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The ''11-limit'' consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11|14/11]], [[11/8|11/8]], [[27/22|27/22]] and [[99/98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.
{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]].  


{| class="wikitable"
The 11-limit is a [[rank and codimension|rank-5]] system, and can be modeled in a 4-dimensional [[lattice]], with the primes 3, 5, 7, and 11 represented by each dimension. The prime 2 does not appear in the typical 11-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a fifth dimension is needed.
 
These things are contained by the 11-limit, but not the 7-limit:
* The [[11-odd-limit]];
* Mode 6 of the harmonic or subharmonic series.
 
The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.
 
While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.
 
== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy ([[monotonicity limit]] ≥ 11 and decreasing [[TE error]]): {{EDOs| 12, 15, 19, 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]].
 
Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] < 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on.
 
: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.
 
== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
|-
! | Ratio
| 12/11
! colspan="2" |[[Kite's color notation|interval name]]
| 1u2
! | harmonic solfege
| lu 2nd
| fu-sol
|-
|-
| | 12/11
| 11/10
|1u2
| 1og2
|lu 2nd
| logu 2nd
| | fu-sol
| mi-fu
|-
|-
| | 11/10
| 10/9
|1og2
| y2
|logu 2nd
| yo 2nd
| | mi-fu
| re-mi
|-
|-
| | 10/9
| 9/8
|y2
| w2
|yo 2nd
| wa 2nd
| | re-mi
| do-re
|-
|-
| | 9/8
| 8/7
|w2
| r2
|wa 2nd
| ru 2nd
| | do-re
| ta-do
|-
|-
| | 8/7
| 7/6
|r2
| z3
|ru 2nd
| zo 3rd
| | ta-do
| sol-ta
|-
|-
| | 7/6
| 6/5
|z3
| g3
|zo 3rd
| gu 3rd
| | sol-ta
| mi-sol, ti-re
|-
|-
| | 6/5
| 11/9
|g3
| 1o3
|gu 3rd
| ilo 3rd
| | mi-sol, ti-re
| re-fu
|-
|-
| | 11/9
| 5/4
|1o3
| y3
|ilo 3rd
| yo 3rd
| | re-fu
| do-mi
|-
|-
| | 5/4
| 14/11
|y3
| 1uz4
|yo 3rd
| luzo 4th
| | do-mi
| fu-ta
|-
|-
| | 14/11
| 9/7
|1uz4
| r3
|luzo 4th
| ru 3rd
| | fu-ta
| ta-re
|-
|-
| | 9/7
| 4/3
|r3
| w4
|ru 3rd
| wa 4th
| | ta-re
| do-fa
|-
|-
| | 4/3
| 11/8
|w4
| 1o4
|wa 4th
| ilo 4th
| | do-fa
| do-fu
|-
|-
| | 11/8
| 7/5
|1o4
| zg5
|ilo 4th
| zogu 5th
| | do-fu
| mi-ta
|-
|-
| | 7/5
| 10/7
|zg5
| ry4
|zogu 5th
| ruyo 4th
| | mi-ta
| ta-mi
|-
|-
| | 10/7
| 16/11
|ry4
| 1u5
|ruyo 4th
| lu 5th
| | ta-mi
| fu-do
|-
|-
| | 16/11
| 3/2
|1u5
| w5
|lu 5th
| wa 5th
| | fu-do
| do-sol
|-
|-
| | 3/2
| 14/9
|w5
| z6
|wa 5th
| zo 6th
| | do-sol
| re-ta
|-
|-
| | 14/9
| 11/7
|z6
| 1or5
|zo 6th
| loru 5th
| | re-ta
| ta-fu
|-
|-
| | 11/7
| 8/5
|1or5
| g6
|loru 5th
| gu 6th
| | ta-fu
| mi-do
|-
| | 8/5
|g6
|gu 6th
| | mi-do
|-
|-
| | 18/11
| 18/11
|1u6
| 1u6
|lu 6th
| lu 6th
| | fu-re
| fu-re
|-
|-
| | 5/3
| 5/3
|y6
| y6
|yo 6th
| yo 6th
| | sol-mi
| sol-mi
|-
|-
| | 12/7
| 12/7
|r6
| r6
|ru 6th
| ru 6th
| | ta-sol
| ta-sol
|-
|-
| | 7/4
| 7/4
|z7
| z7
|zo 7th
| zo 7th
| | do-ta
| do-ta
|-
|-
| | 16/9
| 16/9
|w7
| w7
|wa 7th
| wa 7th
| | re-do
| re-do
|-
|-
| | 9/5
| 9/5
|g7
| g7
|gu 7th
| gu 7th
| | mi-re
| mi-re
|-
|-
| | 20/11
| 20/11
|1uy7
| 1uy7
|luyo 7th
| luyo 7th
| | fu-mi
| fu-mi
|-
|-
| | 11/6
| 11/6
|1o7
| 1o7
|ilo 7th
| ilo 7th
| | sol-fu
| sol-fu
|-
|-
| | 2/1
| 2/1
|w8
| w8
|wa 8ve
| wa 8ve
| | do-do
| do-do
|}
|}
While the [[7-limit|7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo|12edo]], the 11-limit introduces neutral intervals, [[Superfourth|superfourth]]s and [[Subfifth|subfifth]]s, which fall in between major, minor and perfect [[interval_category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.
Relative to their size, [[EDO|edo]]s which do (relatively) well in supporting 11-limit intervals are: [[1edo|1edo]], [[2edo|2edo]], [[3edo|3edo]], [[4edo|4edo]], [[5edo|5edo]], [[6edo|6edo]], [[7edo|7edo]], [[9edo|9edo]], [[10edo|10edo]], [[12edo|12edo]], [[15edo|15edo]], [[22edo|22edo]], [[26edo|26edo]], [[31edo|31edo]], [[41edo|41edo]], [[63edo|63edo]], [[72edo|72edo]], [[87edo|87edo]], [[109edo|109edo]], [[161edo|161edo]].


[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]
[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]


==Intervals==
=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:
Here are all the 15-odd-limit intervals of 11:


{| class="wikitable"
{| class="wikitable center-all right-2"
|-
!Ratio
! |[[Kite's color notation|Interval name]]
!Site ([[cents|¢]])
! |
! colspan="2" |[[Color name]]
! | Ratio
! | Cents Value
|-
|-
|[[12/11]]
|150.637
|1u2
|1u2
|lu 2nd
|lu 2nd
| | [[12/11|12/11]]
| | 150.637
|-
|-
|1og2
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|logu 2nd
| | [[11/10|11/10]]
| | 165.004
|-
|-
|[[11/9]]
|347.408
|1o3
|1o3
|ilo 3rd
|ilo 3rd
| | [[11/9|11/9]]
| | 347.408
|-
|-
|1uz4
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|lu 4th
| | [[14/11|14/11]]
| | 417.508
|-
|-
|1uy4
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|luyo 4th
| | [[15/11|15/11]]
| | 536.951
|-
|-
|[[11/8]]
|551.318
|1o4
|1o4
|ilo 4th
|ilo 4th
| | [[11/8|11/8]]
| | 551.318
|-
|-
|[[16/11]]
|648.682
|1u5
|1u5
|lu 5th
|lu 5th
| | [[16/11|16/11]]
| | 648.682
|-
|-
|1og5
|[[22/15]]
|663.049
| 1og5
|logu 5th
|logu 5th
| | [[22/15|22/15]]
| | 663.049
|-
|-
|1or5
|[[11/7]]
|782.492
| 1or5
|loru 5th
|loru 5th
| | [[11/7|11/7]]
| | 782.492
|-
|-
|[[18/11]]
|852.592
|1u6
|1u6
|lu 6th
|lu 6th
| | [[18/11|18/11]]
| | 852.592
|-
|-
|1uy7
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|luyo 7th
| | [[20/11|20/11]]
| | 1034.996
|-
|-
|[[11/6]]
|1049.363
|1o7
|1o7
|ilo 7th
|ilo 7th
| | [[11/6|11/6]]
| | 1049.363
|}
|}
See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]


=Music=
== Music ==
[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave_Hill|Dave Hill]]
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)
 
; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]
 
; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]
 
; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)
 
; [[Dave Hill]]
* [https://ralphdavidhill.bandcamp.com/track/study-3 ''Study #3'']
* [https://ralphdavidhill.bandcamp.com/track/brief-11-limit-ratio-composition ''Brief 11-ratio composition'']
 
; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet


[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill
; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)


[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris_Vaisvil|Chris Vaisvil]]
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']


[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew_Heathwaite|Andrew Heathwaite]]
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)


=See also=
== See also ==
[[Harmonic_Limit|Harmonic Limit]]
* [[Gallery of just intervals]]


[[Category:11-limit]]
[[Category:11-limit| ]] <!-- main article -->
[[Category:example]]
[[Category:Lists of intervals]]
[[Category:interval]]
[[Category:Listen]]
[[Category:limit]]
[[Category:Rank 5]]
[[Category:listen]]
[[Category:prime_limit]]
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