The classic Nivi sari drape
By far the most popular drape in India - This is probably the drape that most people consider to be "the sari". Most likely you will have only seen this sari drape worn in the west. The nivi has an elegant long line of 7 to 9 pleats folded at the front waist. They have a lovely way of sweeping out when you walk. The long pallu draped over the shoulder and hanging down to wrist length gives more of a sweeping vertical element, lending an instant elegance. While modern Indian women in big metros want to appear so and drape their sarees Nivi rather than their mother's or grandmother's regional style, in many regions you still see other kinds of traditional wraps and not only on older women.
To arrange the sari on the front pull it up with your right hand and fit it tightly into the armpit, then drape it diagonally over your left shoulder. You can pleat the shoulder four or five times, leaving the last pleat to hang mid upper arm - we don't want to be too indiscreet. For slippery fabrics pinning these in place is a really good idea. Pin through all layers of the pleats on the shoulder from the inside of the choli, just behind the shoulder seam.
Another option is to wrap the remaining fabric towards the back, don't go all the way around, instead throw the pallu over the right shoulder - this makes a pretty effect having the pallu on the front of your body. This is called the Gujarati style drape and you see it often in Rajasthan and Gujarat.
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