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 [[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 the 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, [[5edo|5]], [[7edo|7]], [[12edo|12]], [[17edo|17]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[306edo|306]], ...

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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 [[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 the 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, [[5edo\|5]], [[7edo\|7]], [[12edo\|12]], [[17edo\|17]], [[29edo\|29]], [[41edo\|41]], [[53edo\|53]], [[94edo\|94]], [[147edo\|147]], [[200edo\|200]], [[253edo\|253]], [[306edo\|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]], ...