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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]]

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"