Xenharmonic Wiki

13-limit: Difference between revisions

← Older edit
VisualWikitext
Avoid framing it as a constraint, and improve categories
→Edo approximation: calibrate the list according to my research results
 
(30 intermediate revisions by 8 users not shown)
Line 1: Line 1:
The '''13-limit''' or 13-prime-limit consists of [[just intonation]] intervals such that the highest [[prime number]] in all ratios is 13. Thus, [[40/39]] would be allowable, since 40 is 2 × 2 × 2 × 5 and 39 is 3 × 13, but 34/33 would not be allowable, since 34 is 2 × 17, and [[17-limit|17]] is a prime number higher than 13. An interval doesn't need to contain a 13 to be considered within the 13-limit. For instance, [[3/2]] is considered part of the 13-limit, since the primes 2 and 3 are smaller than 13. Also, an interval with a 13 in it is not necessarily within the 13-limit. [[23/13]] is not within the 13-limit, since [[23-limit|23]] is a prime number higher than 13.
{{Prime limit navigation|13}}
The '''13-limit''' or 13-prime-limit consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 13. Thus, [[40/39]] would be within the 13-limit, since 40 is {{nowrap|2 × 2 × 2 × 5}} and 39 is {{nowrap|3 × 13}}, but [[34/33]] would not, since 34 is {{nowrap|2 × 17}}, and [[17-limit|17]] is a prime number higher than 13. The 13-limit is the 6th [[prime limit]] and is a superset of the [[11-limit]] and a subset of the [[17-limit]].  


The 13-prime-limit can be modeled in a 5-dimensional lattice, with the primes 3, 5, 7, 11, and 13 represented by each dimension. The prime 2 does not appear in the typical 13-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a sixth dimension is needed.
The 13-limit is a [[rank and codimension|rank-6]] system, and can be modeled in a 5-dimensional [[lattice]], with the primes 3, 5, 7, 11, and 13 represented by each dimension. The prime 2 does not appear in the typical 13-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a sixth dimension is needed.


Examples of [[EDO]]s which represent 13-limit intervals well include: {{EDOs|26, 37, 46, 50, 87, 130, 183, 207, 217, 224, 270, 494, 851, 1075, 1282, 1578, 2159, 2190, 2684, 3265, 3535, 4573, 5004, 5585, 6079, 8269, 8539, 13854, 14124, 16808, 20203, 22887, 28742, 32007, 37011, 50434, 50928, 51629, 54624, 56202, 59467, 64471, 65052, ... . }}
These things are contained by the 13-limit, but not the 11-limit:
* The [[13-odd-limit|13-]] and [[15-odd-limit]];
* Mode 7 and 8 of the harmonic or subharmonic series.
 
== Edo approximation ==
[[Edo]]s which represent 13-limit intervals better ([[monotonicity limit]] ≥ 13 and decreasing [[TE error]]): {{EDOs| 15, 17c, 19, 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]].
 
Here is a list of edos which tunes the 13-limit well relative to their size ({{nowrap|[[TE relative error]] < 5.5%}}): {{EDOs| 31, 41, 46, 53, 58, 72, 87, 94, 103, 111, 121, 130, 140, 152f, 159, 183, 190, 198, 212, 217, 224, 270, 282, 296, 301, 311, 320, 328, 342f, 354, 364, 369f, 373, 383, 400, 414, 422, 431, 441, 460, 472, 494 }}, 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 ==
== Intervals ==
Here are all the 15-odd-limit intervals of 13:


Here are all the 15-odd-limit intervals of 13:
{| class="wikitable"
{| class="wikitable"
|-
! Ratio
! Ratio
! Cents Value
! Cents value
! colspan="2" | [[Color name]]
! colspan="2" | [[Color name]]
! Interval name
! Name
|-
|-
| 14/13
| 14/13
Line 18: Line 29:
| 3uz2
| 3uz2
| thuzo 2nd
| thuzo 2nd
| tridecimal large semitone <br>tridecimal large limma
| tridecimal supraminor second
|-
|-
| 13/12
| 13/12
Line 30: Line 41:
| 3uy2
| 3uy2
| thuyo 2nd
| thuyo 2nd
| tridecimal second-third
| tridecimal semifourth
|-
|-
| 13/11
| 13/11
Line 48: Line 59:
| 3og4
| 3og4
| thogu 4th
| thogu 4th
| tridecimal third-fourth
| tridecimal naiadic
|-
|-
| 18/13
| 18/13
Line 66: Line 77:
| 3uy5
| 3uy5
| thuyo 5th
| thuyo 5th
| tridecimal fifth-sixth
| tridecimal cocytic
|-
|-
| 13/8
| 13/8
Line 84: Line 95:
| 3og7
| 3og7
| thogu 7th
| thogu 7th
| tridecimal sixth-seventh
| tridecimal semitwelfth
|-
|-
| 24/13
| 24/13
Line 100: Line 111:


== Music ==
== Music ==
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Venusian Cataclysms] [http://sonic-arts.org/hill/10-passages-ji/02_hill_venusian-cataclysms.mp3 play] by [[Dave Hill]] {{dead link}} (404 error as of 2/5/2020)
 
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Chord Progression on the Harmonic Overtone Series] [http://sonic-arts.org/hill/10-passages-ji/06_hill_chord-progression-on-harmonic-series.mp3 play] by Dave Hill  {{dead link}} (404 error as of 2/5/2020)
; [[E8 Heterotic]]
* [https://www.youtube.com/watch?v=cUR3MsI-mWM ''Justification''] (2022)
 
; [[Francium]]
* [https://www.youtube.com/watch?v=ratGb2qTStQ ''Bicycle Wheels''] (2023)
 
; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Venusian Cataclysms'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/02_hill_venusian-cataclysms.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Chord Progression on the Harmonic Overtone Series'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/06_hill_chord-progression-on-harmonic-series.mp3 play]{{dead link}}
 
; [https://youtube.com/@hojominori?si=gqJP3hzvup2YL0sz Hojo Minori]
* [https://www.youtube.com/watch?v=xSIS2lobkTk ''P`rismatic fut`URE''] (2025)
 
; [[Ben Johnston]]
* ''String Quartet No. 5'' (1979) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-5 Bandcamp] | [https://www.youtube.com/watch?v=jOpQwiEB4g0 YouTube] – performed by Kepler Quartet
* ''String Quartet No. 7'' (1984)
** "Movt. 1" – [https://newworldrecords.bandcamp.com/track/string-quartet-no-7-scurrying-forceful-intense Bandcamp] | [https://www.youtube.com/watch?v=-TdFgtAf5Cg YouTube]
** "Movt. 2" – [https://newworldrecords.bandcamp.com/track/string-quartet-no-7-eerie Bandcamp] | [https://www.youtube.com/watch?v=Tq9cjvgnbAY YouTube]
** "Movt. 3" – [https://newworldrecords.bandcamp.com/track/string-quartet-no-7-with-solemnity Bandcamp] | [https://www.youtube.com/watch?v=jgFQAGyF0Gw YouTube]
:: performed by Kepler Quartet
 
; [[Kaiveran Lugheidh]]
* [https://soundcloud.com/vale-10/unlicensed-copy ''Unlicensed Copy''] (2017) – mostly 7-limit with some erstwhile 13-based chromaticisms
 
; [[Claudi Meneghin]]
* [http://web.archive.org/web/20160412025512/http://soonlabel.com/xenharmonic/archives/2089 ''Canon on a ground''] – in 2.11.13 subgroup
 
; [[Gene Ward Smith]]
* [https://archive.org/details/ThrenodyForTheVictimsOfWolfgangAmadeusMozart ''Threnody for the Victims of Wolfgang Amadeus Mozart''] (archived 2010) – 13-limit JI in [[6079edo]] tuning
* [https://archive.org/details/RoughDiamond ''Rough Diamond''] (archived 2010) a.k.a. ''Diamond in the Rough''<ref>[http://lumma.org/tuning/gws/gene.html xenharmony.org mirror | ''Gene's Music'']</ref> – symphonic con brio using the Partch 13-odd-limit tonality diamond as a scale.
 
; [[User:Tristanbay|Tristan Bay]]
* [https://youtu.be/ouUV2Uwr2qI ''Junp''] – in [[User:Tristanbay/Margo Scale|a 2.3.11/7.13/7 subgroup JI scale]]
 
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=wbZXArV5ffw ''Dying Visions of a Lonesome Machine''] (2021)
* [https://www.youtube.com/watch?v=bA6wr07PiYE ''Avenoir''] (2022)
* [https://www.youtube.com/watch?v=rBS2gGTostA ''I Was a Teenage Boltzmann Brain''] (2022)
* [https://www.youtube.com/watch?v=NwsMMnOTcQ4 ''Atlas Apassionata''] (2022)


== See also ==
== See also ==
* [[Harmonic limit]]
* [[13-odd-limit]]
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]
* [[Tridecimal neutral seventh chord]]
* [[Augmented chords in just intonation, some]] (they are 13-limit)
== Notes ==


[[Category:Just intonation]]
[[Category:Limit]]
[[Category:Prime limit]]
[[Category:13-limit| ]] <!-- main article -->
[[Category:13-limit| ]] <!-- main article -->
[[Category:Interval collection]]
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 6]]
[[Category:Rank 6]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/13-limit"