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{{Infobox ET}}
{{Infobox ET}}
'''[[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.
{{ED intro}}
 
== 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 ==
== Intervals ==
{| class="wikitable"
{| 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
| | [[175/117]], [[323/216]]
| [[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
| | [[85/64|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
| | 117/35
| [[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
|}
|}
== Harmonics ==
{{Harmonics in equal
| steps = 28
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 28
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}


== Regular temperaments ==
== Regular temperaments ==
{{Main| Quindromeda family }}
{{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 {{Optimal ET sequence| 12, 169, 181, 193, 205, 217, 229 }}, and [[241edo|241]] EDOs.
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 ''[[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 ''[[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.
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 ==
== See also ==
* [[12edo|12EDO]] - relative EDO
* [[7edf]] – relative edf
* [[19ed3|19ED3]] - relative ED3
* [[12edo]] – relative edo
* [[31ed6|31ED6]] - relative ED6
* [[19edt]] – relative edt
* [[34ed7|34ED7]] - relative ED7
* [[31ed6]] – relative ed6
* [[40ed10|40ED10]] - relative ED10
* [[34ed7]] – relative ed7
* [[42ed11|42ED11]] - relative ED11
* [[40ed10]] – relative ed10
* [[18/17 equal-step tuning|AS18/17]] - relative [[AS|ambitonal sequence]]
* [[42ed11]] – relative ed11
* [[76ed80]] – close to the zeta-optimized tuning for 12edo
* [[1ed18/17|AS18/17]] – relative [[AS|ambitonal sequence]]


== External links ==
== 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
* [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
* [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: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"