3-limit: Difference between revisions

The '''3-limit''' consists of all [[just intonation]] intervals whose [[Ratio|numerators and denominators]] are both products of the primes 2 and 3. Some examples of 3-limit intervals are [[3/2]], [[4/3]], [[9/8]]. All 3-limit intervals can be written as <math>2^a \cdot 3^b</math>, where ''a'' and ''b'' can be any (positive, negative or zero) integer. When octave-reduced, if b is non-zero, a and b are opposite signs. In other words, one number in the ratio is a power of 2 and the other number is a power of 3. Confining intervals to the 3-limit is known as [[Pythagorean tuning]], and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music, as well as genres derived from it. The 3-limit can be considered a [[Rank-2 temperament|rank-2]] [[temperament]] which [[Tempering out|tempers out]] no [[comma]]s.

[[Edo]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [[wikipedia: Continued fraction|continued fraction]] for the logarithm base 2 of 3. These are {{EDOs| 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, 359, 665, … }} ({{OEIS|A206788}})

Another approach is to find edos which have more accurate approximation to 3 than all smaller edos. This results in {{EDOs|1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867 }}, … ({{OEIS|A060528}})

 

A stricter approach is to find edos with an increasingly stronger [[consistent circle]] of 3/2. These are {{EDOs|1, 12, 53, 665, 190537, … }} (with strengths 1, 2, 3, 11, 28, … respectively)

[[Category:Rank-2 temperaments]]