View source for 7-limit

{{Prime limit navigation|7}}
{{Wikipedia|7-limit tuning}}
The '''7-limit''' or 7-prime-limit consists of [[just intonation|rational intervals]] where 7 is the highest allowable [[prime]] factor, so that every such interval may be written as a [[ratio]] of integers which are products of 2, 3, 5 and 7. The 7-limit is the fourth prime limit and is a superset of the [[5-limit]] and a subset of the [[11-limit]]. Some examples of 7-limit intervals include [[7/4]], [[7/5]], [[7/6]], [[9/7]], [[15/14]], [[21/16]], [[21/20]], [[35/27]], [[49/36]], and so on. 

These things are contained by the 7-limit, but not the 5-limit: 
* The [[7-odd-limit|7-]] and [[9-odd-limit]]; 
* Mode 4 and 5 of the harmonic or subharmonic series. 

The 7-odd-limit is a constraint on the selection of just intervals for a scale or composition such that 7 is the highest allowable odd number, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 7 odd-limit intervals within the octave is [[1/1]], [[8/7]], [[7/6]], [[6/5]], [[5/4]], [[4/3]], [[7/5]], [[10/7]], [[3/2]], [[8/5]], [[5/3]], [[12/7]], [[7/4]], [[2/1]], which is known as the 7-odd-limit [[tonality diamond]].

The phrase "7-limit just intonation" usually refers to the 7-prime-limit and includes primes 2, 3, 5, and 7. When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 5, and 7, which can be represented in [[The Seven Limit Symmetrical Lattices|3-dimensional lattice diagrams]], each prime represented by a different dimension. Lattices describing scales beyond the 7-limit require more than three dimensions, and in the 7-limit, such lattices have unique features which simplify the relations between 7-limit chords.

For a variety of reasons, common-practice music has been somewhat stuck at the 5-limit for centuries, though 7-limit intervals have a characteristic jazzy sound which is at least partially familiar. Music in the 7-limit thus represents a large step forward, although not as much as 11- or [[13-limit]], which usually sound much more exotic.

== Edo approximation ==
Here is a list of [[edo]]s which tunes the 7-limit with more accuracy (decreasing [[TE error]]): {{EDOs| 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612, … }}. 

Here is a list of edos which tunes the 7-limit well relative to their size ([[TE relative error]] < 5%): {{EDOs| 12, 19, 31, 41, 53, 72, 99, 118, 130, 140, 152, 171, 183, 202, 212, 217, 224, 229, 239, 243, 251, 270, 282, 289, 301, 311, 323, 354, 369, 373, 383, 388, 395, 400, 410, 414, 422, 441, 453, 460, 472, 482, 494, 525, 544, 566, 571, 581, 593, 612, … }}. 

== Intervals ==
{{See also| User:Lériendil/Table of 21-odd-limit 7-limit intervals }}

Here is a table of intervals in the 7-prime-limit and 81-odd-limit. 

{| class="wikitable center-1 right-3"
|-
! [[Ratio]]
! [[Monzo]]
! Size in [[cent|¢]]s
! colspan="2" |[[Color name]]
|-
| 1/1
| {{Monzo| 0 }}
| 0.000
| w1
| wa unison
|-
| 81/80
| {{Monzo| -4 4 -1 }}
| 21.506
| g1
| gu comma
|-
| 64/63
| {{Monzo| 6 -2 0 -1 }}
| 27.264
| r1
| ru comma
|-
| 50/49
| {{Monzo| 1 0 2 -2 }}
| 34.976
| rryy-2
| biruyo comma
|-
| 49/48
| {{Monzo| 1 0 2 -2 }}
| 35.697
| zz2
| zozo comma
|-
| 36/35
| {{Monzo| 2 2 -1 -1 }}
| 48.770
| rg1
| rugu comma
|-
| 28/27
| {{Monzo| 2 -3 0 1 }}
| 62.961
| z2
| zo 2nd
|-
| 25/24
| {{Monzo| -3 -1 2 }}
| 70.672
| yy1
| yoyo unison
|-
| 21/20
| {{Monzo| -2 1 -1 1 }}
| 84.467
| zg2
| zogu 2nd
|-
| 16/15
| {{Monzo| 4 -1 -1 }}
| 111.731
| g2
| gu 2nd
|-
| 15/14
| {{Monzo| -1 1 1 -1 }}
| 119.443
| ry1
| ruyo unison
|-
| 27/25
| {{Monzo| 0 3 -2 }}
| 133.238
| gg2
| gugu 2nd
|-
| 49/45
| {{Monzo| 0 -2 -1 2 }}
| 147.428
| zzg3
| zozogu 3rd
|-
| 35/32
| {{Monzo| -5 0 1 1 }}
| 155.140
| zy2
| zoyo 2nd
|-
| 54/49
| {{Monzo| 1 3 0 -2 }}
| 168.213
| rr1
| ruru unison
|-
| 10/9
| {{Monzo| 1 0 2 -2 }}
| 182.404
| y2
| yo 2nd
|-
| 28/25
| {{Monzo| 2 0 -2 1 }}
| 196.198
| zgg3
| zogugu 3rd
|-
| 9/8
| {{Monzo| -3 2 }}
| 203.910
| w2
| wa 2nd
|-
| 8/7
| {{Monzo| 3 0 0 -1 }}
| 231.174
| r2
| ru 2nd
|-
| 81/70
| {{Monzo| -1 4 -1 -1 }}
| 252.680
| rg2
| rugu 2nd
|-
| 7/6
| {{Monzo| -1 -1 0 1 }}
| 266.871
| z3
| zo 3rd
|-
| 75/64
| {{Monzo| -6 1 2 }}
| 274.582
| yy2
| yoyo 2nd
|-
| 32/27
| {{Monzo| 5 -3 }}
| 294.135
| w3
| wa 3rd
|-
| 25/21
| {{Monzo| 0 -1 2 -1 }}
| 301.847
| ryy2
| ruyoyo 2nd
|-
| 6/5
| {{Monzo| 1 1 -1 }}
| 315.641
| g3
| gu 3rd
|-
| 98/81
| {{Monzo| 1 -4 0 2 }}
| 329.832
| zz4
| zozo 4th
|-
| 60/49
| {{Monzo| 2 1 1 -2 }}
| 350.617
| rry2
| ruruyo 2nd
|-
| 49/40
| {{Monzo| -3 0 -1 2 }}
| 351.338
| zzg4
| zozogu 4th
|-
| 100/81
| {{Monzo| 2 -4 2 }}
| 364.807
| yy3
| yoyo 3rd
|-
| 56/45
| {{Monzo| 3 -2 -1 1 }}
| 378.602
| zg4
| zogu 4th
|-
| 5/4
| {{Monzo| -2 0 1 }}
| 386.314
| y3
| yo 3rd
|-
| 63/50
| {{Monzo| -1 2 -2 1 }}
| 400.108
| zgg4
| zogugu 4th
|-
| 81/64
| {{Monzo| -6 4 }}
| 407.820
| Lw3
| lawa 3rd
|-
| 80/63
| {{Monzo| 4 -2 1 -1 }}
| 413.578
| ry3
| ruyo 3rd
|-
| 32/25
| {{Monzo| 5 0 -2 }}
| 427.373
| gg4
| gugu 4th
|-
| 9/7
| {{Monzo| 0 2 0 -1 }}
| 435.084
| r3
| ru 3rd
|-
| 35/27
| {{Monzo| 0 -3 1 1 }}
| 449.275
| zy4
| zoyo 4th
|-
| 64/49
| {{Monzo| 6 0 0 -2 }}
| 462.348
| rr3
| ruru 3rd
|-
| 98/75
| {{Monzo| 1 -1 -2 2 }}
| 463.069
| zzgg5
| bizogu 5th
|-
| 21/16
| {{Monzo| -4 1 0 1 }}
| 470.781
| z4
| zo 4th
|-
| 4/3
| {{Monzo| 2 -1 }}
| 498.045
| w4
| wa 4th
|-
| 75/56
| {{Monzo| -3 1 2 -1 }}
| 505.757
| ryy3
| ruyoyo 3rd
|-
| 27/20
| {{Monzo| -2 3 -1 }}
| 519.551
| g4
| gu 4th
|-
| 49/36
| {{Monzo| -2 -2 0 2 }}
| 533.742
| zz5
| zozo 5th
|-
| 48/35
| {{Monzo| 4 1 -1 -1 }}
| 546.815
| rg4
| rugu 4th
|-
| 112/81
| {{Monzo| 4 -4 0 1 }}
| 561.006
| z5
| zo 5th
|-
| 25/18
| {{Monzo| -1 -2 2 }}
| 568.717
| yy4
| yoyo 4th
|-
| 7/5
| {{Monzo| 0 0 -1 1 }}
| 582.512
| zg5
| zogu 5th
|-
| 45/32
| {{Monzo| -5 2 1 }}
| 590.224
| y4
| yo 4th
|-
| 64/45
| {{Monzo| 6 -2 -1 }}
| 609.776
| g5
| gu 5th
|-
| 10/7
| {{Monzo| 1 0 1 -1 }}
| 617.488
| ry4
| ruyo 4th
|-
| 36/25
| {{Monzo| 2 2 -2 }}
| 631.283
| gg5
| gugu 5th
|-
| 81/56
| {{Monzo| -3 4 0 -1 }}
| 638.994
| r4
| ru 4th
|-
| 35/24
| {{Monzo| -3 -1 1 1 }}
| 653.185
| zy5
| zoyo 5th
|-
| 72/49
| {{Monzo| 3 2 0 -2 }}
| 666.258
| rr4
| ruru 4th
|-
| 40/27
| {{Monzo| 3 -3 1 }}
| 680.449
| y5
| yo 5th
|-
| 112/75
| {{Monzo| 4 -1 -2 1 }}
| 694.243
| zgg6
| zogugu 6th
|-
| 3/2
| {{Monzo| -1 1 }}
| 701.955
| w5
| wa 5th
|-
| 32/21
| {{Monzo| 5 -1 0 -1 }}
| 729.219
| r5
| ru 5th
|-
| 75/49
| {{Monzo| 0 1 2 -2 }}
| 736.931
| rryy4
| biruyo 4th
|-
| 49/32
| {{Monzo| -5 0 0 2 }}
| 737.652
| zz6
| zozo 6th
|-
| 54/35
| {{Monzo| 1 3 -1 -1 }}
| 750.725
| rg5
| rugu 5th
|-
| 14/9
| {{Monzo| 1 -2 0 1 }}
| 764.916
| z6
| zo 6th
|-
| 25/16
| {{Monzo| -4 0 2 }}
| 772.627
| yy5
| yoyo 5th
|-
| 63/40
| {{Monzo| -3 2 -1 1 }}
| 786.422
| zg6
| zogu 6th
|-
| 128/81
| {{Monzo| 7 -4 }}
| 792.180
| sw6
| sawa 6th
|-
| 100/63
| {{Monzo| 2 -2 2 -1 }}
| 799.892
| ryy5
| ruyoyo 5th
|-
| 8/5
| {{Monzo| 3 0 -1 }}
| 813.686
| g6
| gu 6th
|-
| 45/28
| {{Monzo| -2 2 1 -1 }}
| 821.398
| ry5
| ruyo 5th
|-
| 81/50
| {{Monzo| -1 4 -2 }}
| 835.193
| gg6
| gugu 6th
|-
| 80/49
| {{Monzo| 4 0 1 -2 }}
| 848.662
| rry5
| ruruyo 5th
|-
| 49/30
| {{Monzo| -1 -1 -1 2 }}
| 849.383
| zzg7
| zozogu 7th
|-
| 81/49
| {{Monzo| 0 4 0 -2 }}
| 870.168
| rr5
| ruru 5th
|-
| 5/3
| {{Monzo| 0 -1 1 }}
| 884.359
| y6
| yo 6th
|-
| 42/25
| {{Monzo| 1 1 -2 1 }}
| 898.153
| zgg7
| zogugu 7th
|-
| 27/16
| {{Monzo| -4 3 }}
| 905.865
| w6
| wa 6th
|-
| 128/75
| {{Monzo| 7 -1 -2 }}
| 925.418
| gg7
| gugu 7th
|-
| 12/7
| {{Monzo| 2 1 0 -1 }}
| 933.129
| r6
| ru 6th
|-
| 140/81
| {{Monzo| 2 -4 1 1 }}
| 947.320
| zy7
| zoyo 7th
|-
| 7/4
| {{Monzo| -2 0 0 1 }}
| 968.826
| z7
| zo 7th
|-
| 16/9
| {{Monzo| 4 -2 }}
| 996.090
| w7
| wa 7th
|-
| 25/14
| {{Monzo| -1 0 2 -1 }}
| 1003.802
| ryy6
| ruyoyo 6th
|-
| [[9/5]]
| {{Monzo| 0 2 -1 }}
| 1017.596
| g7
| gu 7th
|-
| 49/27
| {{Monzo| 0 -3 0 2 }}
| 1031.787
| zz8
| zozo 8ve
|-
| 64/35
| {{Monzo| 6 0 -1 -1 }}
| 1044.860
| rg7
| rugu 7th
|-
| 90/49
| {{Monzo| 1 2 1 -2 }}
| 1052.572
| rry6
| ruruyo 6th
|-
| 50/27
| {{Monzo| 1 -3 2 }}
| 1066.762
| yy7
| yoyo 7th
|-
| 28/15
| {{Monzo| 2 -1 -1 1 }}
| 1080.557
| zg8
| zogu 8ve
|-
| 15/8
| {{Monzo| -3 1 1 }}
| 1088.269
| y7
| yo 7th
|-
| 40/21
| {{Monzo| 3 -1 1 -1 }}
| 1115.533
| ry7
| ruyo 7th
|-
| 48/25
| {{Monzo| 4 1 -2 }}
| 1129.328
| gg8
| gugu 8ve
|-
| 27/14
| {{Monzo| -1 3 0 -1 }}
| 1137.039
| r7
| ru 7th
|-
| 35/18
| {{Monzo| -1 -2 1 1 }}
| 1151.230
| zy8
| zoyo 8ve
|-
| 96/49
| {{Monzo| 5 1 0 -2 }}
| 1164.303
| rr7
| ruru 7th
|-
| 49/25
| {{Monzo| 0 0 -2 2 }}
| 1165.024
| zzgg9
| bizogu 9th
|-
| 63/32
| {{Monzo| -5 2 0 1 }}
| 1172.736
| z8
| zo 8ve
|-
| 160/81
| {{Monzo| 5 -4 1 }}
| 1178.494
| y8
| yo 8ve
|-
| 2/1
| {{Monzo| 1 }}
| 1200.000
| w8
| wa 8ve
|}

== Subgroups of the 7-limit ==
* [[2.3.7 subgroup]]
* [[2.5.7 subgroup]]
* [[3.5.7 subgroup]]

== Music ==
=== Modern renderings ===
; {{W|Gustav Holst}}
* "Mars" from ''{{w|The Planets}}'' (1914–1917) – [http://chrisvaisvil.com/gustav-holsts-mars-arranged-for-7-limit-ji-orchestra/ blog] | [http://clones.soonlabel.com/public/classical-music/mars-7-limit-kontakt5.mp3 play] – arranged by [[Chris Vaisvil]] (2012)

; {{W|Scott Joplin}}
* ''{{w|Maple Leaf Rag}}'' (1899) – [http://web.archive.org/web/20190412163308/http://soonlabel.com/xenharmonic/archives/2127 play] – arranged by [[Claudi Meneghin]] (2014)

; {{W|Franz Liszt}}
* {{W|Consolations (Liszt)|"Consolation No. 3"}} (1850) – [http://web.archive.org/web/20201127014815/http://micro.soonlabel.com/gene_ward_smith/Others/Kite/Consolation%20%233%20by%20Ken%20Stillwell%20retuned.mp3 play] – Ken Stillwell performance, retuned by [[Kite Giedraitis]] to the [[kite33]] 7-limit JI scale

; {{W|Johann Pachelbel}}
* ''{{w|Pachelbel's Canon|Canon in D}}'' (''c''. 1680–1706) – [https://web.archive.org/web/20201127013008/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Pachelbel_s%20Canon%20in%20D%20-%20Relaxing%20music,%20with%20mountain%20views.mp3 play] | [https://www.youtube.com/watch?v=HzQmaxDIxnc YouTube] – arranged by [[Claudi Meneghin]] (2011)

=== 20th century ===
; [[Ben Johnston]]
* ''String Quartet No. 4'' (1973) – [https://newworldrecords.bandcamp.com/track/crossings-the-ascent-string-quartet-no-4-amazing-grace Bandcamp] | [https://www.youtube.com/watch?v=ReHIe0WDvNs YouTube] – performed by Kepler Quartet

=== 21st century ===
; [[Abnormality]]
* [https://www.youtube.com/watch?v=WuW5COnfOlE ''Just Elevation''] (2023)

; [[Jacob Adler]]
* [https://m.youtube.com/watch?v=IUePyH2C9Y0 ''7-Limit Harmony''] (2024)

; [[Ivor Darreg]]
* [http://web.archive.org/web/20201127014610/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prelude%20%231%20for%207-limit%20JI.mp3 ''Prelude #1 in 7-limit JI'']

; [[dotuXil]]
* [https://dotuxil.bandcamp.com/track/waterpad "waterpad"] from [https://dotuxil.bandcamp.com/album/collected-refractions ''Collected Refractions''] (2024)

; [[E8 Heterotic]]
* [https://youtu.be/mecOmJbqbxU?si=ILByj2hvPIMKRjJT ''Justicar''] (2020)

; [[Francium]]
* [https://www.youtube.com/watch?v=NANoBRyxll8 ''Too Happy For My Mood''] (2023)
* [https://www.youtube.com/watch?v=YcMcychEAoE ''The Bazillionth Party Track''] (2023)
* [https://www.youtube.com/watch?v=qDfIzd_Q-Hc ''Counting to Infinity''] (2025)

; [[Kite Giedraitis]]
* [http://tallkite.com/music/IHearNumbers.html ''I Hear Numbers'']

; [[Ralph Lewis]]
* [http://micro.soonlabel.com/0-praxis/audio/August/august_12_Ruckus.mp3 ''Ruckus From the Quiet Zone'']

; [[Kaiveran Lugheidh]]
* [https://soundcloud.com/vale-10/nostalgic-blue Nostalgic Blue] (2017) – in 2.3.7 subgroup

; [[Nick, The NRG]]
* [https://www.youtube.com/watch?v=6IBM_JX52ck ''Cloudy Dreams''] (2022)

; [[Juhani Nuorvala]]
* [https://www.youtube.com/watch?v=0bXhY83asUo ''The tap dance scene from Flash Flash''] (2019)

; [[Gene Ward Smith]]
* ''Clinton Variations'' (2010) – [http://www.archive.org/details/ClintonVariations detail] | [http://www.archive.org/download/ClintonVariations/clinton.mp3 play]

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/blue-tuning/blue-ji-excluded-by-peers.mp3 ''Excluded by Peers'']
* [http://micro.soonlabel.com/centaur_tuning/Prelude_For_Centaur_Tuned_Piano.mp3 ''Prelude for Centaur Tuned Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=rTvMMwkH2Z8 ''The Antidote for Entropy''] (2022)

== See also ==
* [[Wikipedia: Highly composite number]]

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