Instead of cents, why not degrees?

In contemporary usage, precise tuning measurements are measured in cents, or hundredths of an equal-tempered semitone. An octave is thus measured as 1200 cents, so one cent, as a difference in frequency, is the 1200th root of two, as an ET semitone is the twelfth root of two.

Alternative scale measurements propose include the SI/metric-inspired millioctave (the thousandth root of two), the savart (which uses base-10 logarithms instead of base-2), and my preferred measurement: the arc degree. Here, I think of the octave (the 2:1 ratio of frequencies) as a circle, with each note of the scale repeating for each octave, and one degree being 1/360 of an octave, 3 1/3 cents each. For finer measurements, minutes and seconds could be used, or merely degrees with decimal fractions.

But why divide the octave into 360 degrees? Answer: 360 is a multiple of 72, and 72 equal temperament is an excellent ET approximation of just intonation, especially 11-limit, that just happens to be a multiple of twelve. It is also a multiple of twenty-four, as used in modern Arabic quarter-tone tuning, and thirty-six, used in some Iranian systems. Also, the ET whole tone is divided into sixty equal parts, and sixty is the smallest natural number divisible by all integers from one through six.

Some measurements in degrees:

• octave (2/1): 360°
• equal whole tone: 60°
• equal semitone: 30°
• equal quarter tone: 15°
• just perfect fifth (3/2): 210° 35′ 11″ or 210.59° (12et 210°)
• just perfect fourth (4/3): 124° 24′ 49″ or 149.41° (12et 150°)
• just major third (5/4): 115° 53′ 39″ or 115.89° (12et 120°, 72et 115°)
• just minor third (6/5): 94° 41′ 33″ or 94.69° (12et 90°, 72et 95°)
• just blues minor third (7/6): 80° 3′ 41″ or 80.06° (72et 80°)

Like degrees, minutes and seconds used in arc measurements, and minutes and seconds in time, this is a modern use of sexagesimal (base-sixty) mathematics, used by the ancient Sumerians and Babylonians, and also by the “elves” in my story for Symphony No. 1.