View source for 13-limit

{{Prime limit navigation|13}}
The '''13-limit''' or 13-prime-limit consists of [[just intonation]] [[interval]]s such that the highest [[prime number]] in all ratios is 13. Thus, [[40/39]] would be within the 13-limit, since 40 is 2 × 2 × 2 × 5 and 39 is 3 × 13, but [[34/33]] would not, since 34 is 2 × 17, and [[17-limit|17]] is a prime number higher than 13.

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.

== Edo approximations ==
A list of [[edo]]s which represent 13-limit intervals better: {{EDOs| 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 }} and so on. 

Another list of edos which tunes the 13-limit well relative to their size ([[TE relative error|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 ==
Here are all the 15-odd-limit intervals of 13:

{| class="wikitable"
! Ratio
! Cents Value
! colspan="2" | [[Color name|Color Name]]
! Name
|-
| 14/13
| 128.298
| 3uz2
| thuzo 2nd
| tridecimal supraminor second
|-
| 13/12
| 138.573
| 3o2
| tho 2nd
| tridecimal subneutral second
|-
| 15/13
| 247.741
| 3uy2
| thuyo 2nd
| tridecimal semifourth
|-
| 13/11
| 289.210
| 3o1u3
| tholu 3rd
| tridecimal minor third
|-
| 16/13
| 359.472
| 3u3
| thu 3rd
| tridecimal supraneutral third
|-
| 13/10
| 454.214
| 3og4
| thogu 4th
| tridecimal naiadic
|-
| 18/13
| 563.382
| 3u4
| thu 4th
| tridecimal sub-tritone
|-
| 13/9
| 636.618
| 3o5
| tho 5th
| tridecimal super-tritone
|-
| 20/13
| 745.786
| 3uy5
| thuyo 5th
| tridecimal cocytic
|-
| 13/8
| 840.528
| 3o6
| tho 6th
| tridecimal subneutral sixth
|-
| 22/13
| 910.790
| 3u1o6
| thulo 6th
| tridecimal major sixth
|-
| 26/15
| 952.259
| 3og7
| thogu 7th
| tridecimal semitwelfth
|-
| 24/13
| 1061.427
| 3u7
| thu 7th
| tridecimal supraneutral seventh
|-
| 13/7
| 1071.702
| 3or7
| thoru 7th
| tridecimal submajor seventh
|}

== Music ==
; [[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}} (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'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/06_hill_chord-progression-on-harmonic-series.mp3 play]{{dead link}} (404 error as of 2/5/2020)

== See also ==
* [[Harmonic limit]]
* [[13-odd-limit]]
* [[Gallery of just intervals]]

[[Category:13-limit| ]] <!-- main article -->
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 6]]