414edo: Difference between revisions

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Cleanup; +prime error table
+RTT table and rank-2 temperaments
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'''414edo''' is the [[EDO|equal division of the octave]] into 414 parts of 2.89855 [[cent]]s each.  
'''414edo''' is the [[EDO|equal division of the octave]] into 414 parts of 2.89855 [[cent]]s each.  


414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for 5. It is [[consistent]] to the [[17-odd-limit]], tempering out {{monzo| -36 11 8 }} (submajor comma) and {{monzo|1 -27 18}} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and 138427734375/137438953472 in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], 1275/1274, and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[Ragismic microtemperaments|hemiennealimmal]] and the 13-limit [[Ragismic microtemperaments|quatracot]].
414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for 5. It is [[consistent]] to the [[17-odd-limit]], tempering out {{monzo| -36 11 8 }} (submajor comma) and {{monzo|1 -27 18}} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], 1275/1274, and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[Ragismic microtemperaments|hemiennealimmal]] and the 13-limit [[Ragismic microtemperaments|quatracot]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|414}}
{{Harmonics in equal|414}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| {{monzo| -36 11 8 }}, {{monzo| 1 -27 18 }}
| [{{val| 414 656 961 }}]
| +0.2222
| 0.1575
| 5.43
|-
| 2.3.5.7
| 2401/2400, 4375/4374, {{monzo| -36 11 8 }}
| [{{val| 414 656 961 1162 }}]
| +0.2299
| 0.1371
| 4.73
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 4375/4374, 1366875/1362944
| [{{val| 414 656 961 1162 1432 }}]
| +0.2182
| 0.1248
| 4.30
|-
| 2.3.5.7.11.13
| 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400
| [{{val| 414 656 961 1162 1432 1532 }}]
| +0.1795
| 0.1431
| 4.94
|-
| 2.3.5.7.11.13.17
| 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197
| [{{val| 414 656 961 1162 1432 1532 1692 }}]
| +0.1751
| 0.1329
| 4.58
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 125\414
| 362.31
| 10125/8192
| [[Submajor]] (5-limit)
|-
| 2
| 61\414
| 176.81
| 195/176
| [[Quatracot]]
|-
| 9
| 109\414<br>(17\414)
| 315.94<br>(49.28)
| 6/5<br>(36/35)
| [[Ennealimmal]]
|-
| 18
| 86\414<br>(6\414)
| 249.28<br>(17.39)
| 231/200<br>(99/98)
| [[Hemiennealimmal]]
|-
| 18
| 164\414<br>(3\414)
| 475.36<br>(8.70)
| 1053/800<br>(1287/1280)
| [[Semihemiennealimmal]]
|}


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]

Revision as of 21:52, 26 January 2022

414edo is the equal division of the octave into 414 parts of 2.89855 cents each.

414edo is closely related to 207edo, but the patent vals differ on the mapping for 5. It is consistent to the 17-odd-limit, tempering out [-36 11 8⟩ (submajor comma) and [1 -27 18⟩ (ennealimma) in the 5-limit; 2401/2400, 4375/4374, and [-37 4 12 1⟩ in the 7-limit; 3025/3024, 9801/9800, 41503/41472, and 1265625/1261568 in the 11-limit; 625/624, 729/728, 1575/1573, 2200/2197, and 26411/26364 in the 13-limit; 833/832, 1089/1088, 1225/1224, 1275/1274, and 1701/1700 in the 17-limit. It supports the 11-limit hemiennealimmal and the 13-limit quatracot.

Prime harmonics

Approximation of prime harmonics in 414edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.51 -0.81 -0.71 -0.59 +0.05 -0.61 +1.04 +0.71 -0.59 -0.11
Relative (%) +0.0 -17.4 -27.8 -24.5 -20.5 +1.8 -21.0 +35.8 +24.5 -20.4 -3.7
Steps
(reduced) 		414
(0) 		656
(242) 		961
(133) 		1162
(334) 		1432
(190) 		1532
(290) 		1692
(36) 		1759
(103) 		1873
(217) 		2011
(355) 		2051
(395)

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [-36 11 8⟩, [1 -27 18⟩ [⟨414 656 961]] +0.2222 0.1575 5.43
2.3.5.7 2401/2400, 4375/4374, [-36 11 8⟩ [⟨414 656 961 1162]] +0.2299 0.1371 4.73
2.3.5.7.11 2401/2400, 3025/3024, 4375/4374, 1366875/1362944 [⟨414 656 961 1162 1432]] +0.2182 0.1248 4.30
2.3.5.7.11.13 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400 [⟨414 656 961 1162 1432 1532]] +0.1795 0.1431 4.94
2.3.5.7.11.13.17 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197 [⟨414 656 961 1162 1432 1532 1692]] +0.1751 0.1329 4.58

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 125\414 362.31 10125/8192 Submajor (5-limit)
2 61\414 176.81 195/176 Quatracot
9 109\414
(17\414) 		315.94
(49.28) 		6/5
(36/35) 		Ennealimmal
18 86\414
(6\414) 		249.28
(17.39) 		231/200
(99/98) 		Hemiennealimmal
18 164\414
(3\414) 		475.36
(8.70) 		1053/800
(1287/1280) 		Semihemiennealimmal
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