# limit_{(tuning-theory)}

Limit is a kind of property in a just intonation system.

There are two kinds of limit: odd limits and prime limits.

- Assume all intervals of a JI have the form of a:b, the odd-limit of this JI would be the maximum odd number among all as and bs.
- Assume all intervals of a JI have the form of a:b, when all the as and bs are factorized using the fundamental theorem of arithmetic, the prime-limit of this JI would be the biggest prime number involved in the factorization.
- Why there's no even-limit? Because you can always multiply or divide by 2 - it corresponds to moving by octaves.

- For example, Ptolemy's intense diatonic scale, a scale with intervals C(1/1), D(9/8), E(5/4), F(4/3), G(3/2), A(5/3), B(15/8) and C(2/1), its odd-limit would be 15 (from B's 15/8) and its prime-limit would be 5 (from E 5/4, A 5/3 and B [3*5]/8). The pythagorean tuning is a 3-prime-limit tuning system, but it has (when only considering the notes C~B with no accidentals) a odd-limit of 27.

2024.7.28