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 2^a 3^b, 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 [~~http://en.wikipedia.org/wiki/Pythagorean_tuning~~ Pythagorean tuning], and the Pythagorean tuning used in Europe during the Middle Ages is seed out of which grew the common-practice tradition of Western music.

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 2^a 3^b, 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 seed out of which grew the common-practice tradition of Western music.

[[EDO]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [http://en.wikipedia.org/wiki/Continued_fraction continued fraction] for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306~~...~~, ...

[[EDO]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [http://en.wikipedia.org/wiki/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 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, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, ...

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, ...

3-limit intervals up to odd-limit 19683:

3-limit intervals up to [[odd-limit]] 19683:

{| class="wikitable"

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* [[Harmonic limit]]

* [[Gallery of just intervals]]

* [http://en.wikipedia.org/wiki/Pythagorean_tuning Pythagorean tuning - Wikipedia]

[[Category:3-limit| ]]

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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 2^a 3^b, 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 [http://en.wikipedia.org/wiki/Pythagorean_tuning Pythagorean tuning], and the Pythagorean tuning used in Europe during the Middle Ages is seed out of which grew the common-practice tradition of Western music.
+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 2^a 3^b, 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 seed out of which grew the common-practice tradition of Western music.
-[[EDO]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [http://en.wikipedia.org/wiki/Continued_fraction continued fraction] for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306..., ...
+[[EDO]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [http://en.wikipedia.org/wiki/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 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, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, ...
+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, ...
--limit intervals up to odd-limit 19683:
+-limit intervals up to [[odd-limit]] 19683:
 {| class="wikitable"
 |-
@@ Line 157: / Line 157: @@
 * [[Harmonic limit]]
 * [[Gallery of just intervals]]
+* [http://en.wikipedia.org/wiki/Pythagorean_tuning Pythagorean tuning - Wikipedia]
 [[Category:3-limit| ]] <!-- main article -->