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'''34edo''' divides the octave into 34 equal steps of approximately 35.3 [[cent]]s.  
'''34edo''' divides the octave into 34 equal steps of approximately 35.3 [[cent]]s.  


<br>
== Introduction ==
== Introduction ==
{{Primes in edo|34|columns=11}}
{{Primes in edo|34|columns=11}}
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Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and Downs Notation #Chord names in other EDOs]].
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and Downs Notation #Chord names in other EDOs]].


== Approximations to Just Intonation ==
== JI approximation ==
Like [[17edo]], 34edo contains good approximations of just intervals involving 13, 11, and 3 – specifically, 13/8, 13/12, 13/11, 13/9, 11/9 and their inversions – while failing to closely approximate ratios of 7. 34edo adds ratios of 5 into the mix – including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions – as well as 17 – including 17/16, 18/17, 17/12, 17/11, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the "syntonic comma" of 81/80, from 21.5 cents to 35.3 cents), it is suitable for quasi-5-limit JI but is not a [[meantone]] system. While no number of fifths (frequently ratios of ~3:2) land on major or minor thirds, an even number of major or minor thirds will be the same pitch as a pitch somewhere in the circle of seventeen fifths.
Like [[17edo]], 34edo contains good approximations of just intervals involving 13, 11, and 3 – specifically, 13/8, 13/12, 13/11, 13/9, 11/9 and their inversions – while failing to closely approximate ratios of 7. 34edo adds ratios of 5 into the mix – including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions – as well as 17 – including 17/16, 18/17, 17/12, 17/11, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the "syntonic comma" of 81/80, from 21.5 cents to 35.3 cents), it is suitable for quasi-5-limit JI but is not a [[meantone]] system. While no number of fifths (frequently ratios of ~3:2) land on major or minor thirds, an even number of major or minor thirds will be the same pitch as a pitch somewhere in the circle of seventeen fifths.


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|}
|}


=== Temperament measures ===
== Tuning by ear ==
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 34et.  
In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5 cents, and using a pure 5/4 which is less than 2 cents off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5 cents, or a relative error of less than 10%.


Note: the 34d val is used for lower error.  
== 34edo and logarithmic phi ==
{| class="wikitable center-all"
As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the logarithmic phi – 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo|-6 2 6 0 0 -13}}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].)
! colspan="2" |
 
! 3-limit
== Regular temperament properties ==
! 5-limit
{| class="wikitable center-4 center-5 center-6"
! 7-limit
! rowspan="2" | Subgroup
! 11-limit
! rowspan="2" | [[Comma list]]
! 13-limit
! rowspan="2" | [[Mapping]]
! 17-limit
! rowspan="2" | Optimal<br>8ve stretch (¢)
! 2.3.5.13.17
! colspan="2" | Tuning error
! 2.3.5.11.13.17
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3.5
| -1.24
| 2048/2025, 15625/15552
| [{{val| 34 54 79 }}]
| -1.10
| -1.10
| 1.03
| 2.92
|-
| 2.3.5.7
| 50/49, 64/63, 4375/4374
| [{{val| 34 54 79 96 }}] (34d)
| -2.56
| -2.56
| 2.66
| 7.57
|-
| 2.3.5.7.11
| 50/49, 64/63, 99/98, 243/242
| [{{val| 34 54 79 96 118 }}] (34d)
| -2.82
| -2.82
| 2.44
| 6.93
|-
| 2.3.5.7.11.13
| 50/49, 64/63, 78/77, 99/98, 144/143
| [{{val| 34 54 79 96 118 126 }}] (34d)
| -2.64
| -2.64
| 2.26
| 6.42
|-
| 2.3.5.7.11.13.17
| 50/49, 64/63, 78/77, 85/84, 99/98, 144/143
| [{{val| 34 54 79 96 118 126 139 }}] (34d)
| -2.30
| -2.30
| -1.06
| -1.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 1.24
| 1.03
| 2.66
| 2.44
| 2.26
| 2.26
| 2.26
| 0.94
| 1.35
|-
! [[TE simple badness|relative]] (%)
| 3.51
| 2.92
| 7.57
| 6.93
| 6.42
| 6.41
| 6.41
| 2.65
| 3.83
|}
|}


* 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.  
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.  
* 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.


== Tuning by ear ==
=== Rank-2 temperaments ===
In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5 cents, and using a pure 5/4 which is less than 2 cents off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5 cents, or a relative error of less than 10%.
 
== 34edo and logarithmic phi ==
As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the logarithmic phi – 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo|-6 2 6 0 0 -13}}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].)
 
== 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.
=== 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 MOSes and rank-2 temperaments:
34et supports the following MOSes and rank-2 temperaments:
{| class="wikitable"
{| class="wikitable"
|+ MOSes sorted by period and generator
|+ Rank-2 temperaments by period and generator
|-
|-
! Periods<br>per octave
! Periods<br>per octave
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| 35.294
| 35.294
|  
|  
|  
| [[Gammic]]
|-
|-
| "
| "
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| 176.471
| 176.471
| [[6L 1s]]<br/> [[7L 6s]] <br/> [[7L 13s]] <br/> 7L 20s
| [[6L 1s]]<br/> [[7L 6s]] <br/> [[7L 13s]] <br/> 7L 20s
| [[Tetracot]]/[[Bunya]]/[[Monkey]]
| [[Tetracot]] / [[bunya]] (34d) / [[modus]] (34d) / [[monkey]] (34) / [[wollemia]] (34)
|-
|-
| "
| "
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| 247.059
| 247.059
| [[5L 4s]] <br/> [[5L 9s]] <br/> [[5L 14s]] <br/> [[5L 19s]] <br/>Pathological 5L 24s
| [[5L 4s]] <br/> [[5L 9s]] <br/> [[5L 14s]] <br/> [[5L 19s]] <br/>Pathological 5L 24s
| [[Immunity]]
| [[Immunity]] (34) / [[immunized]] (34d)
|-
|-
| "
| "
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| 317.647
| 317.647
| [[4L 3s]]<br/> [[4L 7s]]<br/> [[4L 11s]]<br/> [[15L 4s]]
| [[4L 3s]]<br/> [[4L 7s]]<br/> [[4L 11s]]<br/> [[15L 4s]]
| [[Hanson]]/[[Keemun]]
| [[Hanson]] / [[keemun]] (34) / [[catalan]] (34d) / [[catakleismic]] (34d)
|-
|-
| "
| "
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| 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
| [[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]]
| [[Würschmidt]] (34d) / [[worschmidt]] (34)
|-
|-
| "
| "
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| 529.412
| 529.412
| [[2L 3s]]<br/> [[2L 5s]]<br/> [[7L 2s]]<br/> [[9L 7s]] <br/> 9L 16s
| [[2L 3s]]<br/> [[2L 5s]]<br/> [[7L 2s]]<br/> [[9L 7s]] <br/> 9L 16s
| [[Chromatic_pairs#Mabila|Mabila]]
| [[Mabila]]
|-
|-
| 2
| 2
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| 105.882
| 105.882
| [[2L 6s]]<br/>[[2L 8s]]<br/> [[10L 2s]]<br/> [[12L 10s]]
| [[2L 6s]]<br/>[[2L 8s]]<br/> [[10L 2s]]<br/> [[12L 10s]]
| [[Srutal]]/[[Pajara]]/[[Diaschismic]]
| [[Srutal]] (34d) / [[pajara]] (34d) / [[diaschismic]] (34)
|-
|-
| "
| "
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| 141.176
| 141.176
| [[2L 6s]]<br/> [[8L 2s]]<br/> [[8L 10s]] <br/> 8L 16s
| [[2L 6s]]<br/> [[8L 2s]]<br/> [[8L 10s]] <br/> 8L 16s
| [[Fifive]]
| [[Fifive]] / [[crepuscular]] (34d) / [[fifives]] (34)
|-
|-
| "
| "
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=== Commas ===
=== Commas ===
34-EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val|34 54 79 95 118 126}}.)
34-EDO [[tempers out]] the following [[comma]]s. This assumes the [[patent val]] {{val| 34 54 79 95 118 126 }}.


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
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* https://myspace.com/microstick -- somehow broken (if you scroll to right, you'll find the songs, playing them, you can't hear anything)
* https://myspace.com/microstick -- somehow broken (if you scroll to right, you'll find the songs, playing them, you can't hear anything)


[[Category:34edo]]
[[Category:34edo| ]] <!-- main article -->
[[Category:34et]]
[[Category:diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:keemun]]
[[Category:Diaschismic]]
[[Category:kleismic]]
[[Category:Keemun]]
[[Category:listen]]
[[Category:Kleismic]]
[[Category:pajara]]
[[Category:Pajara]]
[[Category:selenium]]
[[Category:Selenium]]
[[Category:Oneirotonic]]
[[Category:Oneirotonic]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/34edo"