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== Regular temperaments ==
== 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&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&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 {{Val list| 12-, 169-, 181-, 193-, 205-, 217-, 229- }}, and [[241edo]].


 
Equating 225/224 with 256/255 leads ''[[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 ''[[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">[[High badness temperaments #Quindromeda|Quindromeda]] (12&amp;193)</font>
'''5-limit'''<br>
Comma: {{monzo|56 -28 -5}}<br>
Mapping: [{{val|1 2 0}}, {{val|0 -5 28}}]<br>
POTE generator: ~4428675/4194304 = 99.526<br>
Vals: 12, 169, 181, 193, 205, 217, 422<br>
Badness: 0.399849<br><br>
'''2.3.5.17.19 subgroup'''<br>
Comma list: 1216/1215, 1445/1444, 6144/6137<br>
Gencom: [2 18/17; 1216/1215 1445/1444 6144/6137]<br>
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">[[Ragismic microtemperaments #Quindro|Quindro]] (205&amp;217)</font>
'''7-limit'''<br>
Comma list: 4375/4374, 72057594037927936/71489976421753125<br>
Mapping: [{{val|1 2 0 15}}, {{val|0 -5 28 -147}}]<br>
POTE generator: ~4428675/4194304 = 99.529<br>
Vals: 205, 217, 422, 639, 1061<br>
Badness: 0.316850<br><br>
'''11-limit'''<br>
Comma list: 4375/4374, 5632/5625, 25165824/25109315<br>
Mapping: [{{val|1 2 0 15 -5}}, {{val|0 -5 28 -147 102}}]<br>
POTE generator: ~17325/16384 = 99.529<br>
Vals: 205, 217, 422, 639, 1061e<br>
Badness: 0.085219<br><br>
'''13-limit'''<br>
Comma list: 2080/2079, 4375/4374, 5632/5625, 20480/20449<br>
Mapping: [{{val|1 2 0 15 -5 11}}, {{val|0 -5 28 -147 102 -88}}]<br>
POTE generator: ~4096/3861 = 99.529<br>
Vals: 205, 217, 422, 639, 1061ef<br>
Badness: 0.039203<br><br>
'''17-limit'''<br>
Comma list: 1156/1155, 2080/2079, 2431/2430, 4375/4374, 5632/5625<br>
Mapping: [{{val|1 2 0 15 -5 11 5}}, {{val|0 -5 28 -147 102 -88 -11}}]<br>
POTE generator: ~18/17 = 99.529<br>
Vals: 205, 217, 422, 639, 1061ef<br>
Badness: 0.024110<br><br>
'''19-limit'''<br>
Comma list: 1156/1155, 1216/1215, 1445/1444, 2080/2079, 2376/2375, 2431/2430<br>
Mapping: [{{val|1 2 0 15 -5 11 5 4}}, {{val|0 -5 28 -147 102 -88 -11 3}}]<br>
POTE generator: ~18/17 = 99.529<br>
Vals: 205, 217, 422, 639h, 1061efh<br>
Badness: 0.016764<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">[[Lokismic temperaments #Semiquindromeda|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 ==
== See also ==

Revision as of 14:24, 21 November 2021

Division of the 5th harmonic into 28 equal parts (28ED5) is related to 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.

degree cents value corresponding
JI intervals 		comments
0 0.0000 exact 1/1
1 99.5112 18/17
2 199.0224 55/49
3 298.5336 19/16
4 398.0448 34/27 pseudo-5/4
5 497.5560 4/3
6 597.0672 24/17
7 696.5784 meantone fifth
(pseudo-3/2)
8 796.0896 19/12
9 895.6008 57/34 pseudo-5/3
10 995.1120 16/9
11 1094.6232 32/17
12 1194.1344 255/128 pseudo-octave
13 1293.6457 19/9
14 1393.1569 38/17, 85/38 meantone major second plus an octave
15 1492.6681 45/19
16 1592.1793 128/51 pseudo-5/2
17 1691.6905 85/32
18 1791.2017 45/16
19 1890.7129 170/57 pseudo-3/1
20 1990.2241 60/19
21 2089.7353 meantone major sixth plus an octave
(pseudo-10/3)
22 2189.2465 85/24
23 2288.7577 15/4
24 2388.2689 135/34 pseudo-4/1
25 2487.7801 80/19
26 2587.2913 49/11
27 2686.8025 85/18
28 2786.3137 exact 5/1 just major third plus two octaves

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 Template:Val list, and 241edo.

Equating 225/224 with 256/255 leads 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 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

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