347edo: Difference between revisions

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== Theory ==
== Theory ==
The equal temperament [[tempering out|tempers out]] [[3136/3125]] and 420175/419904 in the 7-limit, and provides an excellent tuning for [[sengagen]], the 99 & 248 temperament tempering out both, and the planar [[hemimean]] temperament tempering 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 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"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|550 -347}}
| {{monzo| 550 -347 }}
|{{mapping|347 550}}
| {{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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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)*
! Generator*
! Cents<br>(reduced)*
! Cents*
! Associated<br>Ratio*
! 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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


[[Category:Hemimean]]
[[Category:Hemimean]]
[[Category:Sengagen]]
[[Category:Sengagen]]

Revision as of 12:37, 10 December 2023

← 346edo 347edo 348edo →
Prime factorization 347 (prime)
Step size 3.45821 ¢ 
Fifth 203\347 (702.017 ¢)
Semitones (A1:m2) 33:26 (114.1 ¢ : 89.91 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

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

Approximation of prime harmonics in 347edo
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

Table of rank-2 temperaments by generator
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 it is distinct

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