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'''Division of the 5th harmonic into 28 equal parts''' (28ed5) is related to [[12edo|12 edo]], but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size about 99.5112 cents. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning also has the perfect fourth which is more accurate for 4/3 than that of 12edo, as well as 18/17, 19/16, and 24/17.
{{Infobox ET}}
{{ED intro}}


{| class="wikitable"
== Theory ==
28ed5 is related to [[12edo]], but with the 5/1 rather than the 2/1 being just. This compresses the octave by 5.8656{{c}}, a small but significant deviation. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning also has the perfect fourth which is more accurate for 4/3 than that of 12edo, as well as 18/17, 19/16, and 24/17.
 
=== Harmonics ===
{{Harmonics in equal|28|5|1}}
{{Harmonics in equal|28|5|1|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 28ed5 (continued)}}
 
=== Subsets and supersets ===
Since 28 factors into 2<sup>2</sup> × 7, 28ed5 has subset ed5's {{EDs|equave=5| 2, 4, 7, and 14 }}.
 
== Intervals ==
{| class="wikitable center-1 right-2"
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Approximate ratios
! | comments
|-
|-
| | 0
| 0
| | 0.0000
| 0.0
| | '''exact [[1/1]]'''
| [[1/1]]
| |
|-
|-
| | 1
| 1
| | 99.5112
| 99.5
| | [[18/17]]
| [[18/17]]
| |
|-
|-
| | 2
| 2
| | 199.0224
| 199.0
| | [[55/49]]
| [[9/8]]
| |
|-
|-
| | 3
| 3
| | 298.5336
| 298.5
| | [[19/16]]
| [[6/5]]
| |
|-
|-
| | 4
| 4
| | 398.0448
| 398.0
| | 34/27
| [[5/4]]
| | pseudo-[[5/4]]
|-
|-
| | 5
| 5
| | 497.5560
| 497.6
| | [[4/3]]
| [[4/3]]
| |
|-
|-
| | 6
| 6
| | 597.0672
| 597.1
| | [[24/17]]
| [[7/5]]
| |
|-
|-
| | 7
| 7
| | 696.5784
| 696.6
| |
| [[3/2]]
| | meantone fifth <br>(pseudo-[[3/2]])
|-
|-
| | 8
| 8
| | 796.0896
| 796.1
| | [[19/12]]
| [[8/5]]
| |
|-
|-
| | 9
| 9
| | 895.6008
| 895.6
| | 57/34
| [[5/3]]
| | pseudo-[[5/3]]
|-
|-
| | 10
| 10
| | 995.1120
| 995.1
| | [[16/9]]
| [[7/4]]
| |
|-
|-
| | 11
| 11
| | 1094.6232
| 1094.6
| | [[32/17]]
| [[15/8]]
| |
|-
|-
| | 12
| 12
| | 1194.1344
| 1194.1
| | 255/128
| [[2/1]]
| | pseudo-[[octave]]
|-
|-
| | 13
| 13
| | 1293.6457
| 1293.6
| | [[19/18|19/9]]
| [[17/8]]
| |
|-
|-
| | 14
| 14
| | 1393.1569
| 1393.2
| | [[19/17|38/17]], 85/38
| [[9/4]]
| | meantone major second plus an octave
|-
|-
| | 15
| 15
| | 1492.6681
| 1492.7
| | 45/19
| [[12/5]]
| |
|-
|-
| | 16
| 16
| | 1592.1793
| 1592.2
| | 128/51
| [[5/2]]
| | pseudo-[[5/2]]
|-
|-
| | 17
| 17
| | 1691.6905
| 1691.7
| | 85/32
| [[8/3]]
| |
|-
|-
| | 18
| 18
| | 1791.2017
| 1791.2
| | [[45/32|45/16]]
| [[14/5]]
| |
|-
|-
| | 19
| 19
| | 1890.7129
| 1890.7
| | 170/57
| [[3/1]]
| | pseudo-[[3/1]]
|-
|-
| | 20
| 20
| | 1990.2241
| 1990.2
| | [[30/19|60/19]]
| [[16/5]]
| |
|-
|-
| | 21
| 21
| | 2089.7353
| 2089.7
| |
| [[10/3]]
| | meantone major sixth plus an octave <br>(pseudo-[[10/3]])
|-
|-
| | 22
| 22
| | 2189.2465
| 2189.2
| | 85/24
| [[7/2]]
| |
|-
|-
| | 23
| 23
| | 2288.7577
| 2288.8
| | [[15/4]]
| [[15/4]]
| |
|-
|-
| | 24
| 24
| | 2388.2689
| 2388.3
| | 135/34
| [[4/1]]
| | pseudo-[[4/1]]
|-
|-
| | 25
| 25
| | 2487.7801
| 2487.8
| | [[20/19|80/19]]
| [[17/4]]
| |
|-
|-
| | 26
| 26
| | 2587.2913
| 2587.3
| | [[49/44|49/11]]
| [[9/2]]
| |
|-
|-
| | 27
| 27
| | 2686.8025
| 2686.8
| | 85/18
| [[19/4]]
| |
|-
|-
| | 28
| 28
| | 2786.3137
| 2786.3
| | '''exact [[5/1]]'''
| [[5/1]]
| | just major third plus two octaves
|}
|}


[[Category:Ed5]]
== Regular temperaments ==
[[Category:Edonoi]]
{{Main| Quindromeda family }}
 
28ed5 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[subgroup temperament]] which tempers out 1216/1215, 1445/1444, and 6144/6137, which is a [[cluster temperament]] with 12 clusters of notes in an octave (quindromeda temperament). This temperament is supported by {{EDOs| 12-, 169-, 181-, 193-, 205-, 217-, 229-, and 241edo }}.
 
Equating 225/224 with 256/255 leads to [[quintakwai]] (12 & 193), which tempers out 400/399 (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to [[quintagar]] (12 & 217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.
 
== See also ==
* [[7edf]] – relative edf
* [[12edo]] – relative edo
* [[19edt]] – relative edt
* [[31ed6]] – relative ed6
* [[34ed7]] – relative ed7
* [[40ed10]] – relative ed10
* [[42ed11]] – relative ed11
* [[76ed80]] – close to the zeta-optimized tuning for 12edo
* [[1ed18/17|AS18/17]] – relative [[AS|ambitonal sequence]]
 
== External links ==
* [https://sevish.com/scaleworkshop/index.htm?name=28ed5&data=99.5112040666012%0A199.0224081332025%0A298.5336121998037%0A398.0448162664050%0A497.5560203330062%0A597.0672243996075%0A696.5784284662087%0A796.0896325328099%0A895.6008365994112%0A995.1120406660124%0A1094.6232447326137%0A1194.1344487992149%0A1293.6456528658162%0A1393.1568569324174%0A1492.6680609990187%0A1592.1792650656199%0A1691.6904691322211%0A1791.2016731988224%0A1890.7128772654236%0A1990.2240813320249%0A2089.7352853986261%0A2189.2464894652274%0A2288.7576935318286%0A2388.2688975984298%0A2487.7801016650311%0A2587.2913057316323%0A2686.8025097982336%0A2786.3137138648348&freq=220&midi=57&vert=10&horiz=1 Play 28ed5] – Scale Workshop
* [http://terpstrakeyboard.com/web-app/keys.htm?fundamental=220&right=2&upright=1&size=25&rotation=13.897886248013985&instrument=sawtooth&enum=false&spectrum_colors=false&no_labels=false&scale=!%2028ed5.scl%0A!%20%0A28ed5%0A28%0A!%0A99.5112040666012%0A199.0224081332025%0A298.5336121998037%0A398.0448162664050%0A497.5560203330062%0A597.0672243996075%0A696.5784284662087%0A796.0896325328099%0A895.6008365994112%0A995.1120406660124%0A1094.6232447326137%0A1194.1344487992149%0A1293.6456528658162%0A1393.1568569324174%0A1492.6680609990187%0A1592.1792650656199%0A1691.6904691322211%0A1791.2016731988224%0A1890.7128772654236%0A1990.2240813320249%0A2089.7352853986261%0A2189.2464894652274%0A2288.7576935318286%0A2388.2688975984298%0A2487.7801016650311%0A2587.2913057316323%0A2686.8025097982336%0A2786.3137138648348&names=A%0AA%23%2FBb%0AB%0AC%0AC%23%2FDb%0AD%0AD%23%2FEb%0AE%0AE%23%2FFb%0AF%0AG%0AG%23%2FHb%0AH%0AH%23%2FIb%0AI%0AI%23%2FJb%0AJ%0AK%0AK%23%2FLb%0AL%0AL%23%2FMb%0AM%0AM%23%2FNb%0AN%0AO%0AO%23%2FPb%0AP%0AP%23%2FAb&note_colors=ffffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b Play 28ed5] – Terpstra Keyboard WebApp
 
[[Category:12edo]]
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