians, singers and dancers make up the travelling show, including Anthony and the Swing Sisters, who pay homage to the Andrews Sisters.
The show includes a tribute to Benny Goodman, Harry James and Artie Shaw and, an added attraction, the swing dance group the Hollywood Jive Dancers.
For Anthony, it's about sharing one of America's "greatest cultural gifts to the world".
"The American songbook will live after all of us have passed on. It is its own genre; it is its own style. The connecting point with it is the sentiment behind the lyrics. That's the one thing that never changes from generation to generation. We're always connected by emotional ties. It doesn't matter if you are 15 or 75.
"Duke Ellington once said there are two kinds of music - good music and bad music." And this, concludes Anthony, is good music.
The Tommy Dorsey Orchestra is touring New Zealand from July 10 to August 12.