347edo: Difference between revisions
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{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | |||
The equal temperament [[tempering out|tempers out]] {{monzo| 32 -7 -9 }} ([[escapade comma]]) and {{monzo| 54 -37 2 }} ([[monzisma]]), [[3136/3125]], 420175/419904, and 5250987/5242880 in the 7-limit. It provides an excellent tuning for [[sengagen]], the {{nowrap|99 & 248}} temperament tempering out both 3136/3125 and 420175/419904, and the rank-3 [[hemimean]] temperament tempering out 3136/3125. | |||
Extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. It makes more sense as a 2.3.5.7.13 [[subgroup]] temperament, where it tempers out [[676/675]] and [[4096/4095]], or as a 2.3.5.7.13.19 subgroup temperament, where it tempers out [[1521/1520]] and [[1729/1728]]. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|347}} | |||
=== Subsets and supersets === | |||
347edo is the 69th [[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| 550 -347 }} | |||
| {{mapping| 347 550 }} | |||
| -0.0197 | |||
| 0.0197 | |||
| 0.57 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 32 -7 -9 }}, {{monzo| -22 30 -11 }} | |||
| {{mapping| 347 550 806 }} | |||
| -0.1576 | |||
| 0.1956 | |||
| 5.66 | |||
|- | |||
| 2.3.5.7 | |||
| 3136/3125, 420175/419904, 5250987/5242880 | |||
| {{mapping| 347 550 806 974 }} | |||
| -0.0713 | |||
| 0.2259 | |||
| 6.53 | |||
|} | |||
=== 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 | |||
| 7\347 | |||
| 24.21 | |||
| 686/675 | |||
| [[Sengagen]] | |||
|- | |||
| 1 | |||
| 16\347 | |||
| 55.33 | |||
| 16875/16384 | |||
| [[Escapade]] | |||
|- | |||
| 1 | |||
| 69\347 | |||
| 238.62 | |||
| 147/128 | |||
| [[Tokko]] | |||
|- | |||
| 1 | |||
| 72\347 | |||
| 248.99 | |||
| {{monzo| -26 18 -1 }} | |||
| [[Monzismic]] | |||
|- | |||
| 1 | |||
| 146\347 | |||
| 504.90 | |||
| 104976/78125 | |||
| [[Countermeantone]] | |||
|} | |||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
[[Category:Hemimean]] | [[Category:Hemimean]] | ||
[[Category:Sengagen]] | |||