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The 5-limit consists of all justly tuned intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called 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 intervals can usefully be depicted on a lattice diagram, either as a hexagonal lattice or as a 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 hexagonal tiling.

EDOs which do relatively well in approximating the 5-limit are 2edo, 3edo, 7edo, 9edo, 10edo, 12edo, 19edo, 22edo, 31edo, 34edo, 53edo, 118edo and 289edo.

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.

3-limit interval
interval category
|5-limit interval (81/80)
|Another 5-limit (6561/6400)
ratio
cents value


ratio
cents value
ratio
cents value
1/1
0.000
unison
C
81/80
21.506
6561/6400
43.013
2187/2048
113.685
aug. unison
C#
135/128
92.179
25/24
70.672
256/243
90.225
minor 2nd
Db
16/15
111.731
27/25
133.238
9/8
203.910
major 2nd
D
10/9
182.404
800/729
160.897
19683/16384
317.595
aug. 2nd
D#
1215/1024
296.089
75/64
274.582
32/27
294.135
minor 3rd
Eb
6/5
315.641
243/200
337.148
81/64
407.820
major 3rd
E
5/4
386.314
100/81
364.807
8192/6561
384.360
dim. fourth
Fb
512/405
405.866
32/25
427.373
4/3
498.045
fourth
F
27/20
519.551
2187/1600
541.058
729/512
611.730
aug. fourth
F#
45/32
590.224
25/18
568.717
1024/729
588.270
dim. fifth
Gb
64/45
609.776
36/25
631.283
3/2
701.955
fifth
G
40/27
680.449
3200/2187
658.942
6561/4096
815.640
aug. fifth
G#
405/256
794.134
25/16
772.627
128/81
792.180
minor 6th
Ab
8/5
813.686
81/50
835.193
27/16
905.865
major 6th
A
5/3
884.359
400/243
862.852
32768/19683
882.405
dim. 7th
Bbb
2048/1215
903.911
128/75
925.418
16/9
996.090
minor 7th
Bb
9/5
1017.596
729/400
1039.103
243/128
1109.775
major 7th
B
15/8
1088.269
50/27
1066.762
4096/2187
1086.315
dim. octave
Cb
256/135
1107.821
48/25
1129.328
2/1
1200.000
octave
C
160/81
1178.494
12800/6561
1156.987

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.

See Harmonic Limit

Music

Duodene2 by Chris Vaisvil
Ariel's 12-tone JI by Chris Vaisvil
The Ballad of Jed Clampett by Paul Henning
Do Wah Diddy Diddy by Barry and Greenwich
Symphony 4, first movement by William Copper
Magnificat by William Copper
Catch for Woodwin Quintet by William Copper