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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 [[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]].
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){{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, … }}
[[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, … }} ([[OEIS: A060525]])


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, … }}
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, … }} ([[OEIS: A054540]])


== 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]], 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|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"
Line 41: Line 41:
| 43.013
| 43.013
| Lgg1
| Lgg1
|lagugu 1sn
| lagugu 1sn
|-
|-
| aug. 1sn
| aug. 1sn
Line 56: Line 56:
| '''70.672'''
| '''70.672'''
| yy1
| yy1
|yoyo 1sn
| yoyo 1sn
|-
|-
| minor 2nd
| minor 2nd
Line 71: Line 71:
| 133.238
| 133.238
| gg2
| gg2
|gugu 2nd
| gugu 2nd
|-
|-
| major 2nd
| major 2nd
Line 86: Line 86:
| 160.897
| 160.897
| syy2
| syy2
|sayoyo 2nd
| sayoyo 2nd
|-
|-
| aug. 2nd
| aug. 2nd
Line 101: Line 101:
| '''274.582'''
| '''274.582'''
| yy2
| yy2
|yoyo 2nd
| yoyo 2nd
|-
|-
| minor 3rd
| minor 3rd
Line 116: Line 116:
| 337.148
| 337.148
| gg3
| gg3
|gugu 3rd
| gugu 3rd
|-
|-
| major 3rd
| major 3rd
Line 131: Line 131:
| 364.807
| 364.807
| yy3
| yy3
|yoyo 3rd
| yoyo 3rd
|-
|-
| dim. 4th
| dim. 4th
Line 146: Line 146:
| '''427.373'''
| '''427.373'''
| gg4
| gg4
|gugu 4th
| gugu 4th
|-
|-
| 4th
| 4th
Line 161: Line 161:
| 541.058
| 541.058
| Lgg4
| Lgg4
|lagugu 4th
| lagugu 4th
|-
|-
| aug. 4th
| aug. 4th
Line 176: Line 176:
| '''568.717'''
| '''568.717'''
| yy4
| yy4
|yoyo 4th
| yoyo 4th
|-
|-
| dim. 5th
| dim. 5th
Line 191: Line 191:
| '''631.283'''
| '''631.283'''
| gg5
| gg5
|gugu 5th
| gugu 5th
|-
|-
| 5th
| 5th
Line 206: Line 206:
| 658.942
| 658.942
| syy5
| syy5
|sayoyo 5th
| sayoyo 5th
|-
|-
| aug. fifth
| aug. fifth
Line 221: Line 221:
| '''772.627'''
| '''772.627'''
| yy5
| yy5
|yoyo 5th
| yoyo 5th
|-
|-
| minor 6th
| minor 6th
Line 236: Line 236:
| 835.193
| 835.193
| gg6
| gg6
|gugu 6th
| gugu 6th
|-
|-
| major 6th
| major 6th
Line 251: Line 251:
| 862.852
| 862.852
| yy6
| yy6
|yoyo 6th
| yoyo 6th
|-
|-
| dim. 7th
| dim. 7th
Line 266: Line 266:
| '''925.418'''
| '''925.418'''
| gg7
| gg7
|gugu 7th
| gugu 7th
|-
|-
| minor 7th
| minor 7th
Line 281: Line 281:
| 1039.103
| 1039.103
| Lgg7
| Lgg7
|lagugu 7th
| lagugu 7th
|-
|-
| major 7th
| major 7th
Line 296: Line 296:
| 1066.762
| 1066.762
| yy7
| yy7
|yoyo 7th
| yoyo 7th
|-
|-
| dim. 8ve
| dim. 8ve
Line 311: Line 311:
| '''1129.328'''
| '''1129.328'''
| gg8
| gg8
|gugu 8ve
| gugu 8ve
|-
|-
| octave
| octave
Line 326: Line 326:
| 1156.987
| 1156.987
| syy8
| syy8
|sayoyo 8ve
| sayoyo 8ve
|}
|}


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