217edo: Difference between revisions
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The '''217 equal divisions of the octave''' (''' | {{Infobox ET | ||
| Prime factorization = 7 × 31 | |||
| Step size = 5.52995¢ | |||
| Fifth = 127\217 (702.30¢) | |||
| Semitones = 21:16 (116.13¢ : 88.48¢) | |||
| Consistency = 21 | |||
}} | |||
The '''217 equal divisions of the octave''' ('''217edo'''), or the '''217(-tone) equal temperament''' ('''217tet''', '''217et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 217 parts of about 5.53 [[cent]]s each. | |||
== Theory == | == Theory == | ||
217edo is a strong [[19-limit]] system, the smallest uniquely [[consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]]. It shares the same 5th and 7th [[Harmonic series|harmonics]] with [[31edo]] (217 = 7 × 31), as well as the [[11/9]] interval (supporting the [[31-comma temperaments #Birds|birds temperament]]). However, compared to 31edo, its [[patent val]] differ on the mappings for 3, 11, 13, 17 and 19 – in fact, this edo has a very accurate 13th harmonic, as well as the [[19/15]] interval. | |||
It tempers out the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[escapade comma]], {{monzo| 32 -7 -9 }} in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]] and 823543/819200 in the 7-limit; [[441/440]], [[4000/3993]] | It tempers out the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[escapade comma]], {{monzo| 32 -7 -9 }} in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]] and 823543/819200 in the 7-limit; [[441/440]], [[4000/3993]], 5632/5625, and [[16384/16335]] in the 11-limit; [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]] and [[4096/4095]] in the 13-limit; 595/594, [[833/832]], [[936/935]], 1156/1155, [[1225/1224]], [[1701/1700]] in the 17-limit; 343/342, 476/475, 969/968, [[1216/1215]], [[1445/1444]], [[1521/1520]] and 1540/1539 in the 19-limit. It provides the [[optimal patent val]] for the 11- and 13-limit [[arch]] and the 11- and 13-limit [[cotoneum]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
Revision as of 14:39, 24 October 2021
| ← 216edo | 217edo | 218edo → |
The 217 equal divisions of the octave (217edo), or the 217(-tone) equal temperament (217tet, 217et) when viewed from a regular temperament perspective, is the equal division of the octave into 217 parts of about 5.53 cents each.
Theory
217edo is a strong 19-limit system, the smallest uniquely consistent in the 19-odd-limit and consistent to the 21-odd-limit. It shares the same 5th and 7th harmonics with 31edo (217 = 7 × 31), as well as the 11/9 interval (supporting the birds temperament). However, compared to 31edo, its patent val differ on the mappings for 3, 11, 13, 17 and 19 – in fact, this edo has a very accurate 13th harmonic, as well as the 19/15 interval.
It tempers out the parakleisma, [8 14 -13⟩, and the escapade comma, [32 -7 -9⟩ in the 5-limit; 3136/3125, 4375/4374, 10976/10935 and 823543/819200 in the 7-limit; 441/440, 4000/3993, 5632/5625, and 16384/16335 in the 11-limit; 364/363, 676/675, 1001/1000, 1575/1573, 2080/2079 and 4096/4095 in the 13-limit; 595/594, 833/832, 936/935, 1156/1155, 1225/1224, 1701/1700 in the 17-limit; 343/342, 476/475, 969/968, 1216/1215, 1445/1444, 1521/1520 and 1540/1539 in the 19-limit. It provides the optimal patent val for the 11- and 13-limit arch and the 11- and 13-limit cotoneum.
Prime harmonics
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JI approximation
Selected just intervals
The following table shows how 23-odd-limit intervals are represented in 217EDO. Prime harmonics are in bold; inconsistent intervals are in italic.
| Interval, complement | Error (abs, ¢) |
|---|---|
| 16/13, 13/8 | 0.025 |
| 19/15, 30/19 | 0.028 |
| 10/9, 9/5 | 0.085 |
| 17/13, 26/17 | 0.088 |
| 17/16, 32/17 | 0.114 |
| 24/17, 17/12 | 0.235 |
| 20/19, 19/10 | 0.321 |
| 13/12, 24/13 | 0.324 |
| 4/3, 3/2 | 0.349 |
| 19/18, 36/19 | 0.406 |
| 6/5, 5/3 | 0.434 |
| 23/22, 44/23 | 0.463 |
| 15/11, 22/15 | 0.545 |
| 22/19, 19/11 | 0.573 |
| 18/17, 17/9 | 0.585 |
| 20/17, 17/10 | 0.669 |
| 18/13, 13/9 | 0.673 |
| 9/8, 16/9 | 0.698 |
| 21/16, 32/21 | 0.735 |
| 24/19, 19/12 | 0.755 |
| 26/21, 21/13 | 0.760 |
| 13/10, 20/13 | 0.758 |
| 5/4, 8/5 | 0.783 |
| 21/17, 34/21 | 0.849 |
| 11/10, 20/11 | 0.894 |
| 11/9, 18/11 | 0.979 |
| 19/17, 34/19 | 0.991 |
| 30/23, 23/15 | 1.008 |
| 17/15, 30/17 | 1.018 |
| 23/19, 38/23 | 1.036 |
| 26/19, 19/13 | 1.079 |
| 8/7, 7/4 | 1.084 |
| 19/16, 32/19 | 1.104 |
| 15/13, 26/15 | 1.107 |
| 14/13, 13/7 | 1.109 |
| 16/15, 15/8 | 1.132 |
| 17/14, 28/17 | 1.198 |
| 12/11, 11/6 | 1.328 |
| 23/20, 40/23 | 1.357 |
| 7/6, 12/7 | 1.433 |
| 23/18, 36/23 | 1.442 |
| 21/20, 40/21 | 1.518 |
| 22/17, 17/11 | 1.564 |
| 13/11, 22/13 | 1.652 |
| 11/8, 16/11 | 1.677 |
| 9/7, 14/9 | 1.782 |
| 24/23, 23/12 | 1.791 |
| 21/19, 38/21 | 1.839 |
| 7/5, 10/7 | 1.867 |
| 23/17, 34/23 | 2.027 |
| 26/23, 23/13 | 2.115 |
| 32/23, 23/16 | 2.140 |
| 19/14, 28/19 | 2.188 |
| 15/14, 28/15 | 2.216 |
| 28/23, 23/14 | 2.306 |
| 22/21, 21/11 | 2.412 |
| 23/21, 42/23 | 2.655 |
| 14/11, 11/7 | 2.761 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [344 -217⟩ | [⟨217 344]] | -0.110 | 0.1101 | 1.99 |
| 2.3.5 | [8 14 -13⟩, [32 -7 -9⟩ | [⟨217 344 504]] | -0.186 | 0.1398 | 2.53 |
| 2.3.5.7 | 3136/3125, 4375/4374, 823543/819200 |
[⟨217 344 504 609]] | -0.043 | 0.2757 | 4.99 |
| 2.3.5.7.11 | 441/440, 3136/3125, 4000/3993, 4375/4374 |
[⟨217 344 504 609 751]] | -0.131 | 0.3034 | 5.49 |
| 2.3.5.7.11.13 | 364/363, 441/440, 676/675, 3136/3125, 4375/4374 |
[⟨217 344 504 609 751 803]] | -0.111 | 0.2808 | 5.08 |
| 2.3.5.7.11.13.17 | 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125 |
[⟨217 344 504 609 751 803 887]] | -0.099 | 0.2616 | 4.73 |
| 2.3.5.7.11.13.17.19 | 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215 |
[⟨217 344 504 609 751 803 887 922]] | -0.119 | 0.2504 | 4.53 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 3\217 | 16.59 | 100/99 | Quincy |
| 1 | 5\217 | 27.65 | 64/63 | Arch |
| 1 | 10\217 | 55.30 | 16875/16384 | Escapade |
| 1 | 18\217 | 99.54 | 18/17 | Quintagar / quintoneum / quintasandra |
| 1 | 30\217 | 165.90 | 11/10 | Satin |
| 1 | 33\217 | 182.49 | 10/9 | Mitonic / mineral |
| 1 | 57\217 | 315.21 | 6/5 | Parakleismic / paralytic |
| 1 | 86\217 | 475.58 | 320/243 | Vulture |
| 1 | 90\217 | 497.70 | 4/3 | Gary / cotoneum |
| 1 | 101\217 | 558.53 | 112/81 | Condor |
| 7 | 94\217 (1\217) |
519.82 (5.53) |
27/20 |
Brahmagupta |
| 31 | 90\217 (1\217) |
497.70 (5.53) |
4/3 (243/242) |
Birds |