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'''[[Ed7|Division of the 7th harmonic]] into 34 equal parts''' (34ED7) is related to [[12edo|12 EDO]], but with the 7/1 rather than the 2/1 being just. The octave is about 11.0026 cents compressed and the step size is about 99.0831 cents. It is consistent to the [[11-odd-limit|11-integer-limit]], but not to the 12-integer-limit. In comparison, 12EDO is only consistent up to the [[9-odd-limit|10-integer-limit]].
{{Infobox ET}}
{{ED intro}}


{| class="wikitable"
== Theory ==
34ed7 is related to [[12edo]], but with the 7/1 rather than the 2/1 being just. This compresses the octave by 11.0026{{c}}, a small but significant deviation. It is consistent to the [[integer limit|11-integer-limit]], but not to the 12-integer-limit. In comparison, 12edo is only consistent up to the 10-integer-limit.
 
=== Harmonics ===
{{Harmonics in equal|34|7|1|intervals=integer}}
{{Harmonics in equal|34|7|1|intervals=integer|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 34ed7 (continued)}}
 
=== Subsets and supersets ===
Since 34 factors into 2 × 17, 34ed7 contains [[2ed7]] and [[17ed7]] as subsets.
 
== Intervals ==
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Intervals of 34ed7
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Approximate ratios
! | comments
|-
|-
| | 0
| 0
| | 0.0000
| 0.0
| | '''exact [[1/1]]'''
| [[1/1]]
| |
|-
|-
| | 1
| 1
| | 99.0831
| 99.1
| | [[18/17]]
| [[21/20]]
| |
|-
|-
| | 2
| 2
| | 198.1662
| 198.2
| | 28/25
| [[9/8]]
| |
|-
|-
| | 3
| 3
| | 297.2493
| 297.2
| | [[19/16]]
| [[6/5]]
| |
|-
|-
| | 4
| 4
| | 396.3325
| 396.3
| | 49/39, 34/27
| [[5/4]]
| | pseudo-[[5/4]]
|-
|-
| | 5
| 5
| | 495.4156
| 495.4
| | [[4/3]]
| [[4/3]]
| |
|-
|-
| | 6
| 6
| | 594.4987
| 594.5
| | [[24/17]]
| [[7/5]]
| |
|-
|-
| | 7
| 7
| | 693.5818
| 693.6
| | 136/91
| [[3/2]]
| | pseudo-[[3/2]]
|-
|-
| | 8
| 8
| | 792.6649
| 792.7
| | [[30/19]], [[128/81]]
| [[8/5]]
| |
|-
|-
| | 9
| 9
| | 891.7480
| 891.7
| | 77/46
| [[5/3]]
| | pseudo-[[5/3]]
|-
|-
| | 10
| 10
| | 990.8311
| 990.8
| | 85/48, 39/22
| [[7/4]]
| |
|-
|-
| | 11
| 11
| | 1089.9143
| 1089.9
| | [[15/8]]
| [[15/8]]
| |
|-
|-
| | 12
| 12
| | 1188.9974
| 1189.0
| | 143/72, 175/88
| [[2/1]]
| | pseudo-[[octave]]
|-
|-
| | 13
| 13
| | 1288.0805
| 1288.1
| | [[21/20|21/10]], [[20/19|40/19]]
| [[21/10]]
| |
|-
|-
| | 14
| 14
| | 1387.1636
| 1387.2
| | [[49/44|49/22]]
| [[9/4]]
| |
|-
|-
| | 15
| 15
| | 1486.2467
| 1486.2
| | 33/14
| [[7/3]]
| |
|-
|-
| | 16
| 16
| | 1585.3298
| 1585.3
| | [[5/2]]
| [[5/2]]
| |
|-
|-
| | 17
| 17
| | 1684.4130
| 1684.4
| | 119/45, 45/17
| [[8/3]]
| | pseudo-[[8/3]]
|-
|-
| | 18
| 18
| | 1783.4961
| 1783.5
| | [[14/5]]
| [[14/5]]
| |
|-
|-
| | 19
| 19
| | 1882.5792
| 1882.6
| | 95/32, 98/33
| [[3/1]]
| | pseudo-[[3/1]]
|-
|-
| | 20
| 20
| | 1981.6623
| 1981.7
| | [[11/7|22/7]]
| [[22/7]]
| |
|-
|-
| | 21
| 21
| | 2080.7454
| 2080.7
| | 133/40, [[10/3]]
| [[10/3]]
| |
|-
|-
| | 22
| 22
| | 2179.8285
| 2179.8
| | 88/25
| [[7/2]]
| |
|-
|-
| | 23
| 23
| | 2278.9116
| 2278.9
| | [[28/15|56/15]]
| [[15/4]]
| |
|-
|-
| | 24
| 24
| | 2377.9948
| 2378.0
| | 154/39, [[160/81|320/81]], 336/85
| [[4/1]]
| | pseudo-[[4/1]]
|-
|-
| | 25
| 25
| | 2477.0779
| 2477.1
| | 46/11
| [[21/5]]
| |
|-
|-
| | 26
| 26
| | 2576.1610
| 2576.2
| | 133/30
| [[9/2]]
| |
|-
|-
| | 27
| 27
| | 2675.2441
| 2675.2
| | 169/36
| [[14/3]]
| |
|-
|-
| | 28
| 28
| | 2774.3272
| 2774.3
| | 119/24
| [[5/1]]
| | pseudo-[[5/1]]
|-
|-
| | 29
| 29
| | 2873.4103
| 2873.4
| | [[21/16|21/4]]
| [[16/3]]
| | pseudo-[[16/3]]
|-
|-
| | 30
| 30
| | 2972.4934
| 2972.5
| | 39/7
| [[28/5]]
| |
|-
|-
| | 31
| 31
| | 3071.5766
| 3071.6
| | [[28/19|112/19]]
| [[6/1]]
| | pseudo-[[6/1]]
|-
|-
| | 32
| 32
| | 3170.6597
| 3170.7
| | [[25/16|25/4]]
| [[25/4]]
| |
|-
|-
| | 33
| 33
| | 3269.7428
| 3269.7
| | 119/18
| [[20/3]]
| |
|-
|-
| | 34
| 34
| | 3368.8259
| 3368.8
| | '''exact [[7/1]]'''
| [[7/1]]
| | [[7/4|harmonic seventh]] plus two octaves
|}
|}


== Regular temperaments ==
== Regular temperaments ==
{{See also|16ed5/2 #Regular temperaments}}
{{See also| Quintaleap family }}
 
34ED7 can also be thought of as a [[generator]] of the 11-limit temperament which tempers out 896/891, 1375/1372, and 4375/4356, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintupole'' temperament). This temperament is supported by [[12edo|12EDO]], [[109edo|109EDO]], and [[121edo|121EDO]] among others.
 


; <font style="font-size: 1.15em">[[High badness temperaments #Quintaleap|Quintaleap]] (12&amp;121)</font>
34ed7 can also be thought of as a [[generator]] of the 11-limit temperament which tempers out 896/891, 1375/1372, and 4375/4356, which is a [[cluster temperament]] with 12 clusters of notes in an octave ([[quintupole]] temperament). This temperament is supported by [[12edo]], [[109edo]], and [[121edo]] among others.
'''5-limit'''<br>
Comma: {{monzo|37 -16 -5}} = 137438953472/134521003125<br>
Mapping: [{{val|1 2 1}}, {{val|0 -5 16}}]<br>
POTE generator: ~135/128 = 99.267<br>
Vals: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc<br>
Badness: 0.444506<br><br>
; <font style="font-size: 1.15em">[[Octagar temperaments #Quintupole|Quintupole]] (12&amp;121)</font>
'''7-limit'''<br>
Comma list: 4000/3969, 458752/455625<br>
Mapping: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]<br>
POTE generator: ~135/128 = 99.175<br>
Vals: 12, 97, 109, 121<br>
Badness: 0.111620<br><br>
'''11-limit'''<br>
Comma list: 896/891, 1375/1372, 4375/4356<br>
Mapping: [{{val|1 2 1 0 -1}}, {{val|0 -5 16 34 54}}]<br>
POTE generator: ~132/125 = 99.156<br>
Vals: 12, 109, 121, 351bde, 472bdee<br>
Badness: 0.056501<br><br>
'''13-limit'''<br>
Comma list: 352/351, 364/363, 625/624, 2704/2695<br>
Mapping: [{{val|1 2 1 0 -1 -2}}, {{val|0 -5 16 34 54 69}}]<br>
POTE generator: ~55/52 = 99.165<br>
Vals: 12f, 109, 121<br>
Badness: 0.038431<br><br>
'''17-limit'''<br>
Comma list: 256/255, 352/351, 364/363, 375/374, 442/441<br>
Mapping: [{{val|1 2 1 0 -1 -2 5}}, {{val|0 -5 16 34 54 69 -11}}]<br>
POTE generator: ~18/17 = 99.172<br>
Vals: 12f, 109, 121<br>
Badness: 0.028721<br><br>
'''19-limit'''<br>
Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374<br>
Mapping: [{{val|1 2 1 0 -1 -2 5 4}}, {{val|0 -5 16 34 54 69 -11 3}}]<br>
POTE generator: ~18/17 = 99.164<br>
Vals: 12f, 109, 121<br>
Badness: 0.023818<br><br>


== See also ==
== See also ==
* [[12edo|12EDO]] - relative EDO
* [[12edo]] – relative edo
* [[19ed3|19ED3]] - relative ED3
* [[19edt]] – relative edt
* [[28ed5|28ED5]] - relative ED5
* [[28ed5]] – relative ed5
* [[31ed6|31ED6]] - relative ED6
* [[31ed6]] – relative ed6
* [[40ed10|40ED10]] - relative ED10
* [[40ed10]] – relative ed10
* [[42ed11|42ED11]] - relative ED11
* [[42ed11]] – relative ed11
* [[18/17s equal temperament|AS18/17]] - relative [[AS|ambitonal sequence]]
* [[76ed80]] – close to the zeta-optimized tuning for 12edo
* [[1ed18/17|AS18/17]] – relative [[AS|ambitonal sequence]]


[[Category:Ed7]]
[[Category:12edo]]
[[Category:Edonoi]]
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