677edo: Difference between revisions

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-{{Infobox ET|677edo}}
+{{Infobox ET}}
+{{ED intro}}
-'''677 equal divisions of the octave''' (677edo) is a tuning system with notes that are about 1.77¢ apart. While it does well as a 2.3.5.11 tuning system, it's notable for its high accuracy among EDOs of about its size and lower with the first several [[wikipedia:Metallic_mean|metallic ratios]]. Among those, it tunes [[acoustic phi]] (the golden ratio) and the acoustic copper ratio each with less than 1% relative error. The first nine metallic ratios are all tuned within 20% of an edostep.
+== Theory ==
+While it does well as a 2.3.5.11 tuning system, it's notable for its high accuracy among EDOs of about its size and lower with the first several [[wikipedia:Metallic_mean|metallic ratios]]. Among those, it tunes [[acoustic phi]] (the golden ratio) and the acoustic copper ratio each with less than 1% relative error. The first nine metallic ratios are all tuned within 20% of an edostep.
+=== Prime harmonics ===
 {{Harmonics in equal
 | steps = 677
@@ Line 13: / Line 16: @@
 | intervals = prime
 }}
+=== Subsets and supersets ===
+edo is the 123rd [[prime EDO]].
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo|-1073 677}}
+| {{mapping|677 1073}}
+| +0.0110
+| 0.0110
+| 0.62
+|-
+| 2.3.5
+| {{monzo|38 -2 -15}}, {{monzo|-31 43 -16}}
+| {{mapping|677 1073 1572}}
+| −0.0066
+| 0.0264
+| 1.49
+|-
+| 2.3.5.7
+| 703125/702464, 589824/588245, 14348907/14336000
+| {{mapping|677 1073 1572 1901}}
+| −0.0714
+| 0.1145
+| 6.46
+|-
+| 2.3.5.7.11
+| 3025/3024, 24057/24010, 131072/130977, 759375/758912
+| {{mapping|677 1073 1572 1901 2342}}
+| −0.0535
+| 0.1084
+| 6.12
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 109\677
+| 193.21
+| 262144/234375
+| [[Luna]]
+|-
+| 1
+| 125\677
+| 221.57
+| 8388608/7381125
+| [[Fortune]]
+|-
+| 1
+| 281\677
+| 498.08
+| 4/3
+| [[Counterschismic]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Music ==
+; [[Francium]]
+* "From the Ears of the Eel" from ''Scoop'' (2024) – [https://open.spotify.com/track/163kPfkX9BJ6l7jXbJDVOG Spotify] | [https://francium223.bandcamp.com/track/from-the-ears-of-the-eel Bandcamp] | [https://www.youtube.com/watch?v=WHY3EKLi7Zk YouTube]
+* "The Scallop Disco Accident" from ''The Scallop Disco Accident'' (2025) – [https://open.spotify.com/track/3C4sVxcVJWIZqgbpgxoQo1 Spotify] | [https://francium223.bandcamp.com/track/the-scallop-disco-accident Bandcamp] | [https://www.youtube.com/watch?v=V12N5jC_UfM YouTube]
+* "Make Gecko Expansion" from ''I Want To'' (2025) – [https://open.spotify.com/track/5dbEP5zQQrrO00f3XZiELV Spotify] | [https://francium223.bandcamp.com/track/make-gecko-expansion Bandcamp] | [https://www.youtube.com/watch?v=D55Y0-p_mWU YouTube]
+* "Eggs Of An Oven" from ''Eggs'' (2025) – [https://open.spotify.com/track/0pe7FES4PhYa58UBaQScAS Spotify] | [https://francium223.bandcamp.com/track/eggs-of-an-oven Bandcamp] | [https://www.youtube.com/watch?v=sejOxKdHd8E YouTube]