317edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 15: | Line 15: | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 27: | Line 28: | ||
| {{monzo| 1005 -317 }} | | {{monzo| 1005 -317 }} | ||
| {{mapping| 317 1005 }} | | {{mapping| 317 1005 }} | ||
| | | −0.0799 | ||
| 0.0799 | | 0.0799 | ||
| 2.11 | | 2.11 | ||
| Line 34: | Line 35: | ||
| {{monzo| -53 5 16 }}, {{monzo| 33 -17 9 }} | | {{monzo| -53 5 16 }}, {{monzo| 33 -17 9 }} | ||
| {{mapping| 317 1005 736 }} | | {{mapping| 317 1005 736 }} | ||
| | | −0.0254 | ||
| 0.1009 | | 0.1009 | ||
| 2.67 | | 2.67 | ||
| Line 41: | Line 42: | ||
| 420175/419904, 703125/702464, 33554432/33480783 | | 420175/419904, 703125/702464, 33554432/33480783 | ||
| {{mapping| 317 1005 736 890 }} | | {{mapping| 317 1005 736 890 }} | ||
| | | −0.0422 | ||
| 0.0921 | | 0.0921 | ||
| 2.43 | | 2.43 | ||
| Line 48: | Line 49: | ||
| 6250/6237, 12005/11979, 46656/46585, 151263/151250 | | 6250/6237, 12005/11979, 46656/46585, 151263/151250 | ||
| {{mapping| 317 1005 736 890 1097 }} | | {{mapping| 317 1005 736 890 1097 }} | ||
| | | −0.1126 | ||
| 0.1631 | | 0.1631 | ||
| 4.31 | | 4.31 | ||
| Line 55: | Line 56: | ||
| 1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728 | | 1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728 | ||
| {{mapping| 317 1005 736 890 1097 1173 }} | | {{mapping| 317 1005 736 890 1097 1173 }} | ||
| | | −0.0871 | ||
| 0.1594 | | 0.1594 | ||
| 4.21 | | 4.21 | ||
| Line 62: | Line 63: | ||
| 936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957 | | 936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957 | ||
| {{mapping| 317 1005 736 890 1097 1173 1296 }} | | {{mapping| 317 1005 736 890 1097 1173 1296 }} | ||
| | | −0.1109 | ||
| 0.1587 | | 0.1587 | ||
| 4.19 | | 4.19 | ||
| Line 70: | Line 71: | ||
; [[Francium]] | ; [[Francium]] | ||
* "Narrator Of Myths" from ''Mysteries'' (2023) – [https://open.spotify.com/track/6AYSSw1kVKDDzWVYMyxafv Spotify] | [https://francium223.bandcamp.com/track/narrator-of-myths Bandcamp] | [https://www.youtube.com/watch?v=fNmEjgVonHk YouTube] | * "Narrator Of Myths" from ''Mysteries'' (2023) – [https://open.spotify.com/track/6AYSSw1kVKDDzWVYMyxafv Spotify] | [https://francium223.bandcamp.com/track/narrator-of-myths Bandcamp] | [https://www.youtube.com/watch?v=fNmEjgVonHk YouTube] | ||
* "Strange Shade Of Red" from ''You Are A...'' (2024) – [https://open.spotify.com/track/23kTYEBoGRKNLhVSaIxBpC Spotify] | [https://francium223.bandcamp.com/track/strange-shade-of-red Bandcamp] | [https://www.youtube.com/watch?v=G6h1qN__F6I YouTube] | |||
[[Category:Listen]] | [[Category:Listen]] | ||
Latest revision as of 12:37, 21 February 2025
| ← 316edo | 317edo | 318edo → |
317 equal divisions of the octave (abbreviated 317edo or 317ed2), also called 317-tone equal temperament (317tet) or 317 equal temperament (317et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 317 equal parts of about 3.79 ¢ each. Each step represents a frequency ratio of 21/317, or the 317th root of 2.
Theory
317edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway its steps. To start with, it can be considered in the 2.9.5.7 subgroup, in which it is strong, tempering out 420175/419904, 703125/702464, and 33554432/33480783.
Using the patent val nonetheless, the equal temperament tempers out 78732/78125 (sensipent comma in the 5-limit; 16875/16807, 65625/65536, 589824/588245, and 49009212/48828125 in the 7-limit. In the 11-limit, the 317e val tempers out 540/539, 1375/1372, and 3025/3024, whereas the patent val tempers out 441/440, 4000/3993, and 14700/14641.
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 |