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{{Infobox ET}}
{{Infobox ET}}
'''[[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.
{{ED intro}}


34ED7 is consistent to the 11-[[integer-limit]], but not to the 12-integer-limit. In comparison, 12EDO is only consistent up to the 10-[[integer-limit]].
== Theory ==
34ed7 is related to [[12edo]], 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 [[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)}}


== Intervals ==
== Intervals ==
{| class="wikitable mw-collapsible"
{| class="wikitable center-1 right-2"
|+ Intervals of 34ed7
|+ 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
|}
|}
== Harmonics ==
{{Harmonics in equal|34|7|1|intervals=prime}}
{{Harmonics in equal|34|7|1|intervals=prime|collapsed=1|start=12}}


== Regular temperaments ==
== Regular temperaments ==
{{See also| Quintaleap family }}
{{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 (''[[Quintaleap family #Quintupole|quintupole]]'' temperament). This temperament is supported by [[12edo|12EDO]], [[109edo|109EDO]], and [[121edo|121EDO]] among others.
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.


== 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]]
* [[1ed18/17|AS18/17]] – relative [[AS|ambitonal sequence]]
 
 
{{todo|expand}}
[[Category:Ed7]]
[[Category:Edonoi]]

Revision as of 15:08, 15 January 2025

← 33ed7 34ed7 35ed7 →
Prime factorization 2 × 17
Step size 99.0831 ¢ 
Octave 12\34ed7 (1189 ¢) (→ 6\17ed7)
Twelfth 19\34ed7 (1882.58 ¢)
Consistency limit 11
Distinct consistency limit 6

34 equal divisions of the 7th harmonic (abbreviated 34ed7) is a nonoctave tuning system that divides the interval of 7/1 into 34 equal parts of about 99.1 ¢ each. Each step represents a frequency ratio of 71/34, or the 34th root of 7.

Theory

34ed7 is related to 12edo, 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-integer-limit, but not to the 12-integer-limit. In comparison, 12edo is only consistent up to the 10-integer-limit.

Harmonics

Approximation of harmonics in 34ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -11.0 -19.4 -22.0 -12.0 -30.4 +0.0 -33.0 -38.8 -23.0 +10.2 -41.4
Relative (%) -11.1 -19.6 -22.2 -12.1 -30.7 +0.0 -33.3 -39.1 -23.2 +10.3 -41.8
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(28) 		31
(31) 		34
(0) 		36
(2) 		38
(4) 		40
(6) 		42
(8) 		43
(9)
Approximation of harmonics in 34ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +18.2 -11.0 -31.4 -44.0 +49.2 +49.3 -44.3 -34.0 -19.4 -0.8 +21.3 +46.7
Relative (%) +18.4 -11.1 -31.7 -44.4 +49.7 +49.8 -44.7 -34.3 -19.6 -0.8 +21.5 +47.1
Steps
(reduced) 		45
(11) 		46
(12) 		47
(13) 		48
(14) 		50
(16) 		51
(17) 		51
(17) 		52
(18) 		53
(19) 		54
(20) 		55
(21) 		56
(22)

Intervals

Intervals of 34ed7
# Cents Approximate ratios
0 0.0 1/1
1 99.1 21/20
2 198.2 9/8
3 297.2 6/5
4 396.3 5/4
5 495.4 4/3
6 594.5 7/5
7 693.6 3/2
8 792.7 8/5
9 891.7 5/3
10 990.8 7/4
11 1089.9 15/8
12 1189.0 2/1
13 1288.1 21/10
14 1387.2 9/4
15 1486.2 7/3
16 1585.3 5/2
17 1684.4 8/3
18 1783.5 14/5
19 1882.6 3/1
20 1981.7 22/7
21 2080.7 10/3
22 2179.8 7/2
23 2278.9 15/4
24 2378.0 4/1
25 2477.1 21/5
26 2576.2 9/2
27 2675.2 14/3
28 2774.3 5/1
29 2873.4 16/3
30 2972.5 28/5
31 3071.6 6/1
32 3170.7 25/4
33 3269.7 20/3
34 3368.8 7/1

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, 109edo, and 121edo among others.

See also

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