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{{Infobox ET}}
{{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}}. One step is very close to every third step of [[53edo]]. 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.  
'''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}}.  


=== Integer harmonics ===
== Theory ==
{{Harmonics in equal|28|3}}
 
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 ==
== Intervals ==
{| class="wikitable right-all left-4"
{| class="wikitable right-all left-4"
|-
|-
! Steps
! rowspan="2"| Steps
! [[cent]]s
! rowspan="2"| [[cent]]s
! [[hekt]]s
! rowspan="2"| [[hekt]]s
! notation
! colspan="2"| Approximate ratios
! rowspan="2"| notation{{clarify}}
|-
!3.5.11
!3.8.5.11
|-
|-
| 1
| 1
| 67.927
| 67.927
| 46.429
| 46.429
| [[125/121]]
| [[25/24]]
| Cq/D\\y
| Cq/D\\y
|-
|-
Line 21: Line 48:
| 135.854
| 135.854
| 92.857
| 92.857
| [[27/25]]
| [[88/81]]
| Cp/D\\
| Cp/D\\
|-
|-
Line 26: Line 55:
| 203.781
| 203.781
| 139.286
| 139.286
| [[135/121]]
| [[9/8]]
| Cpq/Dy
| Cpq/Dy
|-
|-
Line 31: Line 62:
| 271.708
| 271.708
| 185.714
| 185.714
| [[729/625]]
| [[64/55]], [[75/64]]
| D
| D
|-
|-
Line 36: Line 69:
| 339.635
| 339.635
| 232.143
| 232.143
| [[11/9]]
| [[40/33]]
| Dp/Ey
| Dp/Ey
|-
|-
Line 41: Line 76:
| 407.562
| 407.562
| 278.571
| 278.571
| [[125/99]]
| [[81/64]]
| E
| E
|-
|-
Line 46: Line 83:
| 475.489
| 475.489
| 325.000
| 325.000
| [[33/25]]
| [[72/55]], [[320/243]]
| Epq/F\\y
| Epq/F\\y
|-
|-
Line 51: Line 90:
| 543.416
| 543.416
| 371.429
| 371.429
| [[15/11]]
| [[11/8]]
| Ep/F\\
| Ep/F\\
|-
|-
Line 56: Line 97:
| 611.343
| 611.343
| 417.857
| 417.857
| [[891/625]]
| [[64/45]]
| Eq/Fy
| Eq/Fy
|-
|-
Line 61: Line 104:
| 679.270
| 679.270
| 464.286
| 464.286
| [[81/55]], [[121/81]]
| [[40/27]]
| F
| F
|-
|-
Line 66: Line 111:
| 747.197
| 747.197
| 510.714
| 510.714
| [[125/81]]
| [[99/64]]
| Fq/Gy
| Fq/Gy
|-
|-
Line 71: Line 118:
| 815.124
| 815.124
| 557.143
| 557.143
| [[121/75]]
| [[8/5]]
| G
| G
|-
|-
Line 76: Line 125:
| 883.051
| 883.051
| 603.571
| 603.571
| [[5/3]]
|
| Gq/H\\y
| Gq/H\\y
|-
|-
Line 81: Line 132:
| 950.978
| 950.978
| 650.000
| 650.000
| [[625/363]], [[1089/625]]
| [[512/297]], [[891/512]]
| Gp/H\\
| Gp/H\\
|-
|-
Line 86: Line 139:
| 1018.904
| 1018.904
| 696.429
| 696.429
| [[9/5]]
|
| Gpq/Hy
| Gpq/Hy
|-
|-
Line 91: Line 146:
| 1086.831
| 1086.831
| 742.857
| 742.857
| [[225/121]]
| [[15/8]]
| H
| H
|-
|-
Line 96: Line 153:
| 1154.758
| 1154.758
| 789.286
| 789.286
| [[243/125]]
| [[64/33]]
| Hq/Jy
| Hq/Jy
|-
|-
Line 101: Line 160:
| 1222.685
| 1222.685
| 835.714
| 835.714
| [[55/27]], [[243/121]]
| [[81/40]]
| J
| J
|-
|-
Line 106: Line 167:
| 1290.612
| 1290.612
| 882.143
| 882.143
| [[625/297]]
| [[135/64]]
| Jq/A\\y
| Jq/A\\y
|-
|-
Line 111: Line 174:
| 1358.539
| 1358.539
| 928.571
| 928.571
| [[11/5]]
| [[24/11]]
| Jp/A\\
| Jp/A\\
|-
|-
Line 116: Line 181:
| 1426.466
| 1426.466
| 975.000
| 975.000
| [[25/11]]
| [[55/24]], [[729/320]]
| Jpq/Ay
| Jpq/Ay
|-
|-
Line 121: Line 188:
| 1494.393
| 1494.393
| 1021.429
| 1021.429
| [[297/125]]
| [[64/27]]
| A
| A
|-
|-
Line 126: Line 195:
| 1562.320
| 1562.320
| 1067.857
| 1067.857
| [[27/11]]
| [[99/40]]
| Aq/B\\y
| Aq/B\\y
|-
|-
Line 131: Line 202:
| 1630.247
| 1630.247
| 1114.286
| 1114.286
| [[625/243]]
| [[64/25]], [[165/64]]
| Ap/B\\
| Ap/B\\
|-
|-
Line 136: Line 209:
| 1698.174
| 1698.174
| 1160.714
| 1160.714
| [[121/45]]
| [[8/3]]
| Apq/By
| Apq/By
|-
|-
Line 141: Line 216:
| 1766.101
| 1766.101
| 1207.143
| 1207.143
| [[25/9]]
| [[243/88]]
| B
| B
|-
|-
Line 146: Line 223:
| 1834.028
| 1834.028
| 1253.571
| 1253.571
| [[363/125]]
| [[72/25]]
| Bq/Cy
| Bq/Cy
|-
|-
Line 151: Line 230:
| 1901.955
| 1901.955
| 1300.000
| 1300.000
| [[3/1]]
|
| C
| C
|}
|}
Line 157: Line 238:
* [http://www.seraph.it/dep/det/NonOctaveMelancholic.mp3 NonOctave Melancholic] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/06940a530481965d791c521ea45012f2-267.html blog entry])
* [http://www.seraph.it/dep/det/NonOctaveMelancholic.mp3 NonOctave Melancholic] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/06940a530481965d791c521ea45012f2-267.html blog entry])


[[Category:Edt]]
{{todo|expand}}
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