7-limit: Difference between revisions

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The '''7-limit''' ~~or 7-prime-limit~~ 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. ~~This~~ is ~~an infinite set~~ and ~~still infinite even if we restrict consideration to~~ a ~~single octave~~. Some examples ~~within the octave~~ 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.

The [[7-odd-limit~~]] refers to a constraint on the selection of [[Just intonation~~|~~just~~]] [[Interval class|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 ~~[[Wikipedia:Tonality diamond|7-limit tonality diamond]]~~.

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 ~~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 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]].

~~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-limit|11-]] ~~or [[13~~-limit]], which ~~usually sound much more exotic~~.

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.

~~A list~~ of ~~[[edo]]s which tunes~~ the 7-limit ~~with more accuracy: {{EDOs| 5, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612 }} and so on~~. ~~Another list of edos which tunes~~ the 7-limit ~~well relative to their size (~~[[~~TE relative error|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 }} and so on.

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.

== ~~List~~ of intervals in the 7-prime-limit and 81-odd-limit ==

== Edo approximation ==

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, … }}.

{{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.

{| class="wikitable center-1 right-3"

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|-

| 49/48

| {{Monzo| 1 0 ~~2 -~~2 }}

| {{Monzo| -4 -1 0 2 }}

| 35.697

| zz2

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| wa 8ve

|}

== Subgroups of the 7-limit ==

* [[2.3.7 subgroup]]

* [[2.5.7 subgroup]]

* [[3.5.7 subgroup]]

== Music ==

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; {{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|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 ===

; [[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)

; [[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]]

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; [[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]]

* [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)

* "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]]

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; [[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]]

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; [[Randy Wells]]

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

~~== See also ==~~

* [[Harmonic limit]]

* [[7-odd-limit]]

* [[Wikipedia: Highly composite number]]

[[Category:7-limit| ]]

[[Category:Rank-4 temperaments]]

[[Category:Lists of intervals]]

[[Category:Lattice]]

[[Category:Listen]]

~~[[Category:Rank 4]]~~

@@ Line 1: / Line 1: @@
 {{Prime limit navigation|7}}
 {{Wikipedia|7-limit tuning}}
-The '''7-limit''' or 7-prime-limit 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. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the octave 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.
-The [[7-odd-limit]] refers to a constraint on the selection of [[Just intonation|just]] [[Interval class|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 [[Wikipedia:Tonality diamond|7-limit tonality diamond]].
+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 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 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]].
-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-limit|11-]] or [[13-limit]], which usually sound much more exotic.
+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.
-A list of [[edo]]s which tunes the 7-limit with more accuracy: {{EDOs| 5, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612 }} and so on. Another list of edos which tunes the 7-limit well relative to their size ([[TE relative error|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 }} and so on.
+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.
-== List of intervals in the 7-prime-limit and 81-odd-limit ==
+== Edo approximation ==
+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, … }}.
+{{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.
 {| class="wikitable center-1 right-3"
@@ Line 45: / Line 57: @@
 |-
 | 49/48
-| {{Monzo| 1 0 2 -2 }}
+| {{Monzo| -4 -1 0 2 }}
 | 35.697
 | zz2
@@ Line 608: / Line 620: @@
 | wa 8ve
 |}
+== Subgroups of the 7-limit ==
+* [[2.3.7 subgroup]]
+* [[2.5.7 subgroup]]
+* [[3.5.7 subgroup]]
 == Music ==
@@ Line 618: / Line 635: @@
 ; {{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|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 ===
+; [[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)
+; [[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]]
@@ Line 634: / Line 664: @@
 ; [[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]]
 * [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)
+* "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]]
@@ Line 647: / Line 682: @@
 ; [[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]]
@@ Line 664: / Line 702: @@
 ; [[Randy Wells]]
 * [https://www.youtube.com/watch?v=rTvMMwkH2Z8 ''The Antidote for Entropy''] (2022)
-== See also ==
-* [[Harmonic limit]]
-* [[7-odd-limit]]
-* [[Wikipedia: Highly composite number]]
 [[Category:7-limit| ]] <!-- main page -->
+[[Category:Rank-4 temperaments]]
 [[Category:Lists of intervals]]
 [[Category:Lattice]]
 [[Category:Listen]]
-[[Category:Rank 4]]