5-limit: Difference between revisions
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The octave equivalence classes of 5-limit or quinquimal intervals can usefully be depicted on a lattice diagram, either as a [http://en.wikipedia.org/wiki/Hexagonal_lattice hexagonal lattice] or as a [http://en.wikipedia.org/wiki/Square_lattice square lattice]; this can be done automatically by [http://www.huygens-fokker.org/scala/ Scala]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [http://en.wikipedia.org/wiki/Hexagonal_tiling hexagonal tiling]. | The octave equivalence classes of 5-limit or quinquimal intervals can usefully be depicted on a lattice diagram, either as a [http://en.wikipedia.org/wiki/Hexagonal_lattice hexagonal lattice] or as a [http://en.wikipedia.org/wiki/Square_lattice square lattice]; this can be done automatically by [http://www.huygens-fokker.org/scala/ Scala]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [http://en.wikipedia.org/wiki/Hexagonal_tiling hexagonal tiling]. | ||
[[EDO]]s which do relatively well in approximating the 5-limit (harmonics 3 and 5) are | [[EDO]]s which do relatively well in approximating the 5-limit (harmonics 3 and 5) are {{EDOs| 2, 3, 7, 9, 10, 12, 19, 22, 31, 34, 53, 118, 289, 323, 441, 494, 559, 612, 1171, 1783, 2513, 3684, 4296, ... }} | ||
Another approach is to find EDOs which have better approximations for [[5-odd-limit]] intervals than all smaller EDOs. This results in 1, 2, 3, | Another approach is to find EDOs which have better approximations for [[5-odd-limit]] intervals than all smaller EDOs. This results in {{EDOs| 1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, ... }} | ||
== Syntonic Comma Pairs == | == Syntonic Comma Pairs == | ||
A significant interval in 5-limit JI is [[81/80]], the syntonic comma or Didymus' comma, which measures about 21.5¢. Although it rarely appears as an interval in a scale, it represents the difference between many 5-limit intervals and a nearby [[3-limit]] (Pythagorean) interval. 81/80 is tempered out in [[12edo]], [[ | A significant interval in 5-limit JI is [[81/80]], the syntonic comma or Didymus' comma, which measures about 21.5¢. Although it rarely appears as an interval in a scale, it represents the difference between many 5-limit intervals and a nearby [[3-limit]] (Pythagorean) interval. 81/80 is tempered out in [[12edo]], [[meantone]], and many other related systems, meaning that those 5- and 3-limit distinctions are obliterated and one interval stands in for each. Living in a largely 12edo musical culture from birth, we are not accustomed to distinguishing two different major thirds, two different minor seconds, etc. Below is a list of some common intervals involving 3 and 5 which are distinguished by 81/80. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). '''Bold''' fractions are simplest for this interval category. | ||
{| class="wikitable" | {| class="wikitable" | ||