13-limit: Difference between revisions

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* Mode 7 and 8 of the harmonic or subharmonic series.

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". In [[superpyth]]agorean systems, however, these intervals become closer to true neutral intervals than the 2.3.11 ones. In contrast, 2.3.11 intervals are closest to true neutral intervals when the fifth is slightly flat of just. This is somewhat analogous to intervals of primes [[5/1|5]] and [[7/1|7]], where flattening the fifth makes pythagorean intervals approximate ratios of 5 via [[meantone]], and sharpening the fifth makes pythagorean intervals approximate ratios of 7 via superpyth. In both cases, sharpening the fifth to approximate higher-limit intervals does more damage than flattening the fifth.

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".

As prime limits increase, ratios containing different primes over [[3/1|3]] in the numerator and denominator become more and more ~~important~~. 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. An example of a way to use these intervals is to build ~~Root-3rd-P5~~ triads with a 14/11 [[neogothic]] major third or a 13/11 neogothic minor third, leading to the [[22:28:33]] neogothic major triad and the [[22:26:33]] neogothic minor triad. These chords invert to each other if and only if [[364/363]], the minor minthma, is tempered out. Another such ~~chord is~~ [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. ~~These ratios are approximated well in~~ [[~~29edo~~]], and [[~~mystery~~]] ~~temperament makes use of this fact~~.

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.

== Edo approximation ==

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 * Mode 7 and 8 of the harmonic or subharmonic series.
-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". In [[superpyth]]agorean systems, however, these intervals become closer to true neutral intervals than the 2.3.11 ones. In contrast, 2.3.11 intervals are closest to true neutral intervals when the fifth is slightly flat of just. This is somewhat analogous to intervals of primes [[5/1|5]] and [[7/1|7]], where flattening the fifth makes pythagorean intervals approximate ratios of 5 via [[meantone]], and sharpening the fifth makes pythagorean intervals approximate ratios of 7 via superpyth. In both cases, sharpening the fifth to approximate higher-limit intervals does more damage than flattening the fifth.
+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".
-As prime limits increase, ratios containing different primes over [[3/1|3]] in the numerator and denominator become more and more important. 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. An example of a way to use these intervals is to build Root-3rd-P5 triads with a 14/11 [[neogothic]] major third or a 13/11 neogothic minor third, leading to the [[22:28:33]] neogothic major triad and the [[22:26:33]] neogothic minor triad. These chords invert to each other if and only if [[364/363]], the minor minthma, is tempered out. Another such chord is [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. These ratios are approximated well in [[29edo]], and [[mystery]] temperament makes use of this fact.
+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.
 == Edo approximation ==