Do all tuning systems involving dividing the octave?

Question

Four commonly referenced Western tuning systems are Pythagorean tuning, just intonation, equal temperament, and meantone temperament.

From Music, Language, and the Brain, I recently learned there are a wide diversity of tunings that do not follow these systems. However, from what I could tell, although there may or may not be a “logic” to how the ratios in other cultures are picked, all the tuning systems referenced in the book still are built upon dividing an octave, implying that after a few distinct pitch classes, the octave recurs.

Are there people with tuning patterns or systems that don’t involve repeating the octave?

Answer 1

There are a few patterns in Arabic or Persian music that don't have true octave repetition. For example (I don't remember the exact patterns), one pattern looks a bit like: C-D-E-F-G-A-B-C-D-Eb-F-G; the lower octave and upper octave o melodies a slightly different. Of course, the tunes come first; one gets "scales" by sorting notes by frequency.

There are some commas computed by other than octave division. The Pythagorean comma is the difference between 12 fifths and 7 octaves; the syntonic comma is the difference between a Pythagorean major third and the just major third. All these commas can be rewritten as octave-based (though not necessarily the most natural way to look at them.) The Syntonic comma occurs in Guitar tuning also: A guitar is tuned to 4 fourths and a major third which is equivalent to two octaves: 4/3*4/3*4/3*5/4*4/3 spans the guitar strings the ratio is 80/81 (the guitar cannot be exactly tuned to just intervals thus forcing some tempering.)