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]] | |||
|} | |||