@@ Line 1: / Line 1: @@
-'''[[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.
+'''[[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.
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-==28ed5 as a generator==
+== 28ed5 as a generator ==
-ed5 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.
+ED5 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|12EDO]], [[205edo|205EDO]], and [[217edo|217EDO]] among others.
-'''<font style="font-size: 1.25em">5-limit 12&amp;193 (quinsa-quingu)</font>'''
+'''<font style="font-size: 1.35em">Quinsa-quingu (12&amp;193)</font>'''<br>
+'''<font style="font-size: 1.2em">5-limit</font>'''<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>
+'''<font style="font-size: 1.35em">Quintakwai (12&amp;193)</font>'''<br>
+'''<font style="font-size: 1.2em">7-limit</font>'''<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>
+'''<font style="font-size: 1.2em">11-limit</font>'''<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>
+'''<font style="font-size: 1.2em">13-limit</font>'''<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>
+'''<font style="font-size: 1.2em">17-limit</font>'''<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>
+'''<font style="font-size: 1.2em">19-limit</font>'''<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: 1.35em">Quintoneum (12&amp;217)</font>'''<br>
+'''<font style="font-size: 1.2em">7-limit</font>'''<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: 1.2em">11-limit</font>'''<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>
+'''<font style="font-size: 1.35em">Quintasandra (217&amp;229)</font>'''<br>
+'''<font style="font-size: 1.2em">11-limit</font>'''<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>
+'''<font style="font-size: 1.2em">13-limit</font>'''<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>
+'''<font style="font-size: 1.2em">17-limit</font>'''<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>
+'''<font style="font-size: 1.2em">19-limit</font>'''<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: 1.35em">Quintasandroid (229&amp;241)</font>'''<br>
+'''<font style="font-size: 1.2em">11-limit</font>'''<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>
+'''<font style="font-size: 1.2em">13-limit</font>'''<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>
+'''<font style="font-size: 1.2em">17-limit</font>'''<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>
+'''<font style="font-size: 1.2em">19-limit</font>'''<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: 1.35em">Quintasand (12&amp;229)</font>'''<br>
+'''<font style="font-size: 1.2em">13-limit</font>'''<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>
+'''<font style="font-size: 1.2em">17-limit</font>'''<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>
+'''<font style="font-size: 1.2em">19-limit</font>'''<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: 1.35em">Semiquindromeda (12&amp;422)</font>'''
+'''<font style="font-size: 1.2em">7-limit</font>'''<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>
+'''<font style="font-size: 1.2em">11-limit</font>'''<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>
+'''<font style="font-size: 1.2em">13-limit</font>'''<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>
+'''<font style="font-size: 1.2em">17-limit</font>'''<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>
+'''<font style="font-size: 1.2em">19-limit</font>'''<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>
+'''<font style="font-size: 1.35em">Quindromeda (12&amp;193)</font>'''<br>
+'''<font style="font-size: 1.2em">2.3.5.17.19 subgroup</font>'''<br>
+Comma list: 1216/1215, 1445/1444, 6144/6137<br>
+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>
-Comma: |56 -28 -5&gt;
+== See also ==
+* [[12edo]]: relative EDO
-POTE generator: ~4428675/4194304 = 99.526
+* [[19ED3|19ed3]]: relative ED3
+* [[31ed6]]: relative ED6
-Map: [&lt;1 2 0|, &lt;0 -5 28|]
+* [[34ed7]]: relative ED7
+* [[40ed10]]: relative ED10
-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
+* [[42ed11]]: relative ED11
-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
-Map: [&lt;1 2 0 5 4|, &lt;0 -5 28 -11 3|]
-EDOs: 12, 169, 181, 193, 205, 217, 229, 241, 374, 398, 422, 446, 591, 603<br>
-==See also==
-*[[12edo]]: relative EDO
-*[[19ED3|19ed3]]: relative ED3
-*[[31ed6]]: relative ED6
-*[[34ed7]]: relative ED7
-*[[40ed10]]: relative ED10
-*[[42ed11]]: relative ED11
 [[Category:Ed5]]
 [[Category:Edonoi]]

28ed5: Difference between revisions