3-limit: Difference between revisions
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A '''3-limit''' interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. 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. Some examples of 3-limit intervals are [[3/2]], [[4/3]], [[9/8]]. 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. | A '''3-limit''' interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. 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. Some examples of 3-limit intervals are [[3/2]], [[4/3]], [[9/8]]. 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. | ||
[[ | == Edo approximation == | ||
[[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 of 3 base 2. These are 1, 2, 3, [[5edo|5]], [[7edo|7]], [[12edo|12]], [[17edo|17]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[306edo|306]], ... | |||
Another approach is to find | Another approach is to find edos which have more accurate 3 than all smaller edos. This results in 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, [[359edo|359]], [[665edo|665]], 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, ... | ||
== Table of intervals == | |||
3-limit intervals up to [[odd-limit]] 19683: | 3-limit intervals up to [[odd-limit]] 19683: | ||
{| class="wikitable center-1 right-3" | {| class="wikitable center-1 right-3" | ||
|- | |- | ||
! Ratio | ! [[Ratio]] | ||
! [[Monzo]] | ! [[Monzo]] | ||
! Size ([[ | ! Size ([[Cent|¢]]) | ||
! colspan="2" | [[Kite's color notation|Color | ! colspan="2" | [[Kite's color notation|Color Name]] | ||
! colspan="2" | Interval | ! colspan="2" | Interval Category | ||
|- | |- | ||
| [[1/1]] | | [[1/1]] | ||