The D.C. Music Center, a private program begun a dozen years ago to provide inner-city children and adults with music instruction regardless of ability to pay, may close next month because its funding sources have dried up.

"This can't happen to my baby," said Harriet (Honey) Gorodetzky, the founder of the program, who moved to Jacksonville, Fla., four years ago and is no longer connected with the Music Center. "We provided nourishment for the souls of young people, we fed their souls and made them feel beautiful."

Operating out of the basement of All Souls Church, 16th and Harvard streets NW, the center has offered weekly half-hour lessons to more than 7,000 people, mostly youngsters. Classes in piano, violin, cello, guitar, trumpet, trombone, flute, clarinet and drums are given by a part-time staff of 12 professional musicians, music teachers from both private and public schools, and former students. The instructors are paid $12 an hour and students are charged $8 per half-hour: both the salaries and fees are well below average commercial rates.

"We're here to provide a service, not to make money. No one is turned away. If a student cannot pay our full fee, the center will subsidize the cost. In fact we offer full scholarship, or pay the full amount of the lessons, for many of our students," said Cookie Spaeth, the program's director for the past three years.

Spaeth said the center, which has an annual budget of $43,000, has run out of backup money. "We've been operating $3,000 in debt every month for the last three months. We cannot absorb any more losses," she said. "If we don't get funding by next month, we'll be forced to close."

Ironically, the cutback of music instruction in the D.C. public schools prompted the Music Center's board of directors to open its first satellite center last March at the Episcopal Church of the Atonement, 52nd and East Capital streets NE--a further drain on the center's financial resources. Before the present crisis loomed, a third center in Anacostia was being discussed.

Elizabeth Clanton, the unsalaried chairperson of the center's board, began taking lessons at the Music Center in 1975, along with her then-3-year-old son, Okolo. At the 1975 rate of $4 per half hour, Clanton studied piano and guitar, and her son, who studied violin for five years, has gone on to join the D.C. Youth Orchestra's first violin section. "The ridiculously low price afforded us an opportunity that would have been denied us otherwise. This program has been a lifesaver for other parents who wished to provide their children with an alternative to the street, an alternative to idleness," said Clanton.

This year the program was funded by the United Black Fund, the D.C. Commission on the Arts and Humanities, and the National Endowment for the Arts. The twin problems of sending out money as fast as it is received for long-overdue bills and the preparation of mountains of paperwork to receive long-delayed funding installments are doing the center in. Whether additional monies already applied for will be received from UBF or the city agency will not be determined until later this year, and by that time the center may have gone under.

Spaeth, who is not optimistic, said quite candidly, "We have taken President Reagan's advice and made appeals to the private sector, but all we've gotten is a big negative shake of the head."

Other financial complications have plagued the center at a time when it is most vulnerable. Last year, according to Spaeth, the center hired a professional fund-raising organization that outlined many ambitious projects and received $5,000 up front. She said the center has retained an attorney and plans to file a lawsuit in an attempt to recoup its money. The other complication is caused by nonpayment of bills by students who say they can pay. Although fewer than half of the 300 students who pass through the center yearly pay full fees, about $8,000 in uncollected bills is outstanding at a time when the center is counting every coin.

The center has many advocates who testify to its effectiveness over the years. The Rev. David Eaton, pastor of All Souls and president of the D.C. School Board, said: "The Music Center itself has been a wonderful instrument, allowing our young people to express themselves in ways that are constructive and meaningful. Our community cannot afford to lose it."

"The loss of a community project such as the D.C. Music Center would indeed be great; . . . it is a unique program," said Millie Batista, executive director of the D.C. Commission on the Arts and Humanities.

Helen Rendon of Northwest Washington cites her son Mark, now 19, as an example of what the center stands for. "This year my son played in a concert at Carnegie Hall," she said. "He now attends the Manhattan School of Music and it all started in 1972 when he began studying drums at the center. He stayed there four years because the center gave him self-esteem, an excellent emotion."

Self-esteem is evident as the youngsters bustle to their classes in classical, folk, jazz and modern music. They may rent instruments from the center for a nominal fee and keep them at home for further practice--instruments donated over the years by area music stores and persons who have seen the results of the center's efforts at regularly scheduled concerts. The center also offers classes in music theory, introduction to music for preschoolers, voice, modern dance and tap dance.

Sean Greene, a 10-year-old fifth-grader at Hearst Elementary School, his cheeks flushed with self-esteem after a hectic, triumphant violin session with his teacher Quincy Meeks, said of the center: "I used to just hang out after school. But these are nice people. I like it. I like music."