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'''[[Ed5|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 is 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
|}
|}


==28ed5 as a generator==
== Regular temperaments ==
28ed5 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|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. The small chroma interval between adjacent notes in each cluster is very versatile, representing 1088/1083 ~ 256/255 ~ 289/288 ~ 324/323 ~ 361/360 all tempered together. This temperament is supported by [[12edo]], [[205edo]], and [[217edo]] among others.
{{Main| Quindromeda family }}
 
'''<font style="font-size: 1.25em">5-limit 12&amp;193 (quinsa-quingu)</font>'''
 
Comma: |56 -28 -5&gt;
 
POTE generator: ~4428675/4194304 = 99.526
 
Map: [&lt;1 2 0|, &lt;0 -5 28|]
 
EDOs: 12, 169, 181, 193, 205, 217, 229, 241, 374, 398, 422, 446, 591, 603, 627, 639, 784, 808, 832, 856, 989, 1001, 1013, 1037, 1049, 1061, 1242
 
Badness: 0.399849<br>
 
'''<font style="font-size: 1.25em">7-limit 12&amp;193</font>'''
 
Commas: 5120/5103, 9765625/9680832
 
POTE generator: ~625/588 = 99.483
 
Map: [&lt;1 2 0 -2|, &lt;0 -5 28 58|]
 
EDOs: 12, 169, 181, 193, 205, 374
 
Badness: 0.155536<br>
 
'''<font style="font-size: 1.15em">11-limit 12&amp;193</font>'''
 
Commas: 1375/1372, 4375/4356, 5120/5103
 
POTE generator: ~35/33 = 99.472
 
Map: [&lt;1 2 0 -2 -4|, &lt;0 -5 28 58 90|]
 
EDOs: 12, 169e, 181, 193, 205e, 374
 
Badness: 0.073158<br>
 
'''<font style="font-size: 1.15em">13-limit 12&amp;193</font>'''
 
Commas: 325/324, 1375/1372, 1575/1573, 4096/4095
 
POTE generator: ~35/33 = 99.468
 
Map: [&lt;1 2 0 -2 -4 10|, &lt;0 -5 28 58 90 -76|]
 
EDOs: 12, 181, 193, 374
 
Badness: 0.062737<br>
 
'''<font style="font-size: 1.15em">17-limit 12&amp;193</font>'''
 
Commas: 325/324, 375/374, 595/594, 1275/1274, 4096/4095
 
POTE generator: ~18/17 = 99.469
 
Map: [&lt;1 2 0 -2 -4 10 5|, &lt;0 -5 28 58 90 -76 -11|]
 
EDOs: 12, 181, 193, 374
 
Badness: 0.037855<br>
 
'''<font style="font-size: 1.15em">19-limit 12&amp;193</font>'''
 
Commas: 325/324, 375/374, 400/399, 595/594, 1216/1215, 1275/1274
 
POTE generator: ~18/17 = 99.469
 
Map: [&lt;1 2 0 -2 -4 10 5 4|, &lt;0 -5 28 58 90 -76 -11 3|]
 
EDOs: 12, 181, 193, 374
 
Badness: 0.025861<br>
 
'''<font style="font-size: 1.25em">7-limit 12&amp;229</font>'''
 
Commas: 3136/3125, 33554432/33480783
 
POTE generator: ~200/189 = 99.555
 
Map: [&lt;1 2 0 -3|, &lt;0 -5 28 70|]
 
EDOs: 12, 217, 229, 241, 446
 
Badness: 0.142897<br>
 
'''<font style="font-size: 1.15em">11-limit 12&amp;229</font>'''
 
Commas: 3136/3125, 8019/8000, 15488/15435
 
POTE generator: ~200/189 = 99.570
 
Map: [&lt;1 2 0 -3 -6|, &lt;0 -5 28 70 114|]
 
EDOs: 12, 217e, 229, 241, 446e
 
Badness: 0.093971<br>
 
'''<font style="font-size: 1.15em">13-limit 12&amp;229</font>'''
 
Commas: 1573/1568, 3136/3125, 4096/4095, 4459/4455
 
POTE generator: ~200/189 = 99.556
 
Map: [&lt;1 2 0 -3 -6 11|, &lt;0 -5 28 70 114 -88|]
 
EDOs: 12, 217e, 229, 241f, 446e
 
Badness: 0.100195<br>
 
'''<font style="font-size: 1.15em">17-limit 12&amp;229</font>'''
 
Commas: 561/560, 715/714, 1701/1700, 3136/3125, 4096/4095
 
POTE generator: ~18/17 = 99.556
 
Map: [&lt;1 2 0 -3 -6 11 5|, &lt;0 -5 28 70 114 -88 -11|]
 
EDOs: 12, 217e, 229, 241f, 446e
 
Badness: 0.057851<br>
 
'''<font style="font-size: 1.15em">19-limit 12&amp;229</font>'''
 
Commas: 286/285, 476/475, 561/560, 627/625, 1216/1215, 1729/1728
 
POTE generator: ~18/17 = 99.557
 
Map: [&lt;1 2 0 -3 -6 11 5 4|, &lt;0 -5 28 70 114 -88 -11 3|]
 
EDOs: 12, 217e, 229, 241f, 446e
 
Badness: 0.040410<br>
 
'''<font style="font-size: 1.25em">7-limit 12&amp;422</font>'''
 
Commas: 102760448/102515625, 1220703125/1219784832
 
POTE generator: ~1323/1250 = 99.521
 
Map: [&lt;2 4 0 -5|, &lt;0 -5 28 64|]
 
EDOs: 12, 398, 410, 422, 808, 832, 1242
 
Badness: 0.233140<br>
 
'''<font style="font-size: 1.15em">11-limit 12&amp;422</font>'''
 
Commas: 5632/5625, 9801/9800, 85937500/85766121
 
POTE generator: ~1323/1250 = 99.525
 
Map: [&lt;2 4 0 -5 -10|, &lt;0 -5 28 64 102|]
 
EDOs: 12, 410, 422, 832
 
Badness: 0.093926<br>
 
'''<font style="font-size: 1.15em">13-limit 12f&amp;422</font>'''
 
Commas: 1716/1715, 2080/2079, 5632/5625, 831875/830466
 
POTE generator: ~1323/1250 = 99.523
 
Map: [&lt;2 4 0 -5 -10 -13|, &lt;0 -5 28 64 102 123|]
 
EDOs: 12f, 410, 422, 832
 
Badness: 0.053361<br>
 
'''<font style="font-size: 1.15em">17-limit 12f&amp;422</font>'''
 
Commas: 1716/1715, 2080/2079, 2500/2499, 5632/5625, 15895/15876
 
POTE generator: ~18/17 = 99.522
 
Map: [&lt;2 4 0 -5 -10 -13 10|, &lt;0 -5 28 64 102 123 -11|]
 
EDOs: 12f, 410, 422, 832
 
Badness: 0.034659<br>
 
'''<font style="font-size: 1.15em">19-limit 12f&amp;422</font>'''
 
Commas: 1216/1215, 1445/1444, 1716/1715, 2080/2079, 2376/2375, 2500/2499
 
POTE generator: ~18/17 = 99.523
 
Map: [&lt;2 4 0 -5 -10 -13 10 8|, &lt;0 -5 28 64 102 123 -11 3|]
 
EDOs: 12f, 410, 422, 832h
 
Badness: 0.025439<br>
 
'''<font style="font-size: 1.25em">2.3.5.17.19 subgroup 12&amp;193</font>'''
 
Commas: 1216/1215, 1445/1444, 6144/6137


POTE generator: ~18/17 = 99.524
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 }}.  


Map: [&lt;1 2 0 5 4|, &lt;0 -5 28 -11 3|]
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.


EDOs: 12, 169, 181, 193, 205, 217, 229, 241, 374, 398, 422, 446, 591, 603<br>
== 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]]


==See also==
== External links ==
*[[12edo]]: relative EDO
* [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
*[[19ED3|19ed3]]: relative ED3
* [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
*[[31ed6]]: relative ED6
*[[34ed7]]: relative ED7
*[[40ed10]]: relative ED10
*[[42ed11]]: relative ED11


[[Category:Ed5]]
[[Category:12edo]]
[[Category:Edonoi]]
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