5-limit: Difference between revisions
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The '''5-limit''' consists of all [[just intonation]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called [ | The '''5-limit''' consists of all [[just intonation]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called [[wikipedia: Regular number|regular numbers]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]]. The [[5-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, [[4/3]], [[3/2]], [[8/5]], [[5/3]], [[2/1]]. Approximating these ratios has been basic to Western common-practice music since the Renaissance. | ||
The octave equivalence classes of 5-limit or quinquimal intervals can usefully be depicted on a lattice diagram, either as a [ | The octave equivalence classes of 5-limit or ''quinquimal'' intervals can usefully be depicted on a lattice diagram, either as a [[wikipedia: Hexagonal lattice |hexagonal lattice]] or as a [[wikipedia: Square lattice|square lattice]]; this can be done automatically by [[Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [[wikipedia:Hexagonal tiling|hexagonal tiling]]. | ||
[[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, | [[EDO]]s which do relatively well in approximating the 5-limit (harmonics 3 and 5){{clarify}} 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 {{EDOs| 1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, | Another approach is to find EDOs which have better approximations for [[5-odd-limit]] intervals than all smaller EDOs{{clarify}}. 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 | == 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]], 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 [[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. | ||
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It is important to note that 5-limit music does not mean favoring intervals of 5 over intervals of 3. It means allowing for ''both'' 3's and 5's in generating harmonic material, and so it is an interplay between both. The 5-limit ''includes'' the 3-limit | It is important to note that 5-limit music does not mean favoring intervals of 5 over intervals of 3. It means allowing for ''both'' 3's and 5's in generating harmonic material, and so it is an interplay between both. The 5-limit ''includes'' the 3-limit – a work in 5-limit JI will utilize intervals from both sides of the chart above. | ||
== Music == | == Music == | ||
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* [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] | ||
==See also== | == See also == | ||
* [[5-odd-limit]] | * [[5-odd-limit]] | ||
* [[Harmonic limit]] | |||
[[Category:5-limit| ]] <!-- main article --> | [[Category:5-limit| ]] <!-- main article --> | ||
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[[Category:Sound example]] | [[Category:Sound example]] | ||
[[Category:Prime limit]] | [[Category:Prime limit]] | ||
[[Category:Rank | [[Category:Rank 3]] | ||