13-limit: Difference between revisions

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The 13-limit intervals of the [[2.3.13 subgroup]], such as [[13/12]] and [[16/13]], are close to neutral intervals, but are further from true (hemipythagorean) neutral intervals than [[2.3.11 subgroup]] intervals, and may thus be termed "subneutral" and "superneutral".

The 13- or 15-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 13 or 15, respectively. As prime limits increase, ratios containing different primes over [[3/1|3]] in the numerator and denominator become more and more numerous. Such ratios include [[15/14]], [[14/13]], [[11/10]], [[15/13]], [[13/11]], [[14/11]], [[13/10]], [[15/11]], [[7/5]], and their [[octave complement]]s. In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s. An example of a way to use these intervals is to build {{w|tertian harmony|tertian}} triads such as [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. Another example is the neogothic major and minor triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit but can also be used as they are.

The 13- or 15-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 13 or 15, respectively. As prime limits increase, ratios containing different primes over [[3/1|3]] in the numerator and denominator become more and more numerous. Such ratios include [[15/14]], [[14/13]], [[11/10]], [[15/13]], [[13/11]], [[14/11]], [[13/10]], [[15/11]], [[7/5]], and their [[octave complement]]s. In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s. An example of a way to use these intervals is to build {{w|tertian harmony|tertian}} triads such as [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. Another example is the neogothic major and minor triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit via vanishing of [[364/363]], but can also be used as they are.

== Edo approximation ==

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 The 13-limit intervals of the [[2.3.13 subgroup]], such as [[13/12]] and [[16/13]], are close to neutral intervals, but are further from true (hemipythagorean) neutral intervals than [[2.3.11 subgroup]] intervals, and may thus be termed "subneutral" and "superneutral".
-The 13- or 15-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 13 or 15, respectively. As prime limits increase, ratios containing different primes over [[3/1|3]] in the numerator and denominator become more and more numerous. Such ratios include [[15/14]], [[14/13]], [[11/10]], [[15/13]], [[13/11]], [[14/11]], [[13/10]], [[15/11]], [[7/5]], and their [[octave complement]]s. In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s. An example of a way to use these intervals is to build {{w|tertian harmony|tertian}} triads such as [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. Another example is the neogothic major and minor triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit but can also be used as they are.
+The 13- or 15-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 13 or 15, respectively. As prime limits increase, ratios containing different primes over [[3/1|3]] in the numerator and denominator become more and more numerous. Such ratios include [[15/14]], [[14/13]], [[11/10]], [[15/13]], [[13/11]], [[14/11]], [[13/10]], [[15/11]], [[7/5]], and their [[octave complement]]s. In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s. An example of a way to use these intervals is to build {{w|tertian harmony|tertian}} triads such as [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. Another example is the neogothic major and minor triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit via vanishing of [[364/363]], but can also be used as they are.
 == Edo approximation ==