13-limit: Difference between revisions

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The 13-prime-limit refers to a constraint on selecting just intonation intervals such that the highest [[prime_numbers|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.
The '''13-prime-limit''' refers to a constraint on selecting 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.


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-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.


[[EDO]]s good for 13-limit are 5, 6, 7, 9, 10, 15, 16, 17, 19, 20, 22, 24, 26, 31, 37, 46, 50, 53, 63, 77, 84, 87, 130, 140, 161, 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, ... .
[[EDO]]s good for 13-limit are {{EDOs| 5, 6, 7, 9, 10, 15, 16, 17, 19, 20, 22, 24, 26, 31, 37, 46, 50, 53, 63, 77, 84, 87, 130, 140, 161, 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, ... . }}


== 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" |[[Kite's color notation|Color name]]
! colspan="2" | [[Color name]]
! |Interval name
! Interval name
|-
|-
|14/13
| 14/13
|128.298
| 128.298
|3uz2
| 3uz2
|thuzo 2nd
| thuzo 2nd
|tridecimal large semitone <br>tridecimal large limma
| tridecimal large semitone <br>tridecimal large limma
|-
|-
|13/12
| 13/12
|138.573
| 138.573
|3o2
| 3o2
|tho 2nd
| tho 2nd
|tridecimal subneutral second
| tridecimal subneutral second
|-
|-
|15/13
| 15/13
|247.741
| 247.741
|3uy2
| 3uy2
|thuyo 2nd
| thuyo 2nd
|tridecimal second-third
| tridecimal second-third
|-
|-
|13/11
| 13/11
|289.210
| 289.210
|3o1u3
| 3o1u3
|tholu 3rd
| tholu 3rd
|tridecimal minor third
| tridecimal minor third
|-
|-
|16/13
| 16/13
|359.472
| 359.472
|3u3
| 3u3
|thu 3rd
| thu 3rd
|tridecimal supraneutral third
| tridecimal supraneutral third
|-
|-
|13/10
| 13/10
|454.214
| 454.214
|3og4
| 3og4
|thogu 4th
| thogu 4th
|tridecimal third-fourth
| tridecimal third-fourth
|-
|-
|18/13
| 18/13
|563.382
| 563.382
|3u4
| 3u4
|thu 4th
| thu 4th
|tridecimal sub-tritone
| tridecimal sub-tritone
|-
|-
|13/9
| 13/9
|636.618
| 636.618
|3o5
| 3o5
|tho 5th
| tho 5th
|tridecimal super-tritone
| tridecimal super-tritone
|-
|-
|20/13
| 20/13
|745.786
| 745.786
|3uy5
| 3uy5
|thuyo 5th
| thuyo 5th
|tridecimal fifth-sixth
| tridecimal fifth-sixth
|-
|-
|13/8
| 13/8
|840.528
| 840.528
|3o6
| 3o6
|tho 6th
| tho 6th
|tridecimal subneutral sixth
| tridecimal subneutral sixth
|-
|-
|22/13
| 22/13
|910.790
| 910.790
|3u1o6
| 3u1o6
|thulo 6th
| thulo 6th
|tridecimal major sixth
| tridecimal major sixth
|-
|-
|26/15
| 26/15
|952.259
| 952.259
|3og7
| 3og7
|thogu 7th
| thogu 7th
|tridecimal sixth-seventh
| tridecimal sixth-seventh
|-
|-
|24/13
| 24/13
|1061.427
| 1061.427
|3u7
| 3u7
|thu 7th
| thu 7th
|tridecimal supraneutral seventh
| tridecimal supraneutral seventh
|-
|-
|13/7
| 13/7
|1071.702
| 1071.702
|3or7
| 3or7
|thoru 7th
| thoru 7th
|tridecimal submajor seventh
| tridecimal submajor seventh
|}
|}
See: [[Gallery of Just Intervals]]


=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|Dave Hill]]  (404 error as of 2/5/2020)
 
* [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)


[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 (404 error as of 2/5/2020)
== See also ==


=See also=
* [[Harmonic limit]]
* [[Harmonic limit]]
* [[13-odd-limit]]
* [[13-odd-limit]]
* [[Gallery of Just Intervals]]


[[Category:Limit]]
[[Category:Prime limit]]
[[Category:13-limit]]
[[Category:13-limit]]
[[Category:limit]]
[[Category:Listen]]
[[Category:listen]]
[[Category:Rank 6]]
[[Category:prime_limit]]
[[Category:rank_6]]

Revision as of 17:29, 25 October 2020

The 13-prime-limit refers to a constraint on selecting 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 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 is a prime number higher than 13.

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.

EDOs good for 13-limit are 5, 6, 7, 9, 10, 15, 16, 17, 19, 20, 22, 24, 26, 31, 37, 46, 50, 53, 63, 77, 84, 87, 130, 140, 161, 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, ... .

Intervals

Here are all the 15-odd-limit intervals of 13:

Ratio Cents Value Color name Interval name
14/13 128.298 3uz2 thuzo 2nd tridecimal large semitone
tridecimal large limma
13/12 138.573 3o2 tho 2nd tridecimal subneutral second
15/13 247.741 3uy2 thuyo 2nd tridecimal second-third
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 third-fourth
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 fifth-sixth
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 sixth-seventh
24/13 1061.427 3u7 thu 7th tridecimal supraneutral seventh
13/7 1071.702 3or7 thoru 7th tridecimal submajor seventh

Music

See also

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