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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.  It also supports [[deneb]] temperament in the 3.5.11 subgroup, equating 4 [[11/9]]s with [[11/5]].
'''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}}

Latest revision as of 22:54, 7 November 2026

← 27edt 28edt 29edt →
Prime factorization 22 × 7
Step size 67.927 ¢ 
Octave 18\28edt (1222.69 ¢) (→ 9\14edt)
Consistency limit 3
Distinct consistency limit 3

28edt divides the third harmonic into 28 equal parts of 67.9 cents each. It is the simplest edt which intelligibly doubles the Anti-BP harmonic chain[clarification needed].

Theory

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 log3(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 1953125/1948617. The former means it supports deneb temperament, which equates 3 flat 11/9s to 5/3, and the latter means it supports delta centauri temperament, which equates 9 near-just 5/3s 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 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 octaves (~23 ¢), 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

Approximation of harmonics in 28edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +22.7 +0.0 -22.6 -1.3 +22.7 +27.5 +0.1 +0.0 +21.4 -7.8 -22.6
Relative (%) +33.4 +0.0 -33.2 -1.9 +33.4 +40.5 +0.2 +0.0 +31.5 -11.4 -33.2
Steps
(reduced) 		18
(18) 		28
(0) 		35
(7) 		41
(13) 		46
(18) 		50
(22) 		53
(25) 		56
(0) 		59
(3) 		61
(5) 		63
(7)
Approximation of harmonics in 28edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -25.3 -17.7 -1.3 +22.8 -14.2 +22.7 -3.0 -23.9 +27.5 +14.9 +5.9
Relative (%) -37.2 -26.1 -1.9 +33.6 -20.9 +33.4 -4.4 -35.1 +40.5 +22.0 +8.7
Steps
(reduced) 		65
(9) 		67
(11) 		69
(13) 		71
(15) 		72
(16) 		74
(18) 		75
(19) 		76
(20) 		78
(22) 		79
(23) 		80
(24)

Intervals

Steps cents hekts Approximate ratios notation[clarification needed]
3.5.11 3.8.5.11
1 67.927 46.429 125/121 25/24 Cq/D\\y
2 135.854 92.857 27/25 88/81 Cp/D\\
3 203.781 139.286 135/121 9/8 Cpq/Dy
4 271.708 185.714 729/625 64/55, 75/64 D
5 339.635 232.143 11/9 40/33 Dp/Ey
6 407.562 278.571 125/99 81/64 E
7 475.489 325.000 33/25 72/55, 320/243 Epq/F\\y
8 543.416 371.429 15/11 11/8 Ep/F\\
9 611.343 417.857 891/625 64/45 Eq/Fy
10 679.270 464.286 81/55, 121/81 40/27 F
11 747.197 510.714 125/81 99/64 Fq/Gy
12 815.124 557.143 121/75 8/5 G
13 883.051 603.571 5/3 Gq/H\\y
14 950.978 650.000 625/363, 1089/625 512/297, 891/512 Gp/H\\
15 1018.904 696.429 9/5 Gpq/Hy
16 1086.831 742.857 225/121 15/8 H
17 1154.758 789.286 243/125 64/33 Hq/Jy
18 1222.685 835.714 55/27, 243/121 81/40 J
19 1290.612 882.143 625/297 135/64 Jq/A\\y
20 1358.539 928.571 11/5 24/11 Jp/A\\
21 1426.466 975.000 25/11 55/24, 729/320 Jpq/Ay
22 1494.393 1021.429 297/125 64/27 A
23 1562.320 1067.857 27/11 99/40 Aq/B\\y
24 1630.247 1114.286 625/243 64/25, 165/64 Ap/B\\
25 1698.174 1160.714 121/45 8/3 Apq/By
26 1766.101 1207.143 25/9 243/88 B
27 1834.028 1253.571 363/125 72/25 Bq/Cy
28 1901.955 1300.000 3/1 C

Music

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