28ed5: Difference between revisions

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== 28ed5 as a generator ==

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|12EDO]], [[205edo|205EDO]], and [[217edo|217EDO]] among others.

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). 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.

Equating 225/224 with 256/255 leads ''quintakwai'' (12&193), which tempers out 400/399 in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads ''quintoneum'' (12&217), which tempers out 476/475 in the 2.3.5.7.17.19 subgroup.

'''Quindromeda (12&193)'''

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Comma list: 364/363, 441/440, 3136/3125, 13720/13689

Mapping: [{{val|1 2 0 -3 -5 -7}}, {{val|0 -5 28 70 102 129}}]

POTE generator: ~55/52 = 99.541

POTE generator: ~35/33 = 99.541

Vals: 12f, 205df, 217

Badness: 0.052361

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Vals: 12f, 205df, 217

Badness: 0.025782

'''Quintasandra (217&229)'''

'''[[Hemimean clan #Quintoneum|Quintoneoid]] (12&217)'''

'''13-limit'''

Comma list: 441/440, 1001/1000, 3136/3125, 59150/59049

Mapping: [{{val|1 2 0 -3 -5 11}}, {{val|0 -5 28 70 102 -88}}]

POTE generator: ~35/33 = 99.537

Vals: 12, 205d, 217

Badness: 0.072826

'''17-limit'''

Comma list: 441/440, 595/594, 1001/1000, 2601/2600, 3136/3125

Mapping: [{{val|1 2 0 -3 -5 11 5}}, {{val|0 -5 28 70 102 -88 -11}}]

POTE generator: ~18/17 = 99.537

Vals: 12, 205d, 217

Badness: 0.042339

'''19-limit'''

Comma list: 441/440, 476/475, 595/594, 1001/1000, 1216/1215, 2601/2600

Mapping: [{{val|1 2 0 -3 -5 11 5 4}}, {{val|0 -5 28 70 102 -88 -11 3}}]

POTE generator: ~18/17 = 99.537

Vals: 12, 205d, 217

Badness: 0.028983

'''[[Hemimean clan #Quintoneum|Quintasandra]] (217&229)'''

'''11-limit'''

Comma list: 3136/3125, 19712/19683, 41503/41472

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Vals: 12e, 217, 446, 663ch

Badness: 0.026654

'''Quintasandroid (229&241)'''

'''[[Hemimean clan #Quintoneum|Quintasandroid]] (229&241)'''

'''11-limit'''

Comma list: 3136/3125, 8019/8000, 15488/15435

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Vals: 12f, 217ef, 229, 241, 470cd

Badness: 0.033145

'''Quintasand (12&229)'''

'''[[Hemimean clan #Quintoneum|Quintasand]] (12&229)'''

'''13-limit'''

Comma list: 1573/1568, 3136/3125, 4096/4095, 4459/4455

@@ Line 155: / Line 155: @@
 == 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|12EDO]], [[205edo|205EDO]], and [[217edo|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 (''quindromeda'' temperament). 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.
+Equating 225/224 with 256/255 leads ''quintakwai'' (12&amp;193), which tempers out 400/399 in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads ''quintoneum'' (12&amp;217), which tempers out 476/475 in the 2.3.5.7.17.19 subgroup.
 '''<font style="font-size: 1.35em">Quindromeda (12&amp;193)</font>'''<br>
@@ Line 218: / Line 220: @@
 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: ~55/52 = 99.541<br>
+POTE generator: ~35/33 = 99.541<br>
 Vals: 12f, 205df, 217<br>
 Badness: 0.052361<br><br>
@@ Line 233: / Line 235: @@
 Vals: 12f, 205df, 217<br>
 Badness: 0.025782<br><br>
-'''<font style="font-size: 1.35em">Quintasandra (217&amp;229)</font>'''<br>
+'''<font style="font-size: 1.35em">[[Hemimean clan #Quintoneum|Quintoneoid]] (12&amp;217)</font>'''<br>
+'''<font style="font-size: 1.2em">13-limit</font>'''<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>
+'''<font style="font-size: 1.2em">17-limit</font>'''<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>
+'''<font style="font-size: 1.2em">19-limit</font>'''<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: 1.35em">[[Hemimean clan #Quintoneum|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>
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 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: 1.35em">[[Hemimean clan #Quintoneum|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>
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 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: 1.35em">[[Hemimean clan #Quintoneum|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>