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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 ===
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=== 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
|}
|}
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