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Modulation Analysis Using Al-Shawwā's Technique

This function implements the modulation algorithm based on Sāmī Al-Shawwā's 1946 work "Al-Qawāʿid al-Fannīya fī al-Mūsīqa al-Sharqīya wa al-Gharbīya" (The Artistic Principles of Eastern and Western Music). Al-Shawwā, a Cairo-born Aleppine violinist, provided unique guidelines for maqām modulation based on scale degree relationships within the Arab-Ottoman-Persian note naming framework.

Modulation Rules Implemented:

1. Tonic Correspondence: Modulation between maqāmāt sharing the same tonic (qarār), provided the tonic is classified as an "original" note (aṣlīya)

2. Third-Degree Modulation: Transition where the third degree of the source maqām becomes the tonic of the target, valid only when the third degree is classified as an "original" note

3. Alternative Third-Degree Modulation: If the standard third degree is invalid, Al-Shawwā permits using a niṣf (half-note)/two-parts (2p) scale degree immediately below it, provided that: a) It is the sixth pitch class from the fundamental (qarār) scale degree of the source maqām according to the 24-tone list b) It maintains a distance of two pitch classes from the preceding scale degree within the source maqām

4. Fourth and Fifth-Degree Modulation: Transitions using the fourth or fifth scale degrees of the source maqām as the tonic of the target

5. Sixth-Degree Modulation (No Third): When both the third degree and its alternative are invalid, modulation may occur through the sixth scale degree, provided it is the sixteenth or seventeenth pitch class from the tonic of the source maqām and remains an "original" scale degree

6. Sixth-Degree Modulation (Between Naturals): The sixth scale degree may also be used when it lies between two "natural" scale degrees within the source maqām