677edo: Difference between revisions
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{{Infobox ET| | {{Infobox ET}} | ||
{{EDO intro|677}} | |||
== Theory == | |||
'''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. | '''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. | ||
=== Prime harmonics === | |||
{{Harmonics in equal | {{Harmonics in equal | ||
| steps = 677 | | steps = 677 | ||
| Line 13: | Line 15: | ||
| 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|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" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator<br>(reduced)* | |||
! Cents<br>(reduced)* | |||
! 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]] | |||
|} | |||
Revision as of 18:34, 22 November 2023
| ← 676edo | 677edo | 678edo → |
Theory
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 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
| 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
| Periods per 8ve |
Generator (reduced)* |
Cents (reduced)* |
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 |