The Orange Order has pledged that only hymns will be played outside a Catholic church along the route of a major Belfast parade later this month.
But Sinn Féin has said the move does not go far enough, calling on marchers to meet nationalist residents from the area.
The row comes after a loyalist band played offensive tunes outside St Patrick's Church on Donegall St, and others later defied restrictions passing the venue.
Tensions have simmered over the issue, with unionist politicians attacking Parades Commission rulings imposed on marchers, and loyalists blamed for orchestrating subsequent riots following a republican march.
Now the Orange Order said it was making moves to ease the controversy.
The Order said: "In a respectful and sincere effort to address what we heard and following further conversations with local districts, bands, community representatives, political leaders and locally elected representatives, the Ulster Covenant centenary parade on 29 September when passing St Patrick's, which will be open; will play hymns.
"The return parade in the early evening will show similar respect.
"The Orange Institution is committed to civil and religious liberty for all and developing a programme of mutual understanding that begins to rebuild Christian neighbourliness and a return to normality.
"All we seek is acceptance and respect for our traditions; we will not deny others what we ourselves desire."
It said "quiet, frank and constructive conversations have taken place with the clergy and parishioners" ahead of the forthcoming Covenant parade.
It said it wanted to build relationships by listening to views raised and explaining its position in respect of parading, music and the Parades Commission.
"In furtherance of this outreach we invite Bishop Treanor, the clergy and parishioners of St Patrick's to visit the Covenant exhibition at Schomberg House as our guests. It is hoped that the lines of communication opened by this opportunity will be maintained and developed in the months ahead."
But Sinn Féin's Gerry Kelly said, while he welcomed the move, he believed the Orange Order had to enter direct talks with nationalist residents in the area.
Identifying the pledge over music as a positive move, he added: "But this is not enough.
"The one thing that is not taking place is direct talks with the residents."
He added: "We have an opportunity to move this whole process forward but it cannot be done without direct dialogue."
He said the key to various cultural traditions being celebrated was that respect be shown.
"Surely respect starts with dialogue."