347edo: Difference between revisions
+regular temperament properties |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
The equal temperament [[tempering out|tempers out]] [[3136/3125]] | 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 === | === Prime harmonics === | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|550 -347}} | ! rowspan="2" | [[Comma list]] | ||
|{{mapping|347 550}} | ! 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 | ||
| 0.0197 | | 0.0197 | ||
| 0.57 | | 0.57 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|32 -7 -9}}, {{monzo|-22 30 -11}} | | {{monzo| 32 -7 -9 }}, {{monzo| -22 30 -11 }} | ||
|{{mapping|347 550 806}} | | {{mapping| 347 550 806 }} | ||
| -0.1576 | | -0.1576 | ||
| 0.1956 | | 0.1956 | ||
| 5.66 | | 5.66 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|3136/3125, 420175/419904, 5250987/5242880 | | 3136/3125, 420175/419904, 5250987/5242880 | ||
|{{mapping|347 550 806 974}} | | {{mapping| 347 550 806 974 }} | ||
| -0.0713 | | -0.0713 | ||
| 0.2259 | | 0.2259 | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br> | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|7\347 | | 7\347 | ||
|24.21 | | 24.21 | ||
|686/675 | | 686/675 | ||
|[[Sengagen]] | | [[Sengagen]] | ||
|- | |- | ||
|1 | | 1 | ||
|16\347 | | 16\347 | ||
|55.33 | | 55.33 | ||
|16875/16384 | | 16875/16384 | ||
|[[Escapade]] | | [[Escapade]] | ||
|- | |- | ||
|1 | | 1 | ||
|69\347 | | 69\347 | ||
|238.62 | | 238.62 | ||
|147/128 | | 147/128 | ||
|[[Tokko]] | | [[Tokko]] | ||
|- | |- | ||
|1 | | 1 | ||
|72\347 | | 72\347 | ||
|248.99 | | 248.99 | ||
|{{monzo|-26 18 -1}} | | {{monzo| -26 18 -1 }} | ||
|[[Monzismic]] | | [[Monzismic]] | ||
|- | |- | ||
|1 | | 1 | ||
|146\347 | | 146\347 | ||
|504.90 | | 504.90 | ||
|104976/78125 | | 104976/78125 | ||
|[[Countermeantone]] | | [[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]] | [[Category:Sengagen]] | ||