214edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|214}}
{{EDO intro|214}}
==Theory==
 
214et is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which tempers out the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; [[6144/6125]] (porwell), 16875/16807 (mirkwai), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]] and 1375/1372 in the 11-limit; 1188/1183, [[351/350]] and [[847/845]] in the 13-limit. It can be viewed as a 2.13/5 subgroup temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.
== Theory ==
===Prime harmonics===
214edo is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which [[tempering out|tempers out]] the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; 6144/6125 ([[porwell comma]]), 16875/16807 ([[mirkwai comma]]), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]] and [[1375/1372]] in the 11-limit; [[351/350]], [[847/845]], and [[1188/1183]] in the 13-limit. It can be viewed as a 2.13/5 [[subgroup]] temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.
 
=== Prime harmonics ===
{{Harmonics in equal|214}}
{{Harmonics in equal|214}}
===Subsets and supersets===
 
=== Subsets and supersets ===
214 factors into 2 × 107, with [[2edo]] and [[107edo]] as its subset edos.
214 factors into 2 × 107, with [[2edo]] and [[107edo]] as its subset edos.
==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|-339 214}}
| {{monzo| -339 214 }}
|{{val|214 339}}
| {{mapping| 214 339 }}
| +0.3219
| +0.3219
| 0.3220
| 0.3220
| 5.74
| 5.74
|-
|-
|2.3.5
| 2.3.5
|78732/78125, {{monzo|-49 28 2}}
| 78732/78125, {{monzo| -49 28 2 }}
|{{val|214 339 497}}
| {{mapping| 214 339 497 }}
| +0.1281
| +0.1281
| 0.3797
| 0.3797
| 6.77
| 6.77
|-
|-
|2.3.5.7
| 2.3.5.7
|6144/6125, 16875/16807, 78732/78125
| 6144/6125, 16875/16807, 78732/78125
|{{val|214 339 497 601}}
| {{mapping| 214 339 497 601 }}
| -0.0169
| -0.0169
| 0.4137
| 0.4137
| 7.38
| 7.38
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|540/539, 1375/1372, 3025/3024, 5632/5625
| 540/539, 1375/1372, 3025/3024, 5632/5625
|{{val|214 339 497 601 740}}
| {{mapping| 214 339 497 601 740 }}
| +0.0897
| +0.0897
| 0.4270
| 0.4270
| 7.61
| 7.61
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|540/539, 847/845, 1001/1000, 1375/1372, 5632/5625
| 540/539, 847/845, 1001/1000, 1375/1372, 5632/5625
|{{val|214 339 497 601 740 792}}
| {{mapping| 214 339 497 601 740 792 }}
| +0.0480
| +0.0480
| 0.4008
| 0.4008
| 7.15
| 7.15
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|351/350, 715/714, 936/935, 1275/1274, 5544/5525, 5850/5831
| 351/350, 715/714, 936/935, 1275/1274, 5544/5525, 5850/5831
|{{val|214 339 497 601 740 792 875}}
| {{mapping| 214 339 497 601 740 792 875 }}
| -0.0144
| -0.0144
| 0.4012
| 0.4012
| 7.15
| 7.15
|}
|}
=== 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
|+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
|69\214
| 69\214
|386.92
| 386.92
|5/4
| 5/4
|[[Grendel]]
| [[Grendel]]
|-
|-
|1
| 1
|79\214
| 79\214
|442.99
| 442.99
|9/7
| 9/7
|[[Sensi]]
| [[Sensi]]
|-
|-
|1
| 1
|105\214
| 105\214
|588.79
| 588.79
|7/5
| 7/5
|[[Untriton]] / [[aufo]]
| [[Aufo]]
|-
|-
|2
| 2
|28\214
| 28\214
|157.01
| 157.01
|35/32
| 35/32
|[[Bison]]
| [[Bison]]
|-
|-
|2
| 2
|29\214
| 29\214
|162.62
| 162.62
|1125/1024
| 1125/1024
|[[Kwazy]]
| [[Kwazy]]
|}
|}
 
<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:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 08:31, 3 April 2024

← 213edo 214edo 215edo →
Prime factorization 2 × 107
Step size 5.60748 ¢ 
Fifth 125\214 (700.935 ¢)
Semitones (A1:m2) 19:17 (106.5 ¢ : 95.33 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

214edo is (uniquely) consistent through the 7-odd-limit. The patent val for 214edo is ⟨214 339 497 601 740 792], which tempers out the following commas: 78732/78125 (sensipent comma) and [-51 19 9⟩ (untriton comma) in the 5-limit; 6144/6125 (porwell comma), 16875/16807 (mirkwai comma), 321489/320000 (varunisma), and [22 -1 -10 1⟩ (quasiorwellisma) in the 7-limit; 540/539 and 1375/1372 in the 11-limit; 351/350, 847/845, and 1188/1183 in the 13-limit. It can be viewed as a 2.13/5 subgroup temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.

Prime harmonics

Approximation of prime harmonics in 214edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.02 +0.60 +1.27 -1.79 +0.59 +1.59 -0.32 -0.24 +2.20 -1.11
Relative (%) +0.0 -18.2 +10.7 +22.6 -31.8 +10.6 +28.3 -5.6 -4.2 +39.2 -19.8
Steps
(reduced) 		214
(0) 		339
(125) 		497
(69) 		601
(173) 		740
(98) 		792
(150) 		875
(19) 		909
(53) 		968
(112) 		1040
(184) 		1060
(204)

Subsets and supersets

214 factors into 2 × 107, with 2edo and 107edo as its subset edos.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-339 214⟩ [⟨214 339]] +0.3219 0.3220 5.74
2.3.5 78732/78125, [-49 28 2⟩ [⟨214 339 497]] +0.1281 0.3797 6.77
2.3.5.7 6144/6125, 16875/16807, 78732/78125 [⟨214 339 497 601]] -0.0169 0.4137 7.38
2.3.5.7.11 540/539, 1375/1372, 3025/3024, 5632/5625 [⟨214 339 497 601 740]] +0.0897 0.4270 7.61
2.3.5.7.11.13 540/539, 847/845, 1001/1000, 1375/1372, 5632/5625 [⟨214 339 497 601 740 792]] +0.0480 0.4008 7.15
2.3.5.7.11.13.17 351/350, 715/714, 936/935, 1275/1274, 5544/5525, 5850/5831 [⟨214 339 497 601 740 792 875]] -0.0144 0.4012 7.15

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 69\214 386.92 5/4 Grendel
1 79\214 442.99 9/7 Sensi
1 105\214 588.79 7/5 Aufo
2 28\214 157.01 35/32 Bison
2 29\214 162.62 1125/1024 Kwazy

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

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