214edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|214}}
{{ED intro}}


== Theory ==
== Theory ==
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.
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]], 1375/1372, [[5632/5625]], in the 11-limit; [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]], and [[4096/4095]] in the 13-limit. It can be viewed as a 2.3.5.13.19.23 [[subgroup]] temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
214 factors into 2 × 107, with [[2edo]] and [[107edo]] as its subset edos.
Since 214 factors into {{factorisation|214}}, 214edo contains [[2edo]] and [[107edo]] as its subsets.


== 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]]
! 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]] (%)
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| 6144/6125, 16875/16807, 78732/78125
| 6144/6125, 16875/16807, 78732/78125
| {{mapping| 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, 5632/5625, 72171/71680
| {{mapping| 214 339 497 601 740 }}
| {{mapping| 214 339 497 601 740 }}
| +0.0897
| +0.0897
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|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 540/539, 847/845, 1001/1000, 1375/1372, 5632/5625
| 351/350, 540/539, 847/845, 1375/1372, 4096/4095
| {{mapping| 214 339 497 601 740 792 }}
| {{mapping| 214 339 497 601 740 792 }}
| +0.0480
| +0.0480
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|-
|-
| 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, 540/539, 715/714, 847/845, 936/935, 4096/4095
| {{mapping| 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
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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
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
| 1
| 27\214
| 151.40
| 12/11
| [[Browser]]
|-
|-
| 1
| 1
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| 79\214
| 79\214
| 442.99
| 442.99
| 9/7
| 162/125
| [[Sensi]]
| [[Sensipent]]
|-
|-
| 1
| 1
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| 157.01
| 157.01
| 35/32
| 35/32
| [[Bison]]
| [[Bison]] (214e)
|-
|-
| 2
| 2
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| [[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
<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:Browser]]

Latest revision as of 13:31, 13 March 2026

← 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

214 equal divisions of the octave (abbreviated 214edo or 214ed2), also called 214-tone equal temperament (214tet) or 214 equal temperament (214et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 214 equal parts of about 5.61 ¢ each. Each step represents a frequency ratio of 21/214, or the 214th root of 2.

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, 1375/1372, 5632/5625, in the 11-limit; 351/350, 847/845, 1001/1000, 1188/1183, 1573/1568, and 4096/4095 in the 13-limit. It can be viewed as a 2.3.5.13.19.23 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

Since 214 factors into 2 × 107, 214edo contains 2edo and 107edo as its subsets.

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, 5632/5625, 72171/71680 [⟨214 339 497 601 740]] +0.0897 0.4270 7.61
2.3.5.7.11.13 351/350, 540/539, 847/845, 1375/1372, 4096/4095 [⟨214 339 497 601 740 792]] +0.0480 0.4008 7.15
2.3.5.7.11.13.17 351/350, 540/539, 715/714, 847/845, 936/935, 4096/4095 [⟨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 27\214 151.40 12/11 Browser
1 69\214 386.92 5/4 Grendel
1 79\214 442.99 162/125 Sensipent
1 105\214 588.79 7/5 Aufo
2 28\214 157.01 35/32 Bison (214e)
2 29\214 162.62 1125/1024 Kwazy

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

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