5-limit: Difference between revisions
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The octave equivalence classes of 5-limit 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 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 are [[2edo]], [[3edo]], [[7edo]], [[9edo]], [[10edo]], [[12edo]], [[19edo]], [[22edo]], [[31edo]], [[34edo]], [[53edo]], [[118edo]] | [[EDO]]s which do relatively well in approximating the 5-limit (harmonics 3 and 5) are [[2edo|2]], [[3edo|3]], [[7edo|7]], [[9edo|9]], [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[34edo|34]], [[53edo|53]], [[118edo|118]], [[289edo|289]], [[323edo|323]], [[441edo|441]], [[494edo|494]], [[559edo|559]], [[612edo|612]], [[1171edo|1171]], [[1783edo|1783]], [[2513edo|2513]], [[3684edo|3684]], [[4296edo|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, [[5edo|5]], 7, 12, 19, 31, 34, 53, 118, [[171edo|171]], 289, 323, 441, 612, [[730edo|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]], [[Meantone|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. | |||
A significant interval in 5-limit JI is [[ | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! colspan="4" |wa (3-limit) interval | ! colspan="4" | wa (3-limit) interval | ||
! colspan="2" | interval category | ! colspan="2" | interval category | ||
! colspan="4" |yo or gu (5-limit) interval (81/80) | ! colspan="4" | yo or gu (5-limit) interval (81/80) | ||
! colspan="3" |yoyo or gugu interval (6561/6400) | ! colspan="3" | yoyo or gugu interval (6561/6400) | ||
|- | |- | ||
! ratio | ! ratio | ||
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* [https://soundcloud.com/williamcopper/0511_1 Symphony 4, first movement] by [http://www.williamcopper.com William Copper] | * [https://soundcloud.com/williamcopper/0511_1 Symphony 4, first movement] by [http://www.williamcopper.com William Copper] | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Magnificat0465.mp3 Magnificat] by [http://www.williamcopper.com William Copper] | * [http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Magnificat0465.mp3 Magnificat] by [http://www.williamcopper.com William Copper] | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Catch%20for%20Woodwind%20Quintet-0570.mp3 Catch for Woodwin Quintet] by [http://www.hartenshield.com/william_copper.html William Copper] | * [http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Catch%20for%20Woodwind%20Quintet-0570.mp3 Catch for Woodwin Quintet] by [http://www.hartenshield.com/william_copper.html William Copper] | ||
[[Category:5-limit| ]] <!-- main article --> | |||
[[Category:Interval]] | [[Category:Interval]] | ||
[[Category:Lattice]] | [[Category:Lattice]] | ||
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[[Category:Sound example]] | [[Category:Sound example]] | ||
[[Category:Prime limit]] | [[Category:Prime limit]] | ||
[[Category:Rank | [[Category:Rank three temperament]] | ||