17-limit: Difference between revisions
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In 17-limit [[Just_intonation|Just Intonation]], all ratios in the system will contain no primes higher than 17. The 17-limit adds to the [[13-limit]] a "minor ninth" of about 105¢ -- [[17/16]] -- and several other intervals between the 17th overtone and the lower ones. | In 17-limit [[Just_intonation|Just Intonation]], all ratios in the system will contain no primes higher than 17. The 17-limit adds to the [[13-limit]] a "minor ninth" of about 105¢ -- [[17/16]] -- and several other intervals between the 17th overtone and the lower ones. | ||
The 17-prime-limit can be modeled in a 6-dimensional lattice, with the primes 3, 5, 7, 11, 13, and 17 represented by each dimension. The prime 2 does not appear in the typical 17-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a seventh dimension is need. | |||
[[EDO]]s which provides an excellent tuning for 17-limit intervals are: 46, 58, 72, 80, 94, 111, 121, 149, 159, 183, 217, 253, 270, 282, 301, 311, 320, 354, 364, 383, 388, 400, 414, 422, 436, 441, 460, 494, 525, 535, 581, 597, 615, 624, 639, 643, 653, 684, 692, 718, 742, 764, 771, 776, 814, 860, 867, 882, 894, 908, 925, 935, 954, 981, 995, and 997 among others. | [[EDO]]s which provides an excellent tuning for 17-limit intervals are: 46, 58, 72, 80, 94, 111, 121, 149, 159, 183, 217, 253, 270, 282, 301, 311, 320, 354, 364, 383, 388, 400, 414, 422, 436, 441, 460, 494, 525, 535, 581, 597, 615, 624, 639, 643, 653, 684, 692, 718, 742, 764, 771, 776, 814, 860, 867, 882, 894, 908, 925, 935, 954, 981, 995, and 997 among others. | ||
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==17-limit Intervals== | ==17-limit Intervals== | ||
Here are all the 21-odd-limit intervals of 17: | Here are all the 21-odd-limit intervals of 17: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
| Line 12: | Line 15: | ||
! | Name | ! | Name | ||
|- | |- | ||
| | [[ | | | [[18/17]] | ||
| | 98.955 | | | 98.955 | ||
|17u1 | | | 17u1 | ||
|su | | | su unison | ||
| | small septendecimal semitone | | | small septendecimal semitone | ||
|- | |- | ||
| | [[ | | | [[17/16]] | ||
| | 104.955 | | | 104.955 | ||
|17o2 | | | 17o2 | ||
|so 2nd | | | so 2nd | ||
| | large septendecimal semitone | | | large septendecimal semitone | ||
|- | |- | ||
| | [[ | | | [[17/15]] | ||
| | 216.687 | | | 216.687 | ||
|17og3 | | | 17og3 | ||
|sogu 3rd | | | sogu 3rd | ||
| | septendecimal whole tone | | | septendecimal whole tone | ||
|- | |- | ||
| | [[ | | | [[20/17]] | ||
| | 281.358 | | | 281.358 | ||
|17uy2 | | | 17uy2 | ||
|suyo 2nd | | | suyo 2nd | ||
| | septendecimal minor third | | | septendecimal minor third | ||
|- | |- | ||
| | [[ | | | [[17/14]] | ||
| | 336.130 | | | 336.130 | ||
|17or3 | | | 17or3 | ||
|soru 3rd | | | soru 3rd | ||
| | septendecimal supraminor third | | | septendecimal supraminor third | ||
|- | |- | ||
| | [[ | | | [[21/17]] | ||
| | 365.825 | | | 365.825 | ||
|17uz3 | | | 17uz3 | ||
|suzo 3rd | | | suzo 3rd | ||
| | septendecimal submajor third | | | septendecimal submajor third | ||
|- | |- | ||
| | [[ | | | [[22/17]] | ||
| | 446.363 | | | 446.363 | ||
|17u1o3 | | | 17u1o3 | ||
|sulo 3rd | | | sulo 3rd | ||
| | septendecimal supermajor third | | | septendecimal supermajor third | ||
|- | |- | ||
| | [[ | | | [[17/13]] | ||
| | 464.428 | | | 464.428 | ||
|17o3u4 | | | 17o3u4 | ||
|sothu 4th | | | sothu 4th | ||
| | septendecimal sub-fourth | | | septendecimal sub-fourth | ||
|- | |- | ||
| | [[ | | | [[24/17]] | ||
| | 597.000 | | | 597.000 | ||
|17u4 | | | 17u4 | ||
|su 4th | | | su 4th | ||
| | | | | lesser septendecimal tritone | ||
|- | |- | ||
| | [[ | | | [[17/12]] | ||
| | 603.000 | | | 603.000 | ||
|17o5 | | | 17o5 | ||
|so 5th | | | so 5th | ||
| | | | | greater septendecimal tritone | ||
|- | |- | ||
| | [[ | | | [[26/17]] | ||
| | 735.572 | | | 735.572 | ||
|17u3o5 | | | 17u3o5 | ||
|sutho 5th | | | sutho 5th | ||
| | septendecimal super-fifth | | | septendecimal super-fifth | ||
|- | |- | ||
| | [[ | | | [[17/11]] | ||
| | 753.637 | | | 753.637 | ||
|17o1u6 | | | 17o1u6 | ||
|solu 6th | | | solu 6th | ||
| | septendecimal subminor sixth | | | septendecimal subminor sixth | ||
|- | |- | ||
|[[34/21]] | | | [[34/21]] | ||
|834.175 | | | 834.175 | ||
|17uz6 | | | 17uz6 | ||
|suzo 6th | | | suzo 6th | ||
|septendecimal superminor sixth | | | septendecimal superminor sixth | ||
|- | |- | ||
| | [[ | | | [[28/17]] | ||
| | 863.870 | | | 863.870 | ||
|17uz6 | | | 17uz6 | ||
|suzo 6th | | | suzo 6th | ||
| | septendecimal submajor sixth | | | septendecimal submajor sixth | ||
|- | |- | ||
| | [[ | | | [[17/10]] | ||
| | 918.642 | | | 918.642 | ||
|17og7 | | | 17og7 | ||
|sogu 7th | | | sogu 7th | ||
| | septendecimal major sixth | | | septendecimal major sixth | ||
|- | |- | ||
| | [[ | | | [[30/17]] | ||
| | 983.313 | | | 983.313 | ||
|17uy6 | | | 17uy6 | ||
|suyo 6th | | | suyo 6th | ||
| | septendecimal minor seventh | | | septendecimal minor seventh | ||
|- | |- | ||
| | [[ | | | [[32/17]] | ||
| | 1095.045 | | | 1095.045 | ||
|17u7 | | | 17u7 | ||
|su 7th | | | su 7th | ||
| | small septendecimal major seventh | | | small septendecimal major seventh | ||
|- | |- | ||
| | [[ | | | [[17/9]] | ||
| | 1101.045 | | | 1101.045 | ||
|17o8 | | | 17o8 | ||
|so | | | so octave | ||
| | large septendecimal major seventh | | | large septendecimal major seventh | ||
|} | |} | ||
Revision as of 02:39, 3 May 2019
In 17-limit Just Intonation, all ratios in the system will contain no primes higher than 17. The 17-limit adds to the 13-limit a "minor ninth" of about 105¢ -- 17/16 -- and several other intervals between the 17th overtone and the lower ones.
The 17-prime-limit can be modeled in a 6-dimensional lattice, with the primes 3, 5, 7, 11, 13, and 17 represented by each dimension. The prime 2 does not appear in the typical 17-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a seventh dimension is need.
EDOs which provides an excellent tuning for 17-limit intervals are: 46, 58, 72, 80, 94, 111, 121, 149, 159, 183, 217, 253, 270, 282, 301, 311, 320, 354, 364, 383, 388, 400, 414, 422, 436, 441, 460, 494, 525, 535, 581, 597, 615, 624, 639, 643, 653, 684, 692, 718, 742, 764, 771, 776, 814, 860, 867, 882, 894, 908, 925, 935, 954, 981, 995, and 997 among others.
17-limit Intervals
Here are all the 21-odd-limit intervals of 17:
| Ratio | Cents Value | Color Name | Name | |
|---|---|---|---|---|
| 18/17 | 98.955 | 17u1 | su unison | small septendecimal semitone |
| 17/16 | 104.955 | 17o2 | so 2nd | large septendecimal semitone |
| 17/15 | 216.687 | 17og3 | sogu 3rd | septendecimal whole tone |
| 20/17 | 281.358 | 17uy2 | suyo 2nd | septendecimal minor third |
| 17/14 | 336.130 | 17or3 | soru 3rd | septendecimal supraminor third |
| 21/17 | 365.825 | 17uz3 | suzo 3rd | septendecimal submajor third |
| 22/17 | 446.363 | 17u1o3 | sulo 3rd | septendecimal supermajor third |
| 17/13 | 464.428 | 17o3u4 | sothu 4th | septendecimal sub-fourth |
| 24/17 | 597.000 | 17u4 | su 4th | lesser septendecimal tritone |
| 17/12 | 603.000 | 17o5 | so 5th | greater septendecimal tritone |
| 26/17 | 735.572 | 17u3o5 | sutho 5th | septendecimal super-fifth |
| 17/11 | 753.637 | 17o1u6 | solu 6th | septendecimal subminor sixth |
| 34/21 | 834.175 | 17uz6 | suzo 6th | septendecimal superminor sixth |
| 28/17 | 863.870 | 17uz6 | suzo 6th | septendecimal submajor sixth |
| 17/10 | 918.642 | 17og7 | sogu 7th | septendecimal major sixth |
| 30/17 | 983.313 | 17uy6 | suyo 6th | septendecimal minor seventh |
| 32/17 | 1095.045 | 17u7 | su 7th | small septendecimal major seventh |
| 17/9 | 1101.045 | 17o8 | so octave | large septendecimal major seventh |
To avoid confusion with the solfege syllable So, the so 2nd, 5th and 8ve are sometimes called the iso 2nd, 5th and 8ve.