15-odd-limit: Difference between revisions
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use the proper centered tilde. in some fonts "(~15 .. 41 steps)" looks like "(˜15 .. 41 steps)" which is bad. |
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{{Odd-limit navigation|15}} | |||
{{Odd-limit intro|15}} | |||
This collection of intervals has proven to be useful for illustrating certain characteristics of medium-sized [[edo]]s (~15 to 41 steps). | |||
* [[16/15]], [[15/8]] | * [[1/1]] | ||
* [[15/14]], [[28/15]] | * '''[[16/15]], [[15/8]]''' | ||
* '''[[15/14]], [[28/15]]''' | |||
* [[14/13]], [[13/7]] | * [[14/13]], [[13/7]] | ||
* [[13/12]], [[24/13]] | * [[13/12]], [[24/13]] | ||
| Line 10: | Line 13: | ||
* [[9/8]], [[16/9]] | * [[9/8]], [[16/9]] | ||
* [[8/7]], [[7/4]] | * [[8/7]], [[7/4]] | ||
* [[15/13]], [[26/15]] | * '''[[15/13]], [[26/15]]''' | ||
* [[7/6]], [[12/7]] | * [[7/6]], [[12/7]] | ||
* [[13/11]], [[22/13]] | * [[13/11]], [[22/13]] | ||
| Line 21: | Line 24: | ||
* [[13/10]], [[20/13]] | * [[13/10]], [[20/13]] | ||
* [[4/3]], [[3/2]] | * [[4/3]], [[3/2]] | ||
* [[15/11]], [[22/15]] | * '''[[15/11]], [[22/15]]''' | ||
* [[11/8]], [[16/11]] | * [[11/8]], [[16/11]] | ||
* [[18/13]], [[13/9]] | * [[18/13]], [[13/9]] | ||
* [[7/5]], [[ | * [[7/5]], [[10/7]] | ||
[[ | {| class="wikitable center-all right-2 left-5" | ||
[[Category: | ! Ratio | ||
! Size ([[cents|¢]]) | |||
! colspan="2" | [[Color name]] | |||
! Name(s) | |||
|- | |||
| [[16/15]] | |||
| 111.731 | |||
| g2 | |||
| gu 2nd | |||
| classic diatonic semitone | |||
|- | |||
| [[15/14]] | |||
| 119.443 | |||
| ry1 | |||
| ruyo unison | |||
| septimal diatonic semitone | |||
|- | |||
| [[15/13]] | |||
| 247.741 | |||
| 3uy2 | |||
| thuyo 2nd | |||
| tridecimal supermajor second / tridecimal second-third | |||
|- | |||
| [[15/11]] | |||
| 536.951 | |||
| 1uy4 | |||
| luyo 4th | |||
| undecimal acute fourth | |||
|- | |||
| [[22/15]] | |||
| 663.049 | |||
| 1og5 | |||
| logu 5th | |||
| undecimal grave fifth | |||
|- | |||
| [[26/15]] | |||
| 952.259 | |||
| 3og7 | |||
| thogu 7th | |||
| tridecimal subminor seventh / tridecimal sixth-seventh | |||
|- | |||
| [[28/15]] | |||
| 1080.557 | |||
| zg8 | |||
| zogu octave | |||
| small septimal major seventh | |||
|- | |||
| [[15/8]] | |||
| 1088.269 | |||
| y7 | |||
| yo 7th | |||
| just major seventh | |||
|} | |||
The smallest edo which is consistent in the 15-odd-limit is [[29edo]]. | |||
The one that is distinctly consistent in the same is [[111edo]]. | |||
== See also == | |||
* [[Arto and tendo theory]] | |||
* [[Diamond15]] – as a scale | |||
[[Category:15-odd-limit| ]] <!-- main article --> | |||
Latest revision as of 13:45, 8 October 2025
The 15-odd-limit is the set of all rational intervals which can be written as 2k(a/b) where a, b ≤ 15 and k is an integer. To the 13-odd-limit, it adds 4 pairs of octave-reduced intervals involving 15.
Below is a list of all octave-reduced intervals in the 15-odd-limit. This collection of intervals has proven to be useful for illustrating certain characteristics of medium-sized edos (~15 to 41 steps).
- 1/1
- 16/15, 15/8
- 15/14, 28/15
- 14/13, 13/7
- 13/12, 24/13
- 12/11, 11/6
- 11/10, 20/11
- 10/9, 9/5
- 9/8, 16/9
- 8/7, 7/4
- 15/13, 26/15
- 7/6, 12/7
- 13/11, 22/13
- 6/5, 5/3
- 11/9, 18/11
- 16/13, 13/8
- 5/4, 8/5
- 14/11, 11/7
- 9/7, 14/9
- 13/10, 20/13
- 4/3, 3/2
- 15/11, 22/15
- 11/8, 16/11
- 18/13, 13/9
- 7/5, 10/7
| Ratio | Size (¢) | Color name | Name(s) | |
|---|---|---|---|---|
| 16/15 | 111.731 | g2 | gu 2nd | classic diatonic semitone |
| 15/14 | 119.443 | ry1 | ruyo unison | septimal diatonic semitone |
| 15/13 | 247.741 | 3uy2 | thuyo 2nd | tridecimal supermajor second / tridecimal second-third |
| 15/11 | 536.951 | 1uy4 | luyo 4th | undecimal acute fourth |
| 22/15 | 663.049 | 1og5 | logu 5th | undecimal grave fifth |
| 26/15 | 952.259 | 3og7 | thogu 7th | tridecimal subminor seventh / tridecimal sixth-seventh |
| 28/15 | 1080.557 | zg8 | zogu octave | small septimal major seventh |
| 15/8 | 1088.269 | y7 | yo 7th | just major seventh |
The smallest edo which is consistent in the 15-odd-limit is 29edo.
The one that is distinctly consistent in the same is 111edo.
See also
- Arto and tendo theory
- Diamond15 – as a scale