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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 === | ||
| Line 13: | Line 15: | ||
== 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 | ||
| Line 46: | Line 49: | ||
=== 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 | |||
<nowiki>* | |||
[[Category:Hemimean]] | [[Category:Hemimean]] | ||
[[Category:Sengagen]] | [[Category:Sengagen]] | ||
Latest revision as of 13:33, 13 March 2026
| ← 346edo | 347edo | 348edo → |
347 equal divisions of the octave (abbreviated 347edo or 347ed2), also called 347-tone equal temperament (347tet) or 347 equal temperament (347et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 347 equal parts of about 3.46 ¢ each. Each step represents a frequency ratio of 21/347, or the 347th root of 2.
Theory
The equal temperament tempers out [32 -7 -9⟩ (escapade comma) and [54 -37 2⟩ (monzisma), 3136/3125, 420175/419904, and 5250987/5242880 in the 7-limit. It provides an excellent tuning for sengagen, the 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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.06 | +1.01 | -0.53 | -1.46 | -0.18 | -1.21 | -0.11 | +1.12 | +0.97 | -0.37 |
| Relative (%) | +0.0 | +1.8 | +29.1 | -15.2 | -42.3 | -5.3 | -35.0 | -3.1 | +32.4 | +28.1 | -10.6 | |
| Steps (reduced) |
347 (0) |
550 (203) |
806 (112) |
974 (280) |
1200 (159) |
1284 (243) |
1418 (30) |
1474 (86) |
1570 (182) |
1686 (298) |
1719 (331) | |
Subsets and supersets
347edo is the 69th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [550 -347⟩ | [⟨347 550]] | -0.0197 | 0.0197 | 0.57 |
| 2.3.5 | [32 -7 -9⟩, [-22 30 -11⟩ | [⟨347 550 806]] | -0.1576 | 0.1956 | 5.66 |
| 2.3.5.7 | 3136/3125, 420175/419904, 5250987/5242880 | [⟨347 550 806 974]] | -0.0713 | 0.2259 | 6.53 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated 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 | [-26 18 -1⟩ | Monzismic |
| 1 | 146\347 | 504.90 | 104976/78125 | Countermeantone |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct