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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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|348}}
{{ED intro}}


== 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]].
 
In 348edo, the prime harmonics up to 13 map the same way as in [[87edo]], except the 7th harmonic, which is corrected.


=== Odd harmonics ===
=== Odd harmonics ===
Line 9: Line 13:


=== 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" |[[Comma list|Comma List]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[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]] (%)
|-
|-
|2.9
| 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
|}
|}

Latest revision as of 14:41, 15 December 2025

← 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

348 equal divisions of the octave (abbreviated 348edo or 348ed2), also called 348-tone equal temperament (348tet) or 348 equal temperament (348et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 348 equal parts of about 3.45 ¢ each. Each step represents a frequency ratio of 21/348, or the 348th root of 2.

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.

In 348edo, the prime harmonics up to 13 map the same way as in 87edo, except the 7th harmonic, which is corrected.

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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