348edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|348}} == Theory == 348et is consistent to the 7-limit, but the error of the harmonic 3 is quite large, commending itself as a 2.9.5.7.11.13 subgrou..."
 
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== Theory ==
== Theory ==
348et is consistent to the 7-limit, but the error of the harmonic 3 is quite large, commending itself as a 2.9.5.7.11.13 subgroup temperament. Using the patent val, it tempers out 156250000/155649627 and [[2401/2400]] in the 7-limit. It provides the [[optimal patent val]] for [[subneutral]].
348et is [[consistent]] to the [[7-odd-limit]], but the error of the [[harmonic]] [[3/1|3]] is quite large, commending itself as a 2.9.5.7.11.13 [[subgroup]] temperament.  
 
Using the [[patent val]], it tempers out [[2401/2400]], [[15625/15552]], [[390625/388962]] and 156250000/155649627 and in the 7-limit. It [[support]]s [[quadritikleismic]] and [[subneutral]].


=== Odd harmonics ===
=== Odd harmonics ===
Line 9: Line 11:


=== Subsets and supersets ===
=== Subsets and supersets ===
348 factors into 2<sup>2</sup> × 3 × 29, with subset edos {{EDOs|2, 3, 4, 6, 12, 29, 58, 87, 116, and 174}}. [[696edo]], which doubles it, gives a good correction to the harmonic 3.
Since 348 factors into 2<sup>2</sup> × 3 × 29, 348edo has subset edos {{EDOs| 2, 3, 4, 6, 12, 29, 58, 87, 116, and 174 }}. [[696edo]], which doubles it, gives a good correction to the harmonic 3.


== 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]] (%)
|-
|-
|2.9
| 2.9
|{{monzo|-1103 348}}
| {{monzo| -1103 348 }}
|{{mapping|348 1103}}
| {{mapping| 348 1103 }}
| +0.0728
| +0.0728
| 0.0728
| 0.0728
| 2.11
| 2.11
|-
|-
|2.9.5
| 2.9.5
|32805/32768, {{monzo|52 40 -77}}
| 32805/32768, {{monzo| 7 52 -74 }}
|{{mapping|348 1103 808}}
| {{mapping| 348 1103 808 }}
| +0.0639
| +0.0639
| 0.0608
| 0.0608
|1.76
| 1.76
|-
|-
|2.9.5.7
| 2.9.5.7
|32805/32768, 250047/250000, 247669456896/247165842875
| 32805/32768, 250047/250000, {{monzo| 7 9 -2 -11 }}
|{{mapping|348 1103 808 977}}
| {{mapping| 348 1103 808 977 }}
| +0.0355
| +0.0355
| 0.0721
| 0.0721
| 2.09
| 2.09
|-
|-
|2.9.5.7.11
| 2.9.5.7.11
|9801/9800, 32805/32768, 46656/46585, 250047/250000
| 9801/9800, 32805/32768, 46656/46585, 151263/151250
|{{mapping|348 1103 808 977 1204}}
| {{mapping| 348 1103 808 977 1204 }}
| +0.0049
| +0.0049
| 0.0889
| 0.0889
| 2.58
| 2.58
|-
|-
|2.9.5.7.11.13
| 2.9.5.7.11.13
|729/728, 1575/1573, 4096/4095, 67392/67375, 250047/250000
| 729/728, 1575/1573, 2200/2197, 32805/32768, 31250/31213
|{{mapping|348 1103 808 977 1204 1288}}
| {{mapping| 348 1103 808 977 1204 1288 }}
| -0.0343
| -0.0343
| 0.1194
| 0.1194
| 3.46
| 3.46
|}
|}

Revision as of 14:46, 5 December 2023

← 347edo 348edo 349edo →
Prime factorization 22 × 3 × 29
Step size 3.44828 ¢ 
Fifth 204\348 (703.448 ¢) (→ 17\29)
Semitones (A1:m2) 36:24 (124.1 ¢ : 82.76 ¢)
Dual sharp fifth 204\348 (703.448 ¢) (→ 17\29)
Dual flat fifth 203\348 (700 ¢) (→ 7\12)
Dual major 2nd 59\348 (203.448 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

348et is consistent to the 7-odd-limit, but the error of the harmonic 3 is quite large, commending itself as a 2.9.5.7.11.13 subgroup temperament.

Using the patent val, it tempers out 2401/2400, 15625/15552, 390625/388962 and 156250000/155649627 and in the 7-limit. It supports quadritikleismic and subneutral.

Odd harmonics

Approximation of odd harmonics in 348edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.49 -0.11 +0.14 -0.46 +0.41 +0.85 +1.39 -1.51 -0.96 +1.63 -0.69
Relative (%) +43.3 -3.1 +4.0 -13.4 +11.8 +24.7 +40.2 -43.7 -27.9 +47.4 -20.0
Steps
(reduced) 		552
(204) 		808
(112) 		977
(281) 		1103
(59) 		1204
(160) 		1288
(244) 		1360
(316) 		1422
(30) 		1478
(86) 		1529
(137) 		1574
(182)

Subsets and supersets

Since 348 factors into 22 × 3 × 29, 348edo has subset edos 2, 3, 4, 6, 12, 29, 58, 87, 116, and 174. 696edo, which doubles it, gives a good correction to the harmonic 3.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.9 [-1103 348⟩ [⟨348 1103]] +0.0728 0.0728 2.11
2.9.5 32805/32768, [7 52 -74⟩ [⟨348 1103 808]] +0.0639 0.0608 1.76
2.9.5.7 32805/32768, 250047/250000, [7 9 -2 -11⟩ [⟨348 1103 808 977]] +0.0355 0.0721 2.09
2.9.5.7.11 9801/9800, 32805/32768, 46656/46585, 151263/151250 [⟨348 1103 808 977 1204]] +0.0049 0.0889 2.58
2.9.5.7.11.13 729/728, 1575/1573, 2200/2197, 32805/32768, 31250/31213 [⟨348 1103 808 977 1204 1288]] -0.0343 0.1194 3.46
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