374edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|374}}
{{ED intro}}


== Theory ==
== Theory ==
374et is only consistent to the [[3-odd-limit]]. Omitting the harmonic 5, it is consistent to the [[31-odd-limit]]. Using the patent val, it tempers out 40500000/40353607, 184528125/184473632, [[5120/5103]] and [[2100875/2097152]] in the 7-limit; 1073741824/1071794405, 161280/161051, 820125/819896, 2097152/2096325, 12005/11979, 2621440/2614689, 496125/495616, 1296000/1294139, 1265625/1261568, [[200704/200475]], 5767168/5764801, 1375/1372, 184549376/184528125, 1479016/1476225, 275653125/275365888, [[41503/41472]], 1362944/1361367, 166375/165888, 3294225/3294172 and 322102/321489 in the 11-limit. It [[support]]s [[quintakwai]] and [[quartemka]].
374et is in[[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[5/1|5]] is about halfway between its steps. Omitting the harmonic 5, it is consistent to the [[31-odd-limit]].  
 
Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[5120/5103]], 1071875/1062882, 1500625/1492992, [[2100875/2097152]], and 9765625/9680832 in the 7-limit; 1375/1372, 4375/4356, 12005/11979, and [[41503/41472]] in the 11-limit. It [[support]]s [[quintakwai]] and [[quartemka]].


=== Prime harmonics ===
=== Prime harmonics ===
Line 9: Line 11:


=== Subsets and supersets ===
=== Subsets and supersets ===
374 factors into 2 × 11 × 17, with subset edos {{EDOs|2, 11, 17, 22, 34, and 187}}. [[748edo]], which doubles it, gives a good correction to the harmonic 5.
Since 374 factors into 2 × 11 × 17, 374edo has subset edos {{EDOs| 2, 11, 17, 22, 34, and 187 }}. [[748edo]], which doubles it, gives a good correction to the harmonic 5, but its approximation of harmonic 3 has drifted too far to render it inconsistent in the [[9-odd-limit]].  


== 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" |[[Comma list|Comma List]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | [[Mapping]]
! colspan="2" |Tuning Error
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|593 -374}}
| {{monzo| 593 -374 }}
|{{mapping|374 593}}
| {{mapping| 374 593 }}
| -0.2268
| −0.2268
| 0.2267
| 0.2267
| 7.07
| 7.07
|-
|-
|2.3.7
| 2.3.7
|{{monzo|4 -22 11}}, {{monzo|51 -18 -8}}
| {{monzo| 4 -22 11 }}, {{monzo| 51 -18 -8 }}
|{{mapping|374 593 1050}}
| {{mapping| 374 593 1050 }}
| -0.1699
| −0.1699
| 0.2018
| 0.2018
| 6.29
| 6.29
|-
|-
|2.3.7.11
| 2.3.7.11
|41503/41472, 1362944/1361367, 70493667328/70027449129
| 41503/41472, 1362944/1361367, 70493667328/70027449129
|{{mapping|374 593 1050 1294}}
| {{mapping| 374 593 1050 1294 }}
| -0.1675
| −0.1675
| 0.1748
| 0.1748
| 5.45
| 5.45
|-
|-
|2.3.7.11.13
| 2.3.7.11.13
|41503/41472, 20449/20412, 652288/649539, 10648/10647
| 10648/10647, 20449/20412, 41503/41472, 652288/649539
|{{mapping|374 593 1050 1294 1384}}
| {{mapping| 374 593 1050 1294 138 4}}
| -0.1401
| −0.1401
| 0.1656
| 0.1656
| 5.16
| 5.16
|-
|-
|2.3.7.11.13.17
| 2.3.7.11.13.17
|22528/22491, 2058/2057, 34816/34749, 8624/8619, 8281/8262
| 2058/2057, 8281/8262, 8624/8619, 22528/22491, 34816/34749
|{{mapping|374 593 1050 1294 1384 1529}}
| {{mapping| 374 593 1050 1294 1384 1529 }}
| -0.1546
| −0.1546
| 0.1546
| 0.1546
| 4.82
| 4.82
|-
|-
|2.3.7.11.13.17.19
| 2.3.7.11.13.17.19
|1729/1728, 2912/2907, 22528/22491, 2058/2057, 5929/5928, 34816/34749
| 1729/1728, 2058/2057, 2912/2907, 5929/5928, 22528/22491, 34816/34749
|{{mapping|374 593 1050 1294 1384 1529 1589}}
| {{mapping| 374 593 1050 1294 1384 1529 1589 }}
| -0.1622
| −0.1622
| 0.1444
| 0.1444
| 4.50
| 4.50
|}
|}

Latest revision as of 12:14, 21 February 2025

← 373edo 374edo 375edo →
Prime factorization 2 × 11 × 17
Step size 3.20856 ¢ 
Fifth 219\374 (702.674 ¢)
Semitones (A1:m2) 37:27 (118.7 ¢ : 86.63 ¢)
Consistency limit 3
Distinct consistency limit 3

374 equal divisions of the octave (abbreviated 374edo or 374ed2), also called 374-tone equal temperament (374tet) or 374 equal temperament (374et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 374 equal parts of about 3.21 ¢ each. Each step represents a frequency ratio of 21/374, or the 374th root of 2.

Theory

374et is inconsistent to the 5-odd-limit since harmonic 5 is about halfway between its steps. Omitting the harmonic 5, it is consistent to the 31-odd-limit.

Using the patent val, the equal temperament tempers out 5120/5103, 1071875/1062882, 1500625/1492992, 2100875/2097152, and 9765625/9680832 in the 7-limit; 1375/1372, 4375/4356, 12005/11979, and 41503/41472 in the 11-limit. It supports quintakwai and quartemka.

Prime harmonics

Approximation of prime harmonics in 374edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.72 -1.29 +0.16 +0.55 +0.11 +0.93 +0.88 +0.60 +0.37 +0.42
Relative (%) +0.0 +22.4 -40.1 +4.9 +17.3 +3.6 +28.9 +27.5 +18.8 +11.5 +13.1
Steps
(reduced) 		374
(0) 		593
(219) 		868
(120) 		1050
(302) 		1294
(172) 		1384
(262) 		1529
(33) 		1589
(93) 		1692
(196) 		1817
(321) 		1853
(357)

Subsets and supersets

Since 374 factors into 2 × 11 × 17, 374edo has subset edos 2, 11, 17, 22, 34, and 187. 748edo, which doubles it, gives a good correction to the harmonic 5, but its approximation of harmonic 3 has drifted too far to render it inconsistent in the 9-odd-limit.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [593 -374⟩ [⟨374 593]] −0.2268 0.2267 7.07
2.3.7 [4 -22 11⟩, [51 -18 -8⟩ [⟨374 593 1050]] −0.1699 0.2018 6.29
2.3.7.11 41503/41472, 1362944/1361367, 70493667328/70027449129 [⟨374 593 1050 1294]] −0.1675 0.1748 5.45
2.3.7.11.13 10648/10647, 20449/20412, 41503/41472, 652288/649539 [⟨374 593 1050 1294 138 4]] −0.1401 0.1656 5.16
2.3.7.11.13.17 2058/2057, 8281/8262, 8624/8619, 22528/22491, 34816/34749 [⟨374 593 1050 1294 1384 1529]] −0.1546 0.1546 4.82
2.3.7.11.13.17.19 1729/1728, 2058/2057, 2912/2907, 5929/5928, 22528/22491, 34816/34749 [⟨374 593 1050 1294 1384 1529 1589]] −0.1622 0.1444 4.50
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