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{{Prime limit navigation|7}}
{{Prime limit navigation|7}}
{{Wikipedia|7-limit tuning}}
{{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.  
The '''7-limit''' (a.k.a. ''yaza'' in [[color notation]]) 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:  
These things are contained by the 7-limit, but not the 5-limit:  
Line 9: Line 9:
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 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.
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 [[7-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.
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 ==
== 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 [[edo]]s which tunes the 7-limit with more accuracy ([[monotonicity limit]] ≥ 7 and decreasing [[TE error]]): {{EDOs| 5, 8d, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612, … }}. For a more comprehensive list, see [[Sequence of equal temperaments by error]].  


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, … }}.  
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, … }}.
 
{{Note| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "8d" means taking the second closest approximation of harmonic 7. }}


== Intervals ==
== Intervals ==
Line 55: Line 57:
|-
|-
| 49/48
| 49/48
| {{Monzo| 1 0 2 -2 }}
| {{Monzo| -4 -1 0 2 }}
| 35.697
| 35.697
| zz2
| zz2
Line 633: Line 635:


; {{W|Franz Liszt}}
; {{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|Consolations (Liszt)|"Consolation No. 3"}} (1850) – [https://soundcloud.com/tallkite/liszt-consolation-3-by-ken-1 play] – Ken Stillwell performance, retuned by [[Kite Giedraitis]] to the [[kite33]] 7-limit JI scale


; {{W|Johann Pachelbel}}
; {{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)
* ''{{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)
; Traditional (unknown composer)
* [https://www.youtube.com/shorts/uXxfy6r39hI ''Scarborough Fair''] – arranged by [[Claudi Meneghin]] (2026)


=== 20th century ===
=== 20th century ===
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; [[Jacob Adler]]
; [[Jacob Adler]]
* [https://m.youtube.com/watch?v=IUePyH2C9Y0 ''7-Limit Harmony''] (2024)
* [https://m.youtube.com/watch?v=IUePyH2C9Y0 ''7-Limit Harmony''] (2024)
; [[Amanda Cole]]
* [https://www.youtube.com/watch?v=3-3aXAtE574 ''Lumatone Improvisation in 7-limit just intonation tuning with sine tone drone''] (2024)


; [[Ivor Darreg]]
; [[Ivor Darreg]]
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; [[E8 Heterotic]]
; [[E8 Heterotic]]
* [https://youtu.be/mecOmJbqbxU?si=ILByj2hvPIMKRjJT ''Justicar''] (2020)
* [https://www.youtube.com/watch?v=mecOmJbqbxU ''Justicar''] (2020)
 
; [[Eufalesio]]
* [https://soundcloud.com/eufalesio/mind-ye-a-worse-comelore?in=eufalesio/sets/microtonal-stuff ''Mind Ye A Worse Comelore''] from [https://soundcloud.com/eufalesio/sets/microtonal-stuff ''Microtonal stuff''] (2022)


; [[Francium]]
; [[Francium]]
Line 662: Line 673:
* [https://www.youtube.com/watch?v=YcMcychEAoE ''The Bazillionth Party Track''] (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)
* [https://www.youtube.com/watch?v=qDfIzd_Q-Hc ''Counting to Infinity''] (2025)
* "You Geese" from ''Holy Carp'' (2025) – [https://open.spotify.com/track/5xcKZqwgw2SXSZRf1NQsyT Spotify] | [https://francium223.bandcamp.com/track/you-geese Bandcamp] | [https://www.youtube.com/watch?v=jlLQYfHp69A YouTube]


; [[Kite Giedraitis]]
; [[Kite Giedraitis]]
Line 670: Line 682:


; [[Kaiveran Lugheidh]]
; [[Kaiveran Lugheidh]]
* [https://soundcloud.com/vale-10/nostalgic-blue Nostalgic Blue] (2017) – in 2.3.7 subgroup
* [https://soundcloud.com/vale-10/nostalgic-blue ''Nostalgic Blue''] (2017) – in 2.3.7 subgroup
 
; [[Melanie Martinez]]
* [https://m.youtube.com/watch?v=OKBB1VufWCg ''Training Wheels''] (2015)


; [[Nick, The NRG]]
; [[Nick, The NRG]]
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; [[Randy Wells]]
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=rTvMMwkH2Z8 ''The Antidote for Entropy''] (2022)
* [https://www.youtube.com/watch?v=rTvMMwkH2Z8 ''The Antidote for Entropy''] (2022)
== See also ==
* [[Wikipedia: Highly composite number]]


[[Category:7-limit| ]] <!-- main page -->
[[Category:7-limit| ]] <!-- main page -->
[[Category:Rank-4 temperaments]]
[[Category:Lists of intervals]]
[[Category:Lists of intervals]]
[[Category:Lattice]]
[[Category:Lattice]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Rank 4]]

Latest revision as of 03:15, 6 June 2026

← 5-limit  7-limit  11-limit →

English Wikipedia has an article on:

The 7-limit (a.k.a. yaza in color notation) consists of 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- 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 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 edos which tunes the 7-limit with more accuracy (monotonicity limit ≥ 7 and decreasing TE error): 5, 8d, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612, …. For a more comprehensive list, see Sequence of equal temperaments by error.

Here is a list of edos which tunes the 7-limit well relative to their size (TE relative error < 5%): 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, ….

📝Note: Wart notation is used to specify the val chosen for the edo. In the above list, "8d" means taking the second closest approximation of harmonic 7.

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.

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

Subgroups of the 7-limit

Music

Modern renderings

Gustav Holst
Scott Joplin
Franz Liszt
Johann Pachelbel
Traditional (unknown composer)

20th century

Ben Johnston
  • String Quartet No. 4 (1973) – Bandcamp | YouTube – performed by Kepler Quartet

21st century

Abnormality
Jacob Adler
Amanda Cole
Ivor Darreg
dotuXil
E8 Heterotic
Eufalesio
Francium
Kite Giedraitis
Ralph Lewis
Kaiveran Lugheidh
Melanie Martinez
Nick, The NRG
Juhani Nuorvala
Gene Ward Smith
Chris Vaisvil
Randy Wells
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