Solution to Al-Jurjani´s 1,000 year old signal processing problem on the generation of harmonics in music

Interval (tuning) systems in music reflect history, culture and geography. The classical music of the west is based on the twelve-tone equal temperament (12-TET) interval (tuning) system which became dominant throughout the world. Today, other interval systems are being re-discovered by the west with the advances in information technologies. Many multi-media applications use digital signal processing to; process, synthesize, recognize, classify and share music. The important classification features such as; interval system, mode and maqams provide solutions for the discrepancy of music; east-west; Turkish-Arab-Chinese-Indian; blues-jazz-clasical. In this paper, the theoretical problem of constructing an interval system independent from the harmonics `the fifths´, today known as the N-limit tonality diamond (matrix), is examined. It was first proposed by Ibni Sina´s teacher Al-Jurjani approximately 1,000 years ago and solved for N = 5. In the 19th and 20th centuries, the problem was solved for N = 9 and 13. The missing solutions for N = 3 and 7 and misleading N = 15 solutions make the analysis of such music even harder. In this paper, N-limit tonality diamond problem is solved for increasing N as an application in digital signal processing. These rare and distict intervals are proposed as features for easier identification and classifications.