317edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|317}} | {{EDO intro|317}} | ||
== Theory == | == Theory == | ||
317et tempers out 589824/588245, [[16875/16807]], [[65625/65536]] and 49009212/48828125 in the 7-limit; 1835008/1830125, 14700/14641, 2097152/2096325, [[4000/3993]], 2734375/2725888, 1953125/1951488, 172032/171875, [[441/440]], 5767168/5764801, 825000/823543, 537109375/536870912, 134775333/134217728, 160083/160000, 16808715/16777216, 539055/537824 and 3294225/3294172 in the 11-limit. | 317et is only consistent to the [[5-odd-limit]] and the [[harmonic]] 3 is about halfway its steps. Using the patent val, it tempers out 589824/588245, [[16875/16807]], [[65625/65536]] and 49009212/48828125 in the 7-limit; 1835008/1830125, 14700/14641, 2097152/2096325, [[4000/3993]], 2734375/2725888, 1953125/1951488, 172032/171875, [[441/440]], 5767168/5764801, 825000/823543, 537109375/536870912, 134775333/134217728, 160083/160000, 16808715/16777216, 539055/537824 and 3294225/3294172 in the 11-limit. | ||
===Odd harmonics=== | ===Odd harmonics=== | ||
{{Harmonics in equal|317}} | {{Harmonics in equal|317}} | ||
===Subsets and supersets=== | ===Subsets and supersets=== | ||
317edo is the 66th [[prime edo]]. [[634edo]], which doubles it, gives a good correction to the harmonic 3. | 317edo is the 66th [[prime edo]]. [[634edo]], which doubles it, gives a good correction to the harmonic 3. | ||
==Regular temperament properties== | ==Regular temperament properties== | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
Revision as of 10:08, 31 December 2023
| ← 316edo | 317edo | 318edo → |
Theory
317et is only consistent to the 5-odd-limit and the harmonic 3 is about halfway its steps. Using the patent val, it tempers out 589824/588245, 16875/16807, 65625/65536 and 49009212/48828125 in the 7-limit; 1835008/1830125, 14700/14641, 2097152/2096325, 4000/3993, 2734375/2725888, 1953125/1951488, 172032/171875, 441/440, 5767168/5764801, 825000/823543, 537109375/536870912, 134775333/134217728, 160083/160000, 16808715/16777216, 539055/537824 and 3294225/3294172 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.64 | -0.19 | +0.26 | +0.51 | +1.36 | -0.15 | -1.83 | +1.04 | +1.54 | -1.38 | +0.12 |
| Relative (%) | -43.3 | -5.1 | +6.8 | +13.4 | +36.0 | -3.9 | -48.4 | +27.4 | +40.7 | -36.5 | +3.1 | |
| Steps (reduced) |
502 (185) |
736 (102) |
890 (256) |
1005 (54) |
1097 (146) |
1173 (222) |
1238 (287) |
1296 (28) |
1347 (79) |
1392 (124) |
1434 (166) | |
Subsets and supersets
317edo is the 66th prime edo. 634edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [1005 -317⟩ | [⟨317 1005]] | -0.0799 | 0.0799 | 2.11 |
| 2.9.5 | [-53 5 16⟩, [33 -17 9⟩ | [⟨317 1005 736]] | -0.0254 | 0.1009 | 2.67 |
| 2.9.5.7 | 420175/419904, 703125/702464, 33554432/33480783 | [⟨317 1005 736 890]] | -0.0422 | 0.0921 | 2.43 |
| 2.9.5.7.11 | 6250/6237, 12005/11979, 46656/46585, 151263/151250 | [⟨317 1005 736 890 1097]] | -0.1126 | 0.1631 | 4.31 |
| 2.9.5.7.11.13 | 1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728 | [⟨317 1005 736 890 1097 1173]] | -0.0871 | 0.1594 | 4.21 |
| 2.9.5.7.11.13.17 | 936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957 | [⟨317 1005 736 890 1097 1173 1296]] | -0.1109 | 0.1587 | 4.19 |