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This is the simplest edt which intelligibly doubles the Anti-BP harmonic chain. It is a schismatic temperament, albeit a really strange one due to being a multiple of 4edt and being index-2 in BP.
{{Infobox ET}}
'''28edt''' divides the [[3/1|third harmonic]] into 28 [[equal]] parts of 67.9 [[cent]]s each. It is the simplest [[edt]] which intelligibly doubles the Anti-BP harmonic chain{{clarify}}.  


It is very close to every third step of 53edo.
== Theory ==


{| class="wikitable"
From a no-twos perspective, 28edt has a nearly-pure 5th harmonic (which tritave-reduces to [[5/3]]), with 41/28 being a continued fraction [[convergent]] to log<sub>3</sub>(5). It poorly approximates harmonic 7, but has an acceptable approximation of harmonic 11, so it makes sense to consider it as a [[3.5.11 subgroup]] system. In the 3.5.11 subgroup, it tempers out the commas [[6655/6561]] and [[4-cent comma|1953125/1948617]]. The former means it supports [[deneb]] temperament, which equates 3 flat [[11/9]]s to [[5/3]], and the latter means it supports [[delta centauri]] temperament, which equates 9 near-just [[5/3]]s tritave-reduced to [[11/9]] (although delta centauri is a [[microtemperament]] and larger EDTs like [[71edt]] offer much better tunings of it). [[56edt]] rectifies the approximation of the 7th harmonic.
 
28edt has a very accurate [[8/1|8th harmonic]], and as such, one step is very close to every third step of [[53edo]], and the tritave-reduced 8th harmonic [[8/3]] and tritave-reduced 8th subharmonic [[9/8]] are equally well-approximated. It tempers out the [[schisma]] in the 8.3.5 subgroup, albeit a really strange tuning due to being a multiple of 4edt and being index-2 in BP. It can also be used for the 3.8.5.11 extension of deneb temperament equating +5 generators to 8/3.
 
Similar to how [[14edt]] can be seen as [[9edo]] with significantly [[stretched and compressed tuning|stretched]] [[octave]]s (~23{{c}}), 28edt can be seen as [[18edo]] with the same stretched octave. Note that using the stretched octave will map 8/1 to an inconsistent approximation rather than the aforementioned accurate 8/1. Considering that [[29edt]] is a compressed 18edo, neither 28edt or 29edt are truly equivalent to 18edo (these two basically correspond to 18edo's dual fifths).
 
== Harmonics ==
{{Harmonics in equal
| steps = 28
| num = 3
| denom = 1
| intervals = integer
}}
{{Harmonics in equal
| steps = 28
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}
 
== Intervals ==
{| class="wikitable right-all left-4"
|-
|-
! | Degree
! rowspan="2"| Steps
! | cents
! rowspan="2"| [[cent]]s
!hekts
! rowspan="2"| [[hekt]]s
! | notation
! colspan="2"| Approximate ratios
! rowspan="2"| notation{{clarify}}
|-
|-
| | 1/28
!3.5.11
| | 67.93
!3.8.5.11
|46.429
| | Cq/D\\y
|-
|-
| | 2/28
| 1
| | 135.85
| 67.927
|92.857
| 46.429
| | Cp/D\\
| [[125/121]]
| [[25/24]]
| Cq/D\\y
|-
|-
| | 3/28
| 2
| | 203.78
| 135.854
|139.286
| 92.857
| | Cpq/Dy
| [[27/25]]
| [[88/81]]
| Cp/D\\
|-
|-
| | 4/28
| 3
| | 271.71
| 203.781
|185.714
| 139.286
| | D
| [[135/121]]
| [[9/8]]
| Cpq/Dy
|-
|-
| | 5/28
| 4
| | 339.635
| 271.708
|232.143
| 185.714
| | Dp/Ey
| [[729/625]]
| [[64/55]], [[75/64]]
| D
|-
|-
| | 6/28
| 5
| | 407.56
| 339.635
|278.571
| 232.143
| | E
| [[11/9]]
| [[40/33]]
| Dp/Ey
|-
|-
| | 7/28
| 6
| | 475.49
| 407.562
|325
| 278.571
| | Epq/F\\y
| [[125/99]]
| [[81/64]]
| E
|-
|-
| | 8/28
| 7
| | 543.42
| 475.489
|371.429
| 325.000
| | Ep/F\\
| [[33/25]]
| [[72/55]], [[320/243]]
| Epq/F\\y
|-
|-
| | 9/28
| 8
| | 611.34
| 543.416
|417.857
| 371.429
| | Eq/Fy
| [[15/11]]
| [[11/8]]
| Ep/F\\
|-
|-
| | 10/28
| 9
| | 679.27
| 611.343
|464.286
| 417.857
| | F
| [[891/625]]
| [[64/45]]
| Eq/Fy
|-
|-
| | 11/28
| 10
| | 747.2
| 679.270
|510.714
| 464.286
| | Fq/Gy
| [[81/55]], [[121/81]]
| [[40/27]]
| F
|-
|-
| | 12/28
| 11
| | 815.12
| 747.197
|557.143
| 510.714
| | G
| [[125/81]]
| [[99/64]]
| Fq/Gy
|-
|-
| | 13/28
| 12
| | 883.05
| 815.124
|603.571
| 557.143
| | Gq/H\\y
| [[121/75]]
| [[8/5]]
| G
|-
|-
| | 14/28
| 13
| | 950.98
| 883.051
|650
| 603.571
| | Gp/H\\
| [[5/3]]
|
| Gq/H\\y
|-
|-
| | 15/28
| 14
| | 1018.9
| 950.978
|696.429
| 650.000
| | Gpq/Hy
| [[625/363]], [[1089/625]]
| [[512/297]], [[891/512]]
| Gp/H\\
|-
|-
| | 16/28
| 15
| | 1086.83
| 1018.904
|742.857
| 696.429
| | H
| [[9/5]]
|
| Gpq/Hy
|-
|-
| | 17/28
| 16
| | 1154.76
| 1086.831
|789.286
| 742.857
| | Hq/Jy
| [[225/121]]
| [[15/8]]
| H
|-
|-
| | 18/28
| 17
| | 1222.685
| 1154.758
|835.714
| 789.286
| | J
| [[243/125]]
| [[64/33]]
| Hq/Jy
|-
|-
| | 19/28
| 18
| | 1290.61
| 1222.685
|882.143
| 835.714
| | Jq/A\\y
| [[55/27]], [[243/121]]
| [[81/40]]
| J
|-
|-
| | 20/28
| 19
| | 1358.54
| 1290.612
|928.571
| 882.143
| | Jp/A\\
| [[625/297]]
| [[135/64]]
| Jq/A\\y
|-
|-
| | 21/28
| 20
| | 1426.47
| 1358.539
|975
| 928.571
| | Jpq/Ay
| [[11/5]]
| [[24/11]]
| Jp/A\\
|-
|-
| | 22/28
| 21
| | 1494.39
| 1426.466
|1021.429
| 975.000
| | A
| [[25/11]]
| [[55/24]], [[729/320]]
| Jpq/Ay
|-
|-
| | 23/28
| 22
| | 1562.32
| 1494.393
|1067.857
| 1021.429
| | Aq/B\\y
| [[297/125]]
| [[64/27]]
| A
|-
|-
| | 24/28
| 23
| | 1630.25
| 1562.320
|1114.286
| 1067.857
| | Ap/B\\
| [[27/11]]
| [[99/40]]
| Aq/B\\y
|-
|-
| | 25/28
| 24
| | 1698.17
| 1630.247
|1160.714
| 1114.286
| | Apq/By
| [[625/243]]
| [[64/25]], [[165/64]]
| Ap/B\\
|-
|-
| | 26/28
| 25
| | 1766.1
| 1698.174
|1207.143
| 1160.714
| | B
| [[121/45]]
| [[8/3]]
| Apq/By
|-
|-
| | 27/28
| 26
| | 1834.03
| 1766.101
|1253.571
| 1207.143
| | Bq/Cy
| [[25/9]]
| [[243/88]]
| B
|-
|-
| | 28/28
| 27
| | 1901.955
| 1834.028
|1300
| 1253.571
| | C
| [[363/125]]
| [[72/25]]
| Bq/Cy
|-
| 28
| 1901.955
| 1300.000
| [[3/1]]
|  
| C
|}
|}


[http://www.seraph.it/dep/det/NonOctaveMelancholic.mp3 NonOctave Melancholic] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/06940a530481965d791c521ea45012f2-267.html blog entry])
== Music ==
* [http://www.seraph.it/dep/det/NonOctaveMelancholic.mp3 NonOctave Melancholic] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/06940a530481965d791c521ea45012f2-267.html blog entry])
 
{{todo|expand}}
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