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'''[[Ed5|Division of the 5th harmonic]] into 28 equal parts''' (28ED5) is related to [[12edo|12EDO]], 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 (''quindromeda'' temperament). This temperament is supported by {{Val list|12, 169, 181, 193, 205, 217, 229}}, and [[241edo|241]] EDOs.
{{Main| Quindromeda family }}


Equating 225/224 with 256/255 leads ''[[Hemifamity temperaments #Quintakwai|quintakwai]]'' (12&amp;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 ''[[Hemimean clan #Quintagar|quintagar]]'' (12&amp;217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.
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.


; <font style="font-size: 1.15em">Quindromeda (12&amp;193)</font>
== See also ==
'''5-limit'''<br>
* [[7edf]] – relative edf
Comma: {{monzo|56 -28 -5}}<br>
* [[12edo]] – relative edo
Mapping: [{{val|1 2 0}}, {{val|0 -5 28}}]<br>
* [[19edt]] – relative edt
POTE generator: ~4428675/4194304 = 99.526<br>
* [[31ed6]] – relative ed6
Vals: 12, 169, 181, 193, 205, 217, 422<br>
* [[34ed7]] – relative ed7
Badness: 0.399849<br><br>
* [[40ed10]] – relative ed10
'''2.3.5.17.19 subgroup'''<br>
* [[42ed11]] – relative ed11
Comma list: 1216/1215, 1445/1444, 6144/6137<br>
* [[76ed80]] – close to the zeta-optimized tuning for 12edo
Gencom: [2 18/17; 1216/1215 1445/1444 6144/6137]<br>
* [[1ed18/17|AS18/17]] – relative [[AS|ambitonal sequence]]
Gencom mapping: [{{val|1 2 0 5 4}}, {{val|0 -5 28 -11 3}}]<br>
POTE generator: ~18/17 = 99.524<br>
Vals: 12, 169, 181, 193, 205, 422<br>
RMS error: 0.0813 cents<br><br>
; <font style="font-size: 1.15em">[[Hemifamity temperaments #Quintakwai|Quintakwai]] (12&amp;193)</font>
'''7-limit'''<br>
Comma list: 5120/5103, 9765625/9680832<br>
Mapping: [{{val|1 2 0 -2}}, {{val|0 -5 28 58}}]<br>
POTE generator: ~625/588 = 99.483<br>
Vals: 12, 169, 181, 193<br>
Badness: 0.155536<br><br>
'''11-limit'''<br>
Comma list: 1375/1372, 4375/4356, 5120/5103<br>
Mapping: [{{val|1 2 0 -2 -4}}, {{val|0 -5 28 58 90}}]<br>
POTE generator: ~35/33 = 99.472<br>
Vals: 12, 181, 193, 374, 567ce<br>
Badness: 0.073158<br><br>
'''13-limit'''<br>
Comma list: 325/324, 1375/1372, 1575/1573, 4096/4095<br>
Mapping: [{{val|1 2 0 -2 -4 10}}, {{val|0 -5 28 58 90 -76}}]<br>
POTE generator: ~35/33 = 99.468<br>
Vals: 12, 181, 193, 374, 567ce, 941bce<br>
Badness: 0.062737<br><br>
'''17-limit'''<br>
Comma list: 325/324, 375/374, 595/594, 1275/1274, 4096/4095<br>
Mapping: [{{val|1 2 0 -2 -4 10 5}}, {{val|0 -5 28 58 90 -76 -11}}]<br>
POTE generator: ~18/17 = 99.469<br>
Vals: 12, 181, 193, 374, 567ce, 941bceg<br>
Badness: 0.037855<br><br>
'''19-limit'''<br>
Comma list: 325/324, 375/374, 400/399, 595/594, 1216/1215, 1275/1274<br>
Mapping: [{{val|1 2 0 -2 -4 10 5 4}}, {{val|0 -5 28 58 90 -76 -11 3}}]<br>
POTE generator: ~18/17 = 99.469<br>
Vals: 12, 181, 193, 374, 567ce, 941bcegh, 1508bccdeegghh<br>
Badness: 0.025861<br><br>
; <font style="font-size: 1.15em">[[Hemifamity temperaments #Quintakwai|Quinkwai]] (12f&amp;181)</font>
'''13-limit'''<br>
Comma list: 352/351, 847/845, 1375/1372, 4375/4356<br>
Mapping: [{{val|1 2 0 -2 -4 -5}}, {{val|0 -5 28 58 90 105}}]<br>
POTE generator: ~35/33 = 99.456<br>
Vals: 12f, 169e, 181<br>
Badness: 0.061873<br><br>
'''17-limit'''<br>
Comma list: 352/351, 375/374, 595/594, 833/832, 1375/1372<br>
Mapping: [{{val|1 2 0 -2 -4 -5 5}}, {{val|0 -5 28 58 90 105 -11}}]<br>
POTE generator: ~18/17 = 99.458<br>
Vals: 12f, 169e, 181, 374ff, 555cff<br>
Badness: 0.043506<br><br>
'''19-limit'''<br>
Comma list: 352/351, 375/374, 400/399, 495/494, 595/594, 1375/1372<br>
Mapping: [{{val|1 2 0 -2 -4 -5 5 4}}, {{val|0 -5 28 58 90 105 -11 3}}]<br>
POTE generator: ~18/17 = 99.459<br>
Vals: 12f, 169e, 181, 374ff, 555cff<br>
Badness: 0.031313<br><br>
; <font style="font-size: 1.15em">[[Hemifamity temperaments #Quintakwai|Quintakwoid]] (12f&amp;193)</font>
'''13-limit'''<br>
Comma list: 364/363, 625/624, 1375/1372, 5120/5103<br>
Mapping: [{{val|1 2 0 -2 -4 -6}}, {{val|0 -5 28 58 90 117}}]<br>
POTE generator: ~35/33 = 99.484<br>
Vals: 12f, 181f, 193<br>
Badness: 0.057357<br><br>
'''17-limit'''<br>
Comma list: 364/363, 375/374, 442/441, 595/594, 5120/5103<br>
Mapping: [{{val|1 2 0 -2 -4 -6 5}}, {{val|0 -5 28 58 90 117 -11}}]<br>
POTE generator: ~18/17 = 99.484<br>
Vals: 12f, 181f, 193<br>
Badness: 0.039289<br><br>
'''19-limit'''<br>
Comma list: 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215<br>
Mapping: [{{val|1 2 0 -2 -4 -6 5 4}}, {{val|0 -5 28 58 90 117 -11 3}}]<br>
POTE generator: ~18/17 = 99.485<br>
Vals: 12f, 181f, 193<br>
Badness: 0.028101<br><br>
; <font style="font-size: 1.15em">[[Hemimean clan #Quintagar|Quintagar]] (12&amp;217)</font>
'''7-limit'''<br>
Comma list: 3136/3125, 33554432/33480783<br>
Mapping: [{{val|1 2 0 -3}}, {{val|0 -5 28 70}}]<br>
POTE generator: ~200/189 = 99.555<br>
Vals: 12, 217, 229, 446, 675c<br>
Badness: 0.142897<br><br>
; <font style="font-size: 1.15em">[[Hemimean clan #Quintagar|Quintoneum]] (12&amp;217)</font>
'''11-limit'''<br>
Comma list: 441/440, 3136/3125, 7168000/7144929<br>
Mapping: [{{val|1 2 0 -3 -5}}, {{val|0 -5 28 70 102}}]<br>
POTE generator: ~35/33 = 99.539<br>
Vals: 12, 205d, 217<br>
Badness: 0.087157<br><br>
'''13-limit'''<br>
Comma list: 364/363, 441/440, 3136/3125, 13720/13689<br>
Mapping: [{{val|1 2 0 -3 -5 -7}}, {{val|0 -5 28 70 102 129}}]<br>
POTE generator: ~35/33 = 99.541<br>
Vals: 12f, 205df, 217<br>
Badness: 0.052361<br><br>
'''17-limit'''<br>
Comma list: 364/363, 441/440, 595/594, 3136/3125, 3757/3750<br>
Mapping: [{{val|1 2 0 -3 -5 -7 5}}, {{val|0 -5 28 70 102 129 -11}}]<br>
POTE generator: ~18/17 = 99.540<br>
Vals: 12f, 205df, 217<br>
Badness: 0.035653<br><br>
'''19-limit'''<br>
Comma list: 364/363, 441/440, 476/475, 595/594, 1216/1215, 3757/3750<br>
Mapping: [{{val|1 2 0 -3 -5 -7 5 4}}, {{val|0 -5 28 70 102 129 -11 3}}]<br>
POTE generator: ~18/17 = 99.541<br>
Vals: 12f, 205df, 217<br>
Badness: 0.025782<br><br>
; <font style="font-size: 1.15em">[[Hemimean clan #Quintagar|Quintoneoid]] (12&amp;217)</font>
'''13-limit'''<br>
Comma list: 441/440, 1001/1000, 3136/3125, 59150/59049<br>
Mapping: [{{val|1 2 0 -3 -5 11}}, {{val|0 -5 28 70 102 -88}}]<br>
POTE generator: ~35/33 = 99.537<br>
Vals: 12, 205d, 217<br>
Badness: 0.072826<br><br>
'''17-limit'''<br>
Comma list: 441/440, 595/594, 1001/1000, 2601/2600, 3136/3125<br>
Mapping: [{{val|1 2 0 -3 -5 11 5}}, {{val|0 -5 28 70 102 -88 -11}}]<br>
POTE generator: ~18/17 = 99.537<br>
Vals: 12, 205d, 217<br>
Badness: 0.042339<br><br>
'''19-limit'''<br>
Comma list: 441/440, 476/475, 595/594, 1001/1000, 1216/1215, 2601/2600<br>
Mapping: [{{val|1 2 0 -3 -5 11 5 4}}, {{val|0 -5 28 70 102 -88 -11 3}}]<br>
POTE generator: ~18/17 = 99.537<br>
Vals: 12, 205d, 217<br>
Badness: 0.028983<br><br>
; <font style="font-size: 1.15em">[[Hemimean clan #Quintagar|Quintasandra]] (217&amp;229)</font>
'''11-limit'''<br>
Comma list: 3136/3125, 19712/19683, 41503/41472<br>
Mapping: [{{val|1 2 0 -3 13}}, {{val|0 -5 28 70 -115}}]<br>
POTE generator: ~200/189 = 99.551<br>
Vals: 12e, 217, 446<br>
Badness: 0.109908<br><br>
'''13-limit'''<br>
Comma list: 2080/2079, 3136/3125, 4096/4095, 19712/19683<br>
Mapping: [{{val|1 2 0 -3 13 11}}, {{val|0 -5 28 70 -115 -88}}]<br>
POTE generator: ~55/52 = 99.548<br>
Vals: 12e, 217, 446, 663c<br>
Badness: 0.067730<br><br>
'''17-limit'''<br>
Comma list: 936/935, 1156/1155, 1377/1375, 3136/3125, 4096/4095<br>
Mapping: [{{val|1 2 0 -3 13 11 5}}, {{val|0 -5 28 70 -115 -88 -11}}]<br>
POTE generator: ~18/17 = 99.548<br>
Vals: 12e, 217, 446, 663c<br>
Badness: 0.038153<br><br>
'''19-limit'''<br>
Comma list: 476/475, 936/935, 1156/1155, 1216/1215, 1377/1375, 1729/1728<br>
Mapping: [{{val|1 2 0 -3 13 11 5 4}}, {{val|0 -5 28 70 -115 -88 -11 3}}]<br>
POTE generator: ~18/17 = 99.547<br>
Vals: 12e, 217, 446, 663ch<br>
Badness: 0.026654<br><br>
; <font style="font-size: 1.15em">[[Hemimean clan #Quintagar|Quintasandroid]] (12&amp;229)</font>
'''11-limit'''<br>
Comma list: 3136/3125, 8019/8000, 15488/15435<br>
Mapping: [{{val|1 2 0 -3 -6}}, {{val|0 -5 28 70 114}}]<br>
POTE generator: ~200/189 = 99.570<br>
Vals: 12, 217e, 229, 470cd, 699cd<br>
Badness: 0.093971<br><br>
'''13-limit'''<br>
Comma list: 351/350, 2080/2079, 3136/3125, 10648/10647<br>
Mapping: [{{val|1 2 0 -3 -6 -8}}, {{val|0 -5 28 70 114 141}}]<br>
POTE generator: ~55/52 = 99.578<br>
Vals: 12f, 217ef, 229, 241, 470cd, 711ccd<br>
Badness: 0.065701<br><br>
'''17-limit'''<br>
Comma list: 351/350, 442/441, 561/560, 3136/3125, 7744/7735<br>
Mapping: [{{val|1 2 0 -3 -6 -8 5}}, {{val|0 -5 28 70 114 141 -11}}]<br>
POTE generator: ~18/17 = 99.574<br>
Vals: 12f, 217ef, 229, 241, 470cd<br>
Badness: 0.046624<br><br>
'''19-limit'''<br>
Comma list: 351/350, 442/441, 476/475, 561/560, 627/625, 6144/6137<br>
Mapping: [{{val|1 2 0 -3 -6 -8 5 4}}, {{val|0 -5 28 70 114 141 -11 3}}]<br>
POTE generator: ~18/17 = 99.575<br>
Vals: 12f, 217ef, 229, 241, 470cd<br>
Badness: 0.033145<br><br>
; <font style="font-size: 1.15em">[[Hemimean clan #Quintagar|Quintasand]] (12&amp;229)</font>
'''13-limit'''<br>
Comma list: 1573/1568, 3136/3125, 4096/4095, 4459/4455<br>
Mapping: [{{val|1 2 0 -3 -6 11}}, {{val|0 -5 28 70 114 -88}}]<br>
POTE generator: ~200/189 = 99.556<br>
Vals: 12, 217e, 229, 446e, 675ceef<br>
Badness: 0.100195<br><br>
'''17-limit'''<br>
Comma list: 561/560, 715/714, 1701/1700, 3136/3125, 4096/4095<br>
Mapping: [{{val|1 2 0 -3 -6 11 5}}, {{val|0 -5 28 70 114 -88 -11}}]<br>
POTE generator: ~18/17 = 99.556<br>
Vals: 12, 217e, 229, 446e, 675ceef<br>
Badness: 0.057851<br><br>
'''19-limit'''<br>
Comma list: 286/285, 476/475, 561/560, 627/625, 1216/1215, 1729/1728<br>
Mapping: [{{val|1 2 0 -3 -6 11 5 4}}, {{val|0 -5 28 70 114 -88 -11 3}}]<br>
POTE generator: ~18/17 = 99.557<br>
Vals: 12, 217e, 229, 446e, 675ceefh<br>
Badness: 0.040410<br><br>
; <font style="font-size: 1.15em">Semiquindromeda (12&amp;422)</font>
'''7-limit'''<br>
Comma list: 102760448/102515625, 1220703125/1219784832<br>
Mapping: [{{val|2 4 0 -5}}, {{val|0 -5 28 64}}]<br>
POTE generator: ~1323/1250 = 99.521<br>
Vals: 12, 398, 410, 422, 832, 1254d, 2086bd<br>
Badness: 0.233140<br><br>
'''11-limit'''<br>
Comma list: 5632/5625, 9801/9800, 85937500/85766121<br>
Mapping: [{{val|2 4 0 -5 -10}}, {{val|0 -5 28 64 102}}]<br>
POTE generator: ~1323/1250 = 99.525<br>
Vals: 12, 410, 422<br>
Badness: 0.093926<br><br>
'''13-limit'''<br>
Comma list: 1716/1715, 2080/2079, 5632/5625, 831875/830466<br>
Mapping: [{{val|2 4 0 -5 -10 -13}}, {{val|0 -5 28 64 102 123}}]<br>
POTE generator: ~1323/1250 = 99.523<br>
Vals: 12f, 410, 422, 1254df, 1676bdff, 2098bcddff<br>
Badness: 0.053361<br><br>
'''17-limit'''<br>
Comma list: 1716/1715, 2080/2079, 2500/2499, 5632/5625, 15895/15876<br>
Mapping: [{{val|2 4 0 -5 -10 -13 10}}, {{val|0 -5 28 64 102 123 -11}}]<br>
POTE generator: ~18/17 = 99.522<br>
Vals: 12f, 410, 422, 832, 1254df, 1676bdff<br>
Badness: 0.034659<br><br>
'''19-limit'''<br>
Comma list: 1216/1215, 1445/1444, 1716/1715, 2080/2079, 2376/2375, 2500/2499<br>
Mapping: [{{val|2 4 0 -5 -10 -13 10 8}}, {{val|0 -5 28 64 102 123 -11 3}}]<br>
POTE generator: ~18/17 = 99.523<br>
Vals: 12f, 410, 422, 1254dfhh, 1676bdffhh<br>
Badness: 0.025439<br><br>


== 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]]

Latest revision as of 13:26, 10 June 2025

← 27ed5 28ed5 29ed5 →
Prime factorization 22 × 7
Step size 99.5112 ¢ 
Octave 12\28ed5 (1194.13 ¢) (→ 3\7ed5)
Twelfth 19\28ed5 (1890.71 ¢)
Consistency limit 10
Distinct consistency limit 6

28 equal divisions of the 5th harmonic (abbreviated 28ed5) is a nonoctave tuning system that divides the interval of 5/1 into 28 equal parts of about 99.5 ¢ each. Each step represents a frequency ratio of 51/28, or the 28th root of 5.

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 ¢, 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

Approximation of harmonics in 28ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.9 -11.2 -11.7 +0.0 -17.1 +14.6 -17.6 -22.5 -5.9 +28.2 -23.0
Relative (%) -5.9 -11.3 -11.8 +0.0 -17.2 +14.6 -17.7 -22.6 -5.9 +28.3 -23.1
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(0) 		31
(3) 		34
(6) 		36
(8) 		38
(10) 		40
(12) 		42
(14) 		43
(15)
Approximation of harmonics in 28ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +37.5 +8.7 -11.2 -23.5 -28.9 -28.3 -22.4 -11.7 +3.3 +22.3 +44.8 -28.8
Relative (%) +37.7 +8.7 -11.3 -23.6 -29.0 -28.5 -22.6 -11.8 +3.3 +22.4 +45.1 -29.0
Steps
(reduced) 		45
(17) 		46
(18) 		47
(19) 		48
(20) 		49
(21) 		50
(22) 		51
(23) 		52
(24) 		53
(25) 		54
(26) 		55
(27) 		55
(27)

Subsets and supersets

Since 28 factors into 22 × 7, 28ed5 has subset ed5's 2, 4, 7, and 14.

Intervals

# Cents Approximate ratios
0 0.0 1/1
1 99.5 18/17
2 199.0 9/8
3 298.5 6/5
4 398.0 5/4
5 497.6 4/3
6 597.1 7/5
7 696.6 3/2
8 796.1 8/5
9 895.6 5/3
10 995.1 7/4
11 1094.6 15/8
12 1194.1 2/1
13 1293.6 17/8
14 1393.2 9/4
15 1492.7 12/5
16 1592.2 5/2
17 1691.7 8/3
18 1791.2 14/5
19 1890.7 3/1
20 1990.2 16/5
21 2089.7 10/3
22 2189.2 7/2
23 2288.8 15/4
24 2388.3 4/1
25 2487.8 17/4
26 2587.3 9/2
27 2686.8 19/4
28 2786.3 5/1

Regular temperaments

Main article: 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 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

External links

Retrieved from "https://en.xen.wiki/index.php?title=28ed5&oldid=199124"