"It's a damaru." That was my line and I was sticking to it – never mind that I didn't know what a damaru was. Five of us were in a Mumbai sound studio recording the dialogue of an ad for an Indian internet site. I delivered my one line with, as I perceived it, panache. A few minutes later I watched as it was matched to action already filmed. A damaru, I realised, is a two-headed drum, ubiquitous in the subcontinent. I began to appreciate how exacting the art of dubbing and synching is. Fortunately our recording was a play session: we were on a guided tour around Mumbai and were taking a behind-the-scenes look at the city's multi-billion dollar film industry.
This year India's largest city has been celebrating 100 years of movie making. In May 1913 Raja Harishchandra, a flickering black-and-white film about a mythical Hindu king, opened at the Coronation Theatre in Mumbai (then Bombay) – and Bollywood was effectively born. That first film was silent, 40 minutes long, and had just a few women's parts, all played by men. Quite some world apart from the riotously glitzy, singing-and-dancing blockbusters of today – such as Yeh Jawaani Hai Deewani (This Youth is Crazy) which is nearly three hours long, includes nine swirling song scenes and is currently the highest-grossing Bollywood production of 2013. I'd enjoyed something of a Bollywood fest on the nine-hour flight from Heathrow. Jet Airways is marking the centenary with an extra-special host of Bollywood offerings among its inflight entertainment options – and I arrived in Mumbai bug-eyed.
Head still reeling the next morning, I set off on a city tour. I was staying at the landmark Oberoi, on the waterfront of Marine Drive in Mumbai's southern sector. Originally a fishing settlement, this area is where the sprawling city originated, developed largely thanks to Portuguese traders. It was the British, though, who left lasting marks, and we immediately headed for some of the most famous sights relatively nearby. Firstly the Gateway of India, the high-Victorian archway built to commemorate George V's visit in 1911. On we drove to other flamboyant remnants of the Raj: the High Court Buildings, looking as if they've escaped the pages of a Harry Potter story; the Victoria (or Chhatrapati Shivaji) Terminus (ditto), one of India's busiest stations with a footfall of 100,000 passengers a day; and Crawford Market selling a colourful if eccentric mix of fruit, veg and pets (mangos to macaws). It was in part designed by Lockwood Kipling, father of Rudyard, who was born in an adjacent house.
We moved westwards to Malabar Hill, home to Mumbai's most expensive real estate, and paused by the Hanging Gardens, a terraced green space created over one of the city's reservoirs. Just beyond are the Towers of Silence, actually more like large wells, where Mumbai's Parsi community has for centuries laid out its dead. Until about 15 years ago, my guide Anish told me, the deceased would be consumed within hours by the neighbourhood vultures, but their numbers have dwindled alarmingly – from 7,000 to around 800 – and now the Parsis have started a vulture breeding programme.
We edged on, in thick traffic, stopping to take in striking views of Dhobi Ghat, Mumbai's traditional laundry district. Here you gaze over row upon row of washing pens above which hang white linens, lines of blue jeans (given an "ageing" clean after manufacture) and more.
We pressed on northwards over the Bandra-Worli Sealink. The stupendous bridge opened in 2009 linking south Mumbai with suburbs in the north. Just the other side is Bollywood territory. Many of the really big blockbusters are made in Film City in the far northern suburb of Goregaon, Anish explained, but although there are plans to develop visitor facilities at the site, for the moment it's difficult for tourists to get entry.
No matter: much of Bollywood production takes place in small studios dotted around the Bandra district, and many of them welcome visitors by appointment. It's in the Bandra area, he added, that a lot of the stars live – the likes of Shah Rukh Khan and Deepika Padukone.
First Bollywood stop was one of Bandra's small sound studios, where we experimented with recording – and I improved my knowledge of Indian drums. Then we continued north. We passed the sandy stretch of Juhu Beach and adjacent Juhu Airport where, Anish pointed out, the opening of Slumdog Millionaire was filmed. We stopped briefly at the Marriott on Juhu Tara Road, where the Bollywood crowd comes for Sunday brunch. As we moved on, Anish waved a hand at a leafy avenue, "that's where Amitabh Bachchan lives", his voice tinged with awe. One of Bollywood's screen legends, Bachchan recently took a Hollywood part: gambler Meyer Wolfsheim in Baz Luhrmann's version of The Great Gatsby.
Our final port of call was a studio of furnished sets which also offers a small Bollywood museum. It would be pot luck as to what we'd find in production, Anish explained as we set off to explore S J Studios. As a grand finale, I had been hoping for music, dancing, or at least high-blown drama. The reality, as in much of life, was a great deal more pedestrian: a couple of TV series were in the making. We walked through a hospital set-up, complete with patients and nurses, and then entered a sitting-room studio where a rehearsal was taking place. Two men were in stagey conversation. This, Anish whispered, was a telling scene in Hum Ne Li Hai Shapath, a popular detective show. Was the man in the checked shirt the detective, I asked. Anish looked at me with pity. "Well, actually madam," came the reply "he's an actor."
Harriet O'Brien travelled to Mumbai with Jet Airways (0808 101 1199; jetairways.com) which flies from Heathrow; return fares from £548.
Oberoi Mumbai (00 91 22 6632 5757; oberoihotels.com). Doubles start at R28,000 (£307).
Jetair Tours (00 91 11 43234406; jetairtoursindia.com) offers day tours in Mumbai, including a "Bollywood experience", from £62 per person.