5-limit
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 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.
Terminology
Due to their historical significance, 5-limit intervals go by various names, including classic(al)^{[1]}, pental^{[2]}, or quinquimal^{[citation needed]}.
Recently, composers Catherine Lamb and Marc Sabat have adopted quintal for the HC5^{[citation needed]} since the corresponding Latin numerals are used to refer to higher prime limits such as septimal for the 7-limit and undecimal for the 11-limit. Pental is less consistent due to its Greek origins. However, that creates a conflict of usage as quintal has been the adjective associated with the fifth diatonic degree. (Quintal harmony does not mean 5-limit harmony, but harmony with chords stacked by fifths – cf. secundal harmony, tertian harmony, quartal harmony.)
A finite set of 5-limit intervals are labeled just, especially when the interval in question is the simplest in the category. For example, 5/4 is known as the just major third^{[3]}. Indeed, just intonation traditionally meant specifically the 5-limit version thereof. Even so, justness is not to be generalized to all 5-limit intervals, nor can we assume all just intervals 5-limit in contemporary usage.
The term ptolemaic could also refer to the 5-limit^{[4]}. On this wiki it is part of the Pythagorean-commatic interval naming system, and refers specifically to intervals that contain a single factor of harmonic 5. We distinguish multi-order 5-limit intervals by diptolemaic, triptolemaic, and so on.
Edo approximation
A list of edos with progressively better tunings (TE error) for 5-limit intervals: 2, 3, 4, 5, 7, 12, 19, 31, 34, 46, 53, 118, 171, 289, 323, 388, 441, 559, 612, 1171, 1783, 2513, 3684, 4296, …
Another list of edos that do relatively well in approximating harmonics 3 and 5: 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^{[5]})
Another list of edos that do relatively well in approximating 5-odd-limit intervals: 1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, … (OEIS: A054540^{[6]})
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. The syntonic comma is tempered out in 12edo, meantone, and many other related systems, meaning that those 5-limit 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, and so forth- however, perhaps a handful of people in the Xenharmonic community are at least starting to take the idea of such a distinction more seriously. 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.
interval
category |
wa (3-limit) interval | yo or gu (5-limit) interval (81/80) | yoyo or gugu interval (6561/6400) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
ratio | cents | Color name | ratio | cents | Color name | ratio | cents | Color name | |||||
unison | C | 1/1 | 0.000 | w1 | wa unison | 81/80 | 21.506 | g1 | gu comma | 6561/6400 | 43.013 | Lgg1 | lagugu 1sn |
aug. 1sn | C# | 2187/2048 | 113.685 | Lw1 | lawa 1sn | 135/128 | 92.179 | Ly1 | layo 1sn | 25/24 | 70.672 | yy1 | yoyo 1sn |
minor 2nd | Db | 256/243 | 90.225 | sw2 | sawa 2nd | 16/15 | 111.731 | g2 | gu 2nd | 27/25 | 133.238 | gg2 | gugu 2nd |
major 2nd | D | 9/8 | 203.910 | w2 | wa 2nd | 10/9 | 182.404 | y2 | yo 2nd | 800/729 | 160.897 | syy2 | sayoyo 2nd |
aug. 2nd | D# | 19683/16384 | 317.595 | Lw2 | lawa 2nd | 1215/1024 | 296.089 | Ly2 | layo 2nd | 75/64 | 274.582 | yy2 | yoyo 2nd |
minor 3rd | Eb | 32/27 | 294.135 | w3 | wa 3rd | 6/5 | 315.641 | g3 | gu 3rd | 243/200 | 337.148 | gg3 | gugu 3rd |
major 3rd | E | 81/64 | 407.820 | Lw3 | lawa 3rd | 5/4 | 386.314 | y3 | yo 3rd | 100/81 | 364.807 | yy3 | yoyo 3rd |
dim. 4th | Fb | 8192/6561 | 384.360 | sw4 | sawa 4th | 512/405 | 405.866 | sg4 | sagu 4th | 32/25 | 427.373 | gg4 | gugu 4th |
4th | F | 4/3 | 498.045 | w4 | wa 4th | 27/20 | 519.551 | g4 | gu 4th | 2187/1600 | 541.058 | Lgg4 | lagugu 4th |
aug. 4th | F# | 729/512 | 611.730 | Lw4 | lawa 4th | 45/32 | 590.224 | y4 | yo 4th | 25/18 | 568.717 | yy4 | yoyo 4th |
dim. 5th | Gb | 1024/729 | 588.270 | sw5 | sawa 5th | 64/45 | 609.776 | g5 | gu 5th | 36/25 | 631.283 | gg5 | gugu 5th |
5th | G | 3/2 | 701.955 | w5 | wa 5th | 40/27 | 680.449 | y5 | yo 5th | 3200/2187 | 658.942 | syy5 | sayoyo 5th |
aug. fifth | G# | 6561/4096 | 815.640 | Lw5 | lawa 5th | 405/256 | 794.134 | Ly5 | layo 5th | 25/16 | 772.627 | yy5 | yoyo 5th |
minor 6th | Ab | 128/81 | 792.180 | sw6 | sawa 6th | 8/5 | 813.686 | g6 | gu 6th | 81/50 | 835.193 | gg6 | gugu 6th |
major 6th | A | 27/16 | 905.865 | w6 | wa 6th | 5/3 | 884.359 | y6 | yo 6th | 400/243 | 862.852 | yy6 | yoyo 6th |
dim. 7th | Bbb | 32768/19683 | 882.405 | sw7 | sawa 7th | 2048/1215 | 903.911 | sg7 | sagu 7th | 128/75 | 925.418 | gg7 | gugu 7th |
minor 7th | Bb | 16/9 | 996.090 | w7 | wa 7th | 9/5 | 1017.596 | g7 | gu 7th | 729/400 | 1039.103 | Lgg7 | lagugu 7th |
major 7th | B | 243/128 | 1109.775 | Lw7 | lawa 7th | 15/8 | 1088.269 | y7 | yo 7th | 50/27 | 1066.762 | yy7 | yoyo 7th |
dim. 8ve | Cb | 4096/2187 | 1086.315 | sw8 | sawa 8ve | 256/135 | 1107.821 | sg8 | sagu 8ve | 48/25 | 1129.328 | gg8 | gugu 8ve |
octave | C | 2/1 | 1200.000 | w8 | wa 8ve | 160/81 | 1178.494 | y8 | yo 8ve | 12800/6561 | 1156.987 | syy8 | sayoyo 8ve |
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
Modern renderings
- Do Wah Diddy Diddy (unknown arranger)
- The Ballad of Jed Clampett (unknown arranger)
20th century
- String Quartet No. 2 (1964)
- performed by Kepler Quartet
21st century
- William Copper (site 1 site 2^{[dead link]})
- "Movt. 1" of Symphony No. 4 "Atlas" (2016)
- Magnificat (2014) – play | SoundCloud
- Catch for Woodwind Quintet (2014) – play | SoundCloud
- Microtonal Tetris (2023)
See also
Notes
- ↑ A note on the naming of musical intervals by Dave Keenan
- ↑ Gallery of Just Intervals « Music & Techniques by Chris Vaisvil
- ↑ The Helmholtz-Ellis JI Pitch Notation (HEJI) by Marc Sabat and Thomas Nicholson from Plainsound Music Edition
- ↑ Fundamental Principles of Just Intonation and Microtonal Composition by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."
- ↑ The description reads: a list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to four of the simple ratios of musical harmony: 5/4, 4/3, 3/2 and 8/5. It is yet to be found out what criterion is used.
- ↑ The description reads: a list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the six simple ratios of musical harmony: 6/5, 5/4, 4/3, 3/2, 8/5 and 5/3. It is yet to be found out what criterion is used.