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→Approximations to Just Intonation: preserves the structure of 34edo > contains a copy of 34edo
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merge mos box with box of rank-2 temperaments
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* 34et has a lower relative error than any previous ETs in the 5-limit. The next ET that does better in this subgroup is 53.  
* 34et has a lower relative error than any previous ETs in the 5-limit. The next ET that does better in this subgroup is 53.  
* 34et is most prominent in the 2.3.5.13.17 and 2.3.5.11.13.17 subgroups. The next ET that does better in these subgroups is 217 and 87, respectively.
* 34et is most prominent in the 2.3.5.13.17 and 2.3.5.11.13.17 subgroups. The next ET that does better in these subgroups is 217 and 87, respectively.
== MOS scales ==
34edo supports the following MOSes:
{| class="wikitable"
|+ MOSes sorted by period and generator
|-
! Periods<br>per octave
! Generator
! Cents
! MOSes
|-
| 1
| 1\34
| 35.294
|
|-
| "
| 3\34
| 105.88
| [[11L 1s]]<br/> [[11L 12s]]
|-
| "
| 5\34
| 176.471
| [[6L 1s]]<br/> [[7L 6s]] <br/> [[7L 13s]] <br/> 7L 20s
|-
| "
| 7\34
| 247.059
| [[5L 4s]] <br/> [[5L 9s]] <br/> [[5L 14s]] <br/> [[5L 19s]] <br/>Pathological 5L 24s
|-
| "
| 9\34
| 317.647
| [[4L 3s]]<br/> [[4L 7s]]<br/> [[4L 11s]]<br/> [[15L 4s]]
|-
| "
| 11\34
| 388.235
| [[3L 7s]]<br/> [[3L 10s]]<br/> [[3L 13s]]<br/> [[3L 16s]]<br/> [[3L 19s]] <br/>[[3L 22s]]<br/> Pathological [[3L 25s]] <br/> Pathological 3L 28s
|-
| "
| 13\34
| 458.824
| [[3L 2s]]<br/> [[5L 3s]]<br/> [[8L 5s]]<br/> [[13L 8s]]
|-
| "
| 15\34
| 529.412
| [[2L 3s]]<br/> [[2L 5s]]<br/> [[7L 2s]]<br/> [[9L 7s]] <br/> 9L 16s
|-
| 2
| 2\34
| 70.588
| [[16L 2s]]
|-
| "
| 3\34
| 105.882
| [[2L 6s]]<br/>[[2L 8s]]<br/> [[10L 2s]]<br/> [[12L 10s]]
|-
| "
| 4\34
| 141.176
| [[2L 6s]]<br/> [[8L 2s]]<br/> [[8L 10s]] <br/> 8L 16s
|-
| "
| 5\34
| 176.471
| [[6L 2s]]<br/> [[6L 8s]]<br/> [[14L 6s]]
|-
| "
| 6\34
| 211.765
| [[4L 2s]] <br/> [[6L 4s]] <br/> [[6L 10s]] <br/> [[6L 16s]] <br/> Pathological 6L 22s
|-
| "
| 7\34
| 247.059
| [[4L 2s]] <br/> [[4L 6s]] <br/> [[10L 4s]] <br/> [[10L 14s]]
|-
| "
| 8\34
| 282.353
| [[2L 2s]] <br/> [[4L 2s]] <br/> [[4L 6s]] <br/> [[4L 10s]] <br/> [[4L 14s]] <br/> 4L 18s <br/> 4L 22s <br/> Pathological 4L 26s
|}


== Tuning by ear ==
== Tuning by ear ==
Line 690: Line 604:
== 34edo as a regular temperament ==
== 34edo as a regular temperament ==
In the 5-limit, 34edo supports [[hanson]], [[srutal]], [[tetracot]], [[würschmidt]] and [[vishnu]] temperaments. It does less well in the [[7-limit]], with two mappings possible for [[7/4]]: a flat one from the [[patent val]], and a sharp one from the 34d val. By way of the patent val 34 supports [[keemun]] temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit [[pajara]] temperament. In the [[11-limit]], 34de supports 11-limit [[pajaric]], and in fact is quite close to the [[POTE tuning]]; it adds [[4375/4374]] to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports [[semaphore]] on the 2.3.7 subgroup.
In the 5-limit, 34edo supports [[hanson]], [[srutal]], [[tetracot]], [[würschmidt]] and [[vishnu]] temperaments. It does less well in the [[7-limit]], with two mappings possible for [[7/4]]: a flat one from the [[patent val]], and a sharp one from the 34d val. By way of the patent val 34 supports [[keemun]] temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit [[pajara]] temperament. In the [[11-limit]], 34de supports 11-limit [[pajaric]], and in fact is quite close to the [[POTE tuning]]; it adds [[4375/4374]] to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports [[semaphore]] on the 2.3.7 subgroup.
=== Rank two temperaments ===
=== MOSes and rank two temperaments ===
* [[List of 34edo rank two temperaments by badness]]
* [[List of 34edo rank two temperaments by badness]]
* [[List of edo-distinct 34d rank two temperaments]]
* [[List of edo-distinct 34d rank two temperaments]]


34edo supports the following rank-2 temperaments:
34edo supports the following MOSes:
{| class="wikitable"
{| class="wikitable"
|+ Temperaments sorted by generator
|+ MOSes sorted by period and generator
|-
|-
! Periods<br>per octave
! Periods<br>per octave
! Generator
! Generator
! Cents
! Cents
! MOSes
! Temperaments
! Temperaments
|-
|-
Line 706: Line 621:
| 1\34
| 1\34
| 35.294
| 35.294
|
|  
|  
|-
|-
| "
| "
| 3\34
| 3\34
| 105.882
| 105.88
| [[11L 1s]]<br/> [[11L 12s]]
|  
|  
|-
|-
Line 716: Line 633:
| 5\34
| 5\34
| 176.471
| 176.471
| [[6L 1s]]<br/> [[7L 6s]] <br/> [[7L 13s]] <br/> 7L 20s
| [[Tetracot]]/[[Bunya]]/[[Monkey]]
| [[Tetracot]]/[[Bunya]]/[[Monkey]]
|-
|-
Line 721: Line 639:
| 7\34
| 7\34
| 247.059
| 247.059
| [[5L 4s]] <br/> [[5L 9s]] <br/> [[5L 14s]] <br/> [[5L 19s]] <br/>Pathological 5L 24s
| [[Immunity]]
| [[Immunity]]
|-
|-
Line 726: Line 645:
| 9\34
| 9\34
| 317.647
| 317.647
| [[4L 3s]]<br/> [[4L 7s]]<br/> [[4L 11s]]<br/> [[15L 4s]]
| [[Hanson]]/[[Keemun]]
| [[Hanson]]/[[Keemun]]
|-
|-
Line 731: Line 651:
| 11\34
| 11\34
| 388.235
| 388.235
| [[3L 7s]]<br/> [[3L 10s]]<br/> [[3L 13s]]<br/> [[3L 16s]]<br/> [[3L 19s]] <br/>[[3L 22s]]<br/> Pathological [[3L 25s]] <br/> Pathological 3L 28s
| [[Wuerschmidt]]/[[Worschmidt]]
| [[Wuerschmidt]]/[[Worschmidt]]
|-
|-
Line 736: Line 657:
| 13\34
| 13\34
| 458.824
| 458.824
| [[Chromatic_pairs#Petrtri|Petrtri]]
| [[3L 2s]]<br/> [[5L 3s]]<br/> [[8L 5s]]<br/> [[13L 8s]]
| [[Petrtri]]
|-
|-
| "
| "
| 15\34
| 15\34
| 529.412
| 529.412
| [[2L 3s]]<br/> [[2L 5s]]<br/> [[7L 2s]]<br/> [[9L 7s]] <br/> 9L 16s
| [[Chromatic_pairs#Mabila|Mabila]]
| [[Chromatic_pairs#Mabila|Mabila]]
|-
|-
| 2
| 2
| 1\34
| 35.294
|
|-
| "
| 2\34
| 2\34
| 70.588
| 70.588
| [[16L 2s]]
| [[Vishnu]]
| [[Vishnu]]
|-
|-
Line 756: Line 675:
| 3\34
| 3\34
| 105.882
| 105.882
| [[2L 6s]]<br/>[[2L 8s]]<br/> [[10L 2s]]<br/> [[12L 10s]]
| [[Srutal]]/[[Pajara]]/[[Diaschismic]]
| [[Srutal]]/[[Pajara]]/[[Diaschismic]]
|-
|-
Line 761: Line 681:
| 4\34
| 4\34
| 141.176
| 141.176
| [[2L 6s]]<br/> [[8L 2s]]<br/> [[8L 10s]] <br/> 8L 16s
| [[Fifive]]
| [[Fifive]]
|-
|-
Line 766: Line 687:
| 5\34
| 5\34
| 176.471
| 176.471
| [[6L 2s]]<br/> [[6L 8s]]<br/> [[14L 6s]]
|  
|  
|-
|-
Line 771: Line 693:
| 6\34
| 6\34
| 211.765
| 211.765
| [[4L 2s]] <br/> [[6L 4s]] <br/> [[6L 10s]] <br/> [[6L 16s]] <br/> Pathological 6L 22s
|  
|  
|-
|-
Line 776: Line 699:
| 7\34
| 7\34
| 247.059
| 247.059
| [[4L 2s]] <br/> [[4L 6s]] <br/> [[10L 4s]] <br/> [[10L 14s]]
|  
|  
|-
|-
Line 781: Line 705:
| 8\34
| 8\34
| 282.353
| 282.353
|  
| [[2L 2s]] <br/> [[4L 2s]] <br/> [[4L 6s]] <br/> [[4L 10s]] <br/> [[4L 14s]] <br/> 4L 18s <br/> 4L 22s <br/> Pathological 4L 26s
|-
| 17
| 1\34
| 35.294
|  
|  
|}
|}


=== Commas ===
=== Commas ===
Retrieved from "https://en.xen.wiki/w/34edo"