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
{{Harmonics in equal
| steps = 677
| steps = 677
Line 13: Line 16:
| intervals = prime
| intervals = prime
}}
}}
=== Subsets and supersets ===
677edo 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]

Latest revision as of 11:54, 28 April 2025

← 676edo 677edo 678edo →
Prime factorization 677 (prime)
Step size 1.77253 ¢ 
Fifth 396\677 (701.92 ¢)
Semitones (A1:m2) 64:51 (113.4 ¢ : 90.4 ¢)
Consistency limit 11
Distinct consistency limit 11

677 equal divisions of the octave (abbreviated 677edo or 677ed2), also called 677-tone equal temperament (677tet) or 677 equal temperament (677et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 677 equal parts of about 1.77 ¢ each. Each step represents a frequency ratio of 21/677, or the 677th root of 2.

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 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

Approximation of prime harmonics in 677edo
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) +0.00 -0.03 +0.10 +0.75 -0.06 -0.35 -0.38 +0.27 -0.80
Relative (%) +0.0 -2.0 +5.5 +42.1 -3.5 -19.8 -21.2 +15.3 -45.1
Steps
(reduced) 		677
(0) 		1073
(396) 		1572
(218) 		1901
(547) 		2342
(311) 		2505
(474) 		2767
(59) 		2876
(168) 		3062
(354)

Subsets and supersets

677edo is the 123rd prime EDO.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1073 677⟩ [⟨677 1073]] +0.0110 0.0110 0.62
2.3.5 [38 -2 -15⟩, [-31 43 -16⟩ [⟨677 1073 1572]] −0.0066 0.0264 1.49
2.3.5.7 703125/702464, 589824/588245, 14348907/14336000 [⟨677 1073 1572 1901]] −0.0714 0.1145 6.46
2.3.5.7.11 3025/3024, 24057/24010, 131072/130977, 759375/758912 [⟨677 1073 1572 1901 2342]] −0.0535 0.1084 6.12

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Francium
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