13-limit: Difference between revisions

The '''13-limit''' or 13-prime-limit consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 13. Thus, [[40/39]] would be within the 13-limit, since 40 is {{nowrap|2 × 2 × 2 × 5}} and 39 is {{nowrap|3 × 13}}, but [[34/33]] would not, since 34 is {{nowrap|2 × 17}}, and [[17-limit|17]] is a prime number higher than 13. The 13-limit is the 6th [[prime limit]] and is a superset of the [[11-limit]] and a subset of the [[17-limit]].  

The 13-limit is a [[rank and codimension|rank-6]] system, and can be modeled in a 5-dimensional [[lattice]], with the primes 3, 5, 7, 11, and 13 represented by each dimension. The prime 2 does not appear in the typical 13-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a sixth dimension is needed.

* 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.

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.

== Notes ==