User:BudjarnLambeth/Basal subgroup: Difference between revisions

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BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
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Document a concept that occured to me, based on CompactStar's half prime subgroups
 
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,265 edits
m BudjarnLambeth moved page Basal subgroup to User:BudjarnLambeth/Basal subgroup
 
(7 intermediate revisions by the same user not shown)
Line 2: Line 2:


It is built on top of the concept of a [[half-prime subgroup]], so it is recommended to study half-prime subgroups first before exploring basal subgroups.
It is built on top of the concept of a [[half-prime subgroup]], so it is recommended to study half-prime subgroups first before exploring basal subgroups.
The basal subgroup for equal divisions of n/m is named BSGn/m{{idiosyncratic}}. This makes it easy to refer to a specific BSG.




''Example'':  
''Example'':  


The basal subgroup of '''[[edo|ed2/1]]''' (BSG2/1) is 2.3.5.7.11.13... ''a.k.a.'' 2/1 . 3/1 . 5/1 . 7/1 . 11/1 . 13/1...
The basal subgroup of '''[[edo|ed2/1]]''' - - - BSG2/1 - - - is 2.3.5.7.11.13... ''a.k.a.'' 2/1 . 3/1 . 5/1 . 7/1 . 11/1 . 13/1...


== Prime basal subgroups ==
== Prime basal subgroups ==
Line 16: Line 18:


'''List:'''
'''List:'''
* The basal subgroup of '''[[edt|ed3/1]]''' (BSG3/1) is 3/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[edt|ed3/1]]''' - - - BSG3/1 - - - is 3/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed5|ed5/1]]''' (BSG5/1) is 5/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
* The basal subgroup of '''[[ed5|ed5/1]]''' - - - BSG5/1 - - - is 5/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
* The basal subgroup of '''[[ed7|ed7/1]]''' (BSG7/1) is 7/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
* The basal subgroup of '''[[ed7|ed7/1]]''' - - - BSG7/1 - - - is 7/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
* The basal subgroup of '''[[ed11|ed11/1]]''' (BSG11/1) is 11/1 . 13/1 . 17/1 .  19/1 . 23/1 . 29/1...
* The basal subgroup of '''[[ed11|ed11/1]]''' - - - BSG11/1 - - - is 11/1 . 13/1 . 17/1 .  19/1 . 23/1 . 29/1...
* The basal subgroup of '''[[ed13|ed13/1]]''' (BSG13/1) is 13/1 . 17/1 . 19/1 . 23/1 . 29/1 . 31/1...
* The basal subgroup of '''[[ed13|ed13/1]]''' - - - BSG13/1 - - - is 13/1 . 17/1 . 19/1 . 23/1 . 29/1 . 31/1...
* The basal subgroup of '''[[ed17|ed17/1]]''' (BSG17/1) is 17/1 . 19/1 . 23/1 . 29/1 . 31/1 . 37/1...
* The basal subgroup of '''[[ed17|ed17/1]]''' - - - BSG17/1 - - - is 17/1 . 19/1 . 23/1 . 29/1 . 31/1 . 37/1...
* The basal subgroup of '''[[ed19|ed19/1]]''' (BSG19/1) is 19/1 . 23/1 . 29/1 . 31/1 . 37/1 . 41/1...
* The basal subgroup of '''[[ed19|ed19/1]]''' - - - BSG19/1 - - - is 19/1 . 23/1 . 29/1 . 31/1 . 37/1 . 41/1...
* The basal subgroup of '''[[ed23|ed23/1]]''' (BSG23/1) is 23/1 . 29/1 . 31/1 . 37/1 . 41/1 . 43/1...
* The basal subgroup of '''[[ed23|ed23/1]]''' - - - BSG23/1 - - - is 23/1 . 29/1 . 31/1 . 37/1 . 41/1 . 43/1...
* The basal subgroup of '''[[ed29|ed29/1]]''' (BSG29/1) is 29/1 . 31/1 . 37/1 . 41/1 . 43/1 . 47/1...
* The basal subgroup of '''[[ed29|ed29/1]]''' - - - BSG29/1 - - - is 29/1 . 31/1 . 37/1 . 41/1 . 43/1 . 47/1...
''and so on...''
''and so on...''


=== ed3/2, ed5/2, ed7/2, ed11/2... ===
=== ed3/2, ed5/2, ed7/2, ed11/2... ===
A.k.a. "[[half-prime subgroup]]s".
A.k.a. "[[half-prime subgroup]]s".


To find BSGn/2:
To find BSGn/2:
Line 37: Line 38:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed3/2]]''' (BSG3/2) is 3/2 . 5/2 . 7/2 . 11/2 . 13/2 . 17/2...
* The basal subgroup of '''[[ed3/2]]''' - - - BSG3/2 - - - is 3/2 . 5/2 . 7/2 . 11/2 . 13/2 . 17/2...
* The basal subgroup of '''[[ed5/2]]''' (BSG5/2) is 5/2 . 7/2 . 11/2 . 13/2 . 17/2 . 19/2...
* The basal subgroup of '''[[ed5/2]]''' - - - BSG5/2 - - - is 5/2 . 7/2 . 11/2 . 13/2 . 17/2 . 19/2...
* The basal subgroup of '''[[ed7/2]]''' (BSG7/2) is 7/2 . 11/2 . 13/2 . 17/2 . 19/2 . 23/2...
* The basal subgroup of '''[[ed7/2]]''' - - - BSG7/2 - - - is 7/2 . 11/2 . 13/2 . 17/2 . 19/2 . 23/2...
* The basal subgroup of '''[[ed11/2]]''' (BSG11/2) is 11/2 . 13/2 . 17/2 .  19/2 . 23/2 . 29/2...
* The basal subgroup of '''[[ed11/2]]''' - - - BSG11/2 - - - is 11/2 . 13/2 . 17/2 .  19/2 . 23/2 . 29/2...
* The basal subgroup of '''[[ed13/2]]''' (BSG13/2) is 13/2 . 17/2 . 19/2 . 23/2 . 29/2 . 31/2...
* The basal subgroup of '''[[ed13/2]]''' - - - BSG13/2 - - - is 13/2 . 17/2 . 19/2 . 23/2 . 29/2 . 31/2...
* The basal subgroup of '''[[ed17/2]]''' (BSG17/2) is 17/2 . 19/2 . 23/2 . 29/2 . 31/2 . 37/2...
* The basal subgroup of '''[[ed17/2]]''' - - - BSG17/2 - - - is 17/2 . 19/2 . 23/2 . 29/2 . 31/2 . 37/2...
* The basal subgroup of '''[[ed19/2]]''' (BSG19/2) is 19/2 . 23/2 . 29/2 . 31/2 . 37/2 . 41/2...
* The basal subgroup of '''[[ed19/2]]''' - - - BSG19/2 - - - is 19/2 . 23/2 . 29/2 . 31/2 . 37/2 . 41/2...
* The basal subgroup of '''[[ed23/2]]''' (BSG23/2) is 23/2 . 29/2 . 31/2 . 37/2 . 41/2 . 43/2...
* The basal subgroup of '''[[ed23/2]]''' - - - BSG23/2 - - - is 23/2 . 29/2 . 31/2 . 37/2 . 41/2 . 43/2...
* The basal subgroup of '''[[ed29/2]]''' (BSG29/2) is 29/2 . 31/2 . 37/2 . 41/2 . 43/2 . 47/2...
* The basal subgroup of '''[[ed29/2]]''' - - - BSG29/2 - - - is 29/2 . 31/2 . 37/2 . 41/2 . 43/2 . 47/2...
''and so on...''
''and so on...''


=== ed5/3, ed7/3, ed11/3, ed17/3... ===
=== ed5/3, ed7/3, ed11/3, ed17/3... ===
A.k.a. "[[half-prime subgroup|third-prime subgroups]]".
A.k.a. "[[half-prime subgroup|third-prime subgroups]]".


To find BSGn/3:
To find BSGn/3:
Line 58: Line 58:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed5/3]]''' (BSG5/3) is 5/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
* The basal subgroup of '''[[ed5/3]]''' - - - BSG5/3 - - - is 5/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
* The basal subgroup of '''[[ed7/3]]''' (BSG7/3) is 7/3 . 11/3 . 13/3 . 17/3 . 19/3 . 23/3...
* The basal subgroup of '''[[ed7/3]]''' - - - BSG7/3 - - - is 7/3 . 11/3 . 13/3 . 17/3 . 19/3 . 23/3...
* The basal subgroup of '''[[ed11/3]]''' (BSG11/3) is 11/3 . 13/3 . 17/3 .  19/3 . 23/3 . 29.3...
* The basal subgroup of '''[[ed11/3]]''' - - - BSG11/3 - - - is 11/3 . 13/3 . 17/3 .  19/3 . 23/3 . 29.3...
* The basal subgroup of '''[[ed13/3]]''' (BSG13/3) is 13/3 . 17/3 . 19/3 . 23/3 . 29/3 . 31/3...
* The basal subgroup of '''[[ed13/3]]''' - - - BSG13/3 - - - is 13/3 . 17/3 . 19/3 . 23/3 . 29/3 . 31/3...
* The basal subgroup of '''[[ed17/3]]''' (BSG17/3) is 17/3 . 19/3 . 23/3 . 29/3 . 31/3 . 37/3...
* The basal subgroup of '''[[ed17/3]]''' - - - BSG17/3 - - - is 17/3 . 19/3 . 23/3 . 29/3 . 31/3 . 37/3...
* The basal subgroup of '''[[ed19/3]]''' (BSG19/3) is 19/3 . 23/3 . 29/3 . 31/3 . 37/3 . 41/3...
* The basal subgroup of '''[[ed19/3]]''' - - - BSG19/3 - - - is 19/3 . 23/3 . 29/3 . 31/3 . 37/3 . 41/3...
* The basal subgroup of '''[[ed23/3]]''' (BSG23/3) is 23/3 . 29/3 . 31/3 . 37/3 . 41/3 . 43/3...
* The basal subgroup of '''[[ed23/3]]''' - - - BSG23/3 - - - is 23/3 . 29/3 . 31/3 . 37/3 . 41/3 . 43/3...
* The basal subgroup of '''[[ed29/3]]''' (BSG29/3) is 29/3 . 31/3 . 37/3 . 41/3 . 43/3 . 47/3...
* The basal subgroup of '''[[ed29/3]]''' - - - BSG29/3 - - - is 29/3 . 31/3 . 37/3 . 41/3 . 43/3 . 47/3...
''and so on...''
''and so on...''


=== ed5/4, ed7/4, ed11/4, ed17/4... ===
=== ed5/4, ed7/4, ed11/4, ed17/4... ===
A.k.a. "[[half-prime subgroup|quarter-prime subgroups]]".
A.k.a. "[[half-prime subgroup|quarter-prime subgroups]]".


To find BSGn/4:
To find BSGn/4:
Line 78: Line 77:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed5/4]]''' (BSG5/4) is 5/4 . 7/4 . 11/4 . 13/4 . 17/4 . 19/4...
* The basal subgroup of '''[[ed5/4]]''' - - - BSG5/4 - - - is 5/4 . 7/4 . 11/4 . 13/4 . 17/4 . 19/4...
* The basal subgroup of '''[[ed7/4]]''' (BSG7/4) is 7/4 . 11/4 . 13/4 . 17/4 . 19/4 . 23/4...
* The basal subgroup of '''[[ed7/4]]''' - - - BSG7/4 - - - is 7/4 . 11/4 . 13/4 . 17/4 . 19/4 . 23/4...
* The basal subgroup of '''[[ed11/4]]''' (BSG11/4) is 11/4 . 13/4 . 17/4 .  19/4 . 23/4 . 29/4...
* The basal subgroup of '''[[ed11/4]]''' - - - BSG11/4 - - - is 11/4 . 13/4 . 17/4 .  19/4 . 23/4 . 29/4...
* The basal subgroup of '''[[ed13/4]]''' (BSG13/4) is 13/4 . 17/4 . 19/4 . 23/4 . 29/4 . 31/4...
* The basal subgroup of '''[[ed13/4]]''' - - - BSG13/4 - - - is 13/4 . 17/4 . 19/4 . 23/4 . 29/4 . 31/4...
* The basal subgroup of '''[[ed17/4]]''' (BSG17/4) is 17/4 . 19/4 . 23/4 . 29/4 . 31/4 . 37/4...
* The basal subgroup of '''[[ed17/4]]''' - - - BSG17/4 - - - is 17/4 . 19/4 . 23/4 . 29/4 . 31/4 . 37/4...
* The basal subgroup of '''[[ed19/4]]''' (BSG19/4) is 19/4 . 23/4 . 29/4 . 31/4 . 37/4 . 41/4...
* The basal subgroup of '''[[ed19/4]]''' - - - BSG19/4 - - - is 19/4 . 23/4 . 29/4 . 31/4 . 37/4 . 41/4...
* The basal subgroup of '''[[ed23/4]]''' (BSG23/4) is 23/4 . 29/4 . 31/4 . 37/4 . 41/4 . 43/4...
* The basal subgroup of '''[[ed23/4]]''' - - - BSG23/4 - - - is 23/4 . 29/4 . 31/4 . 37/4 . 41/4 . 43/4...
* The basal subgroup of '''[[ed29/4]]''' (BSG29/4) is 29/4 . 31/4 . 37/4 . 41/4 . 43/4 . 47/4...
* The basal subgroup of '''[[ed29/4]]''' - - - BSG29/4 - - - is 29/4 . 31/4 . 37/4 . 41/4 . 43/4 . 47/4...
''and so on...''
''and so on...''


=== ed7/5, ed11/5, ed17/5, ed19/5... ===
=== ed7/5, ed11/5, ed17/5, ed19/5... ===
A.k.a. "[[half-prime subgroup|fifth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|fifth-prime subgroups]]".


To find BSGn/5:
To find BSGn/5:
Line 98: Line 96:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed7/5]]''' (BSG7/5) is 7/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
* The basal subgroup of '''[[ed7/5]]''' - - - BSG7/5 - - - is 7/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
* The basal subgroup of '''[[ed11/5]]''' (BSG11/5) is 11/5 . 13/5 . 17/5 .  19/5 . 23/5 . 29/5...
* The basal subgroup of '''[[ed11/5]]''' - - - BSG11/5 - - - is 11/5 . 13/5 . 17/5 .  19/5 . 23/5 . 29/5...
* The basal subgroup of '''[[ed13/5]]''' (BSG13/5) is 13/5 . 17/5 . 19/5 . 23/5 . 29/5 . 31/5...
* The basal subgroup of '''[[ed13/5]]''' - - - BSG13/5 - - - is 13/5 . 17/5 . 19/5 . 23/5 . 29/5 . 31/5...
* The basal subgroup of '''[[ed17/5]]''' (BSG17/5) is 17/5 . 19/5 . 23/5 . 29/5 . 31/5 . 37/5...
* The basal subgroup of '''[[ed17/5]]''' - - - BSG17/5 - - - is 17/5 . 19/5 . 23/5 . 29/5 . 31/5 . 37/5...
* The basal subgroup of '''[[ed19/5]]''' (BSG19/5) is 19/5 . 23/5 . 29/5 . 31/5 . 37/5 . 41/5...
* The basal subgroup of '''[[ed19/5]]''' - - - BSG19/5 - - - is 19/5 . 23/5 . 29/5 . 31/5 . 37/5 . 41/5...
* The basal subgroup of '''[[ed23/5]]''' (BSG23/5) is 23/5 . 29/5 . 31/5 . 37/5 . 41/5 . 43/5...
* The basal subgroup of '''[[ed23/5]]''' - - - BSG23/5 - - - is 23/5 . 29/5 . 31/5 . 37/5 . 41/5 . 43/5...
* The basal subgroup of '''[[ed29/5]]''' (BSG29/5) is 29/5 . 31/5 . 37/5 . 41/5 . 43/5 . 47/5...
* The basal subgroup of '''[[ed29/5]]''' - - - BSG29/5 - - - is 29/5 . 31/5 . 37/5 . 41/5 . 43/5 . 47/5...
''and so on...''
''and so on...''


=== ed7/6, ed11/6, ed17/6, ed19/6... ===
=== ed7/6, ed11/6, ed17/6, ed19/6... ===
A.k.a. "[[half-prime subgroup|sixth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|sixth-prime subgroups]]".


To find BSGn/6:
To find BSGn/6:
Line 117: Line 114:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed7/6]]''' (BSG7/6) is 7/6 . 11/6 . 13/6 . 17/6 . 19/6 . 23/6...
* The basal subgroup of '''[[ed7/6]]''' - - - BSG7/6 - - - is 7/6 . 11/6 . 13/6 . 17/6 . 19/6 . 23/6...
* The basal subgroup of '''[[ed11/6]]''' (BSG11/6) is 11/6 . 13/6 . 17/6 .  19/6 . 23/6 . 29/6...
* The basal subgroup of '''[[ed11/6]]''' - - - BSG11/6 - - - is 11/6 . 13/6 . 17/6 .  19/6 . 23/6 . 29/6...
* The basal subgroup of '''[[ed13/6]]''' (BSG13/6) is 13/6 . 17/6 . 19/6 . 23/6 . 29/6 . 31/6...
* The basal subgroup of '''[[ed13/6]]''' - - - BSG13/6 - - - is 13/6 . 17/6 . 19/6 . 23/6 . 29/6 . 31/6...
* The basal subgroup of '''[[ed17/6]]''' (BSG17/6) is 17/6 . 19/6 . 23/6 . 29/6 . 31/6 . 37/6...
* The basal subgroup of '''[[ed17/6]]''' - - - BSG17/6 - - - is 17/6 . 19/6 . 23/6 . 29/6 . 31/6 . 37/6...
* The basal subgroup of '''[[ed19/6]]''' (BSG19/6) is 19/6 . 23/6 . 29/6 . 31/6 . 37/6 . 41/6...
* The basal subgroup of '''[[ed19/6]]''' - - - BSG19/6 - - - is 19/6 . 23/6 . 29/6 . 31/6 . 37/6 . 41/6...
* The basal subgroup of '''[[ed23/6]]''' (BSG23/6) is 23/6 . 29/6 . 31/6 . 37/6 . 41/6 . 43/6...
* The basal subgroup of '''[[ed23/6]]''' - - - BSG23/6 - - - is 23/6 . 29/6 . 31/6 . 37/6 . 41/6 . 43/6...
* The basal subgroup of '''[[ed29/6]]''' (BSG29/6) is 29/6 . 31/6 . 37/6 . 41/6 . 43/6 . 47/6...
* The basal subgroup of '''[[ed29/6]]''' - - - BSG29/6 - - - is 29/6 . 31/6 . 37/6 . 41/6 . 43/6 . 47/6...
''and so on...''
''and so on...''


=== ed11/7, ed17/7, ed19/7, ed23/7... ===
=== ed11/7, ed17/7, ed19/7, ed23/7... ===
A.k.a. "[[half-prime subgroup|seventh-prime subgroups]]".
A.k.a. "[[half-prime subgroup|seventh-prime subgroups]]".


To find BSGn/7:
To find BSGn/7:
Line 136: Line 132:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed11/7]]''' (BSG11/7) is 11/7 . 13/7 . 17/7 .  19/7 . 23/7 . 29/7...
* The basal subgroup of '''[[ed11/7]]''' - - - BSG11/7 - - - is 11/7 . 13/7 . 17/7 .  19/7 . 23/7 . 29/7...
* The basal subgroup of '''[[ed13/7]]''' (BSG13/7) is 13/7 . 17/7 . 19/7 . 23/7 . 29/7 . 31/7...
* The basal subgroup of '''[[ed13/7]]''' - - - BSG13/7 - - - is 13/7 . 17/7 . 19/7 . 23/7 . 29/7 . 31/7...
* The basal subgroup of '''[[ed17/7]]''' (BSG17/7) is 17/7 . 19/7 . 23/7 . 29/7 . 31/7 . 37/7...
* The basal subgroup of '''[[ed17/7]]''' - - - BSG17/7 - - - is 17/7 . 19/7 . 23/7 . 29/7 . 31/7 . 37/7...
* The basal subgroup of '''[[ed19/7]]''' (BSG19/7) is 19/7 . 23/7 . 29/7 . 31/7 . 37/7 . 41/7...
* The basal subgroup of '''[[ed19/7]]''' - - - BSG19/7 - - - is 19/7 . 23/7 . 29/7 . 31/7 . 37/7 . 41/7...
* The basal subgroup of '''[[ed23/7]]''' (BSG23/7) is 23/7 . 29/7 . 31/7 . 37/7 . 41/7 . 43/7...
* The basal subgroup of '''[[ed23/7]]''' - - - BSG23/7 - - - is 23/7 . 29/7 . 31/7 . 37/7 . 41/7 . 43/7...
* The basal subgroup of '''[[ed29/7]]''' (BSG29/7) is 29/7 . 31/7 . 37/7 . 41/7 . 43/7 . 47/7...
* The basal subgroup of '''[[ed29/7]]''' - - - BSG29/7 - - - is 29/7 . 31/7 . 37/7 . 41/7 . 43/7 . 47/7...
''and so on...''
''and so on...''


=== ed11/8, ed17/8, ed19/8, ed23/8... ===
=== ed11/8, ed17/8, ed19/8, ed23/8... ===
A.k.a. "[[half-prime subgroup|eighth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|eighth-prime subgroups]]".


To find BSGn/8:
To find BSGn/8:
Line 154: Line 149:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed11/8]]''' (BSG11/8) is 11/8 . 13/8 . 17/8 .  19/8 . 23/8 . 29/8...
* The basal subgroup of '''[[ed11/8]]''' - - - BSG11/8 - - - is 11/8 . 13/8 . 17/8 .  19/8 . 23/8 . 29/8...
* The basal subgroup of '''[[ed13/8]]''' (BSG13/8) is 13/8 . 17/8 . 19/8 . 23/8 . 29/8 . 31/8...
* The basal subgroup of '''[[ed13/8]]''' - - - BSG13/8 - - - is 13/8 . 17/8 . 19/8 . 23/8 . 29/8 . 31/8...
* The basal subgroup of '''[[ed17/8]]''' (BSG17/8) is 17/8 . 19/8 . 23/8 . 29/8 . 31/8 . 37/8...
* The basal subgroup of '''[[ed17/8]]''' - - - BSG17/8 - - - is 17/8 . 19/8 . 23/8 . 29/8 . 31/8 . 37/8...
* The basal subgroup of '''[[ed19/8]]''' (BSG19/8) is 19/8 . 23/8 . 29/8 . 31/8 . 37/8 . 41/8...
* The basal subgroup of '''[[ed19/8]]''' - - - BSG19/8 - - - is 19/8 . 23/8 . 29/8 . 31/8 . 37/8 . 41/8...
* The basal subgroup of '''[[ed23/8]]''' (BSG23/8) is 23/8 . 29/8 . 31/8 . 37/8 . 41/8 . 43/8...
* The basal subgroup of '''[[ed23/8]]''' - - - BSG23/8 - - - is 23/8 . 29/8 . 31/8 . 37/8 . 41/8 . 43/8...
* The basal subgroup of '''[[ed29/8]]''' (BSG29/8) is 29/8 . 31/8 . 37/8 . 41/8 . 43/8 . 47/8...
* The basal subgroup of '''[[ed29/8]]''' - - - BSG29/8 - - - is 29/8 . 31/8 . 37/8 . 41/8 . 43/8 . 47/8...
''and so on...''
''and so on...''


=== ed11/9, ed17/9, ed19/9, ed23/9... ===
=== ed11/9, ed17/9, ed19/9, ed23/9... ===
A.k.a. "[[half-prime subgroup|ninth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|ninth-prime subgroups]]".


To find BSGn/9:
To find BSGn/9:
Line 172: Line 166:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed11/9]]''' (BSG11/9) is 11/9 . 13/9 . 17/9 .  19/9 . 23/9 . 29/9...
* The basal subgroup of '''[[ed11/9]]''' - - - BSG11/9 - - - is 11/9 . 13/9 . 17/9 .  19/9 . 23/9 . 29/9...
* The basal subgroup of '''[[ed13/9]]''' (BSG13/9) is 13/9 . 17/9 . 19/9 . 23/9 . 29/9 . 31/9...
* The basal subgroup of '''[[ed13/9]]''' - - - BSG13/9 - - - is 13/9 . 17/9 . 19/9 . 23/9 . 29/9 . 31/9...
* The basal subgroup of '''[[ed17/9]]''' (BSG17/9) is 17/9 . 19/9 . 23/9 . 29/9 . 31/9 . 37/9...
* The basal subgroup of '''[[ed17/9]]''' - - - BSG17/9 - - - is 17/9 . 19/9 . 23/9 . 29/9 . 31/9 . 37/9...
* The basal subgroup of '''[[ed19/9]]''' (BSG19/9) is 19/9 . 23/9 . 29/9 . 31/9 . 37/9 . 41/9...
* The basal subgroup of '''[[ed19/9]]''' - - - BSG19/9 - - - is 19/9 . 23/9 . 29/9 . 31/9 . 37/9 . 41/9...
* The basal subgroup of '''[[ed23/9]]''' (BSG23/9) is 23/9 . 29/9 . 31/9 . 37/9 . 41/9 . 43/9...
* The basal subgroup of '''[[ed23/9]]''' - - - BSG23/9 - - - is 23/9 . 29/9 . 31/9 . 37/9 . 41/9 . 43/9...
* The basal subgroup of '''[[ed29/9]]''' (BSG29/9) is 29/9 . 31/9 . 37/9 . 41/9 . 43/9 . 47/9...
* The basal subgroup of '''[[ed29/9]]''' - - - BSG29/9 - - - is 29/9 . 31/9 . 37/9 . 41/9 . 43/9 . 47/9...
''and so on...''
''and so on...''


=== ed11/10, ed17/10, ed19/10, ed23/10... ===
=== ed11/10, ed17/10, ed19/10, ed23/10... ===
A.k.a. "[[half-prime subgroup|tenth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|tenth-prime subgroups]]".


To find BSGn/10:
To find BSGn/10:
Line 190: Line 183:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed11/10]]''' (BSG11/10) is 11/10 . 13/10 . 17/10 .  19/10 . 23/10 . 29/10...
* The basal subgroup of '''[[ed11/10]]''' - - - BSG11/10 - - - is 11/10 . 13/10 . 17/10 .  19/10 . 23/10 . 29/10...
* The basal subgroup of '''[[ed13/10]]''' (BSG13/10) is 13/10 . 17/10 . 19/10 . 23/10 . 29/10 . 31/10...
* The basal subgroup of '''[[ed13/10]]''' - - - BSG13/10 - - - is 13/10 . 17/10 . 19/10 . 23/10 . 29/10 . 31/10...
* The basal subgroup of '''[[ed17/10]]''' (BSG17/10) is 17/10 . 19/10 . 23/10 . 29/10 . 31/10 . 37/10...
* The basal subgroup of '''[[ed17/10]]''' - - - BSG17/10 - - - is 17/10 . 19/10 . 23/10 . 29/10 . 31/10 . 37/10...
* The basal subgroup of '''[[ed19/10]]''' (BSG19/10) is 19/10 . 23/10 . 29/10 . 31/10 . 37/10 . 41/10...
* The basal subgroup of '''[[ed19/10]]''' - - - BSG19/10 - - - is 19/10 . 23/10 . 29/10 . 31/10 . 37/10 . 41/10...
* The basal subgroup of '''[[ed23/10]]''' (BSG23/10) is 23/10 . 29/10 . 31/10 . 37/10 . 41/10 . 43/10...
* The basal subgroup of '''[[ed23/10]]''' - - - BSG23/10 - - - is 23/10 . 29/10 . 31/10 . 37/10 . 41/10 . 43/10...
* The basal subgroup of '''[[ed29/10]]''' (BSG29/10) is 29/10 . 31/10 . 37/10 . 41/10 . 43/10 . 47/10...
* The basal subgroup of '''[[ed29/10]]''' - - - BSG29/10 - - - is 29/10 . 31/10 . 37/10 . 41/10 . 43/10 . 47/10...
''and so on...''
''and so on...''


=== ed13/11, ed17/11, ed19/11, ed23/11... ===
=== ed13/11, ed17/11, ed19/11, ed23/11... ===
A.k.a. "[[half-prime subgroup|eleventh-prime subgroups]]".
A.k.a. "[[half-prime subgroup|eleventh-prime subgroups]]".


To find BSGn/11:
To find BSGn/11:
Line 208: Line 200:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed13/11]]''' (BSG13/11) is 13/11 . 17/11 . 19/11 . 23/11 . 29/11 . 31/11...
* The basal subgroup of '''[[ed13/11]]''' - - - BSG13/11 - - - is 13/11 . 17/11 . 19/11 . 23/11 . 29/11 . 31/11...
* The basal subgroup of '''[[ed17/11]]''' (BSG17/11) is 17/11 . 19/11 . 23/11 . 29/11 . 31/11 . 37/11...
* The basal subgroup of '''[[ed17/11]]''' - - - BSG17/11 - - - is 17/11 . 19/11 . 23/11 . 29/11 . 31/11 . 37/11...
* The basal subgroup of '''[[ed19/11]]''' (BSG19/11) is 19/11 . 23/11 . 29/11 . 31/11 . 37/11 . 41/11...
* The basal subgroup of '''[[ed19/11]]''' - - - BSG19/11 - - - is 19/11 . 23/11 . 29/11 . 31/11 . 37/11 . 41/11...
* The basal subgroup of '''[[ed23/11]]''' (BSG23/11) is 23/11 . 29/11 . 31/11 . 37/11 . 41/11 . 43/11...
* The basal subgroup of '''[[ed23/11]]''' - - - BSG23/11 - - - is 23/11 . 29/11 . 31/11 . 37/11 . 41/11 . 43/11...
* The basal subgroup of '''[[ed29/11]]''' (BSG29/11) is 29/11 . 31/11 . 37/11 . 41/11 . 43/11 . 47/11...
* The basal subgroup of '''[[ed29/11]]''' - - - BSG29/11 - - - is 29/11 . 31/11 . 37/11 . 41/11 . 43/11 . 47/11...
''and so on...''
''and so on...''


=== ed13/12, ed17/12, ed29/12, ed23/12... ===
=== ed13/12, ed17/12, ed29/12, ed23/12... ===
A.k.a. "[[half-prime subgroup|twelfth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|twelfth-prime subgroups]]".


To find BSGn/12:
To find BSGn/12:
Line 225: Line 216:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed13/12]]''' (BSG13/12) is 13/12 . 17/12 . 19/12 . 23/12 . 29/12 . 31/12...
* The basal subgroup of '''[[ed13/12]]''' - - - BSG13/12 - - - is 13/12 . 17/12 . 19/12 . 23/12 . 29/12 . 31/12...
* The basal subgroup of '''[[ed17/12]]''' (BSG17/12) is 17/12 . 19/12 . 23/12 . 29/12 . 31/12 . 37/12...
* The basal subgroup of '''[[ed17/12]]''' - - - BSG17/12 - - - is 17/12 . 19/12 . 23/12 . 29/12 . 31/12 . 37/12...
* The basal subgroup of '''[[ed19/12]]''' (BSG19/12) is 19/12 . 23/12 . 29/12 . 31/12 . 37/12 . 41/12...
* The basal subgroup of '''[[ed19/12]]''' - - - BSG19/12 - - - is 19/12 . 23/12 . 29/12 . 31/12 . 37/12 . 41/12...
* The basal subgroup of '''[[ed23/12]]''' (BSG23/12) is 23/12 . 29/12 . 31/12 . 37/12 . 41/12 . 43/12...
* The basal subgroup of '''[[ed23/12]]''' - - - BSG23/12 - - - is 23/12 . 29/12 . 31/12 . 37/12 . 41/12 . 43/12...
* The basal subgroup of '''[[ed29/12]]''' (BSG29/12) is 29/12 . 31/12 . 37/12 . 41/12 . 43/12 . 47/12...
* The basal subgroup of '''[[ed29/12]]''' - - - BSG29/12 - - - is 29/12 . 31/12 . 37/12 . 41/12 . 43/12 . 47/12...
''and so on...''
''and so on...''


=== ed17/13, ed19/13, ed23/13, ed29/13... ===
=== ed17/13, ed19/13, ed23/13, ed29/13... ===
A.k.a. "[[half-prime subgroup|thirteenth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|thirteenth-prime subgroups]]".


To find BSGn/13:
To find BSGn/13:
Line 242: Line 232:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed17/13]]''' (BSG17/13) is 17/13 . 19/13 . 23/13 . 29/13 . 31/13 . 37/13...
* The basal subgroup of '''[[ed17/13]]''' - - - BSG17/13 - - - is 17/13 . 19/13 . 23/13 . 29/13 . 31/13 . 37/13...
* The basal subgroup of '''[[ed19/13]]''' (BSG19/13) is 19/13 . 23/13 . 29/13 . 31/13 . 37/13 . 41/13...
* The basal subgroup of '''[[ed19/13]]''' - - - BSG19/13 - - - is 19/13 . 23/13 . 29/13 . 31/13 . 37/13 . 41/13...
* The basal subgroup of '''[[ed23/13]]''' (BSG23/13) is 23/13 . 29/13 . 31/13 . 37/13 . 41/13 . 43/13...
* The basal subgroup of '''[[ed23/13]]''' - - - BSG23/13 - - - is 23/13 . 29/13 . 31/13 . 37/13 . 41/13 . 43/13...
* The basal subgroup of '''[[ed29/13]]''' (BSG29/13) is 29/13 . 31/13 . 37/13 . 41/13 . 43/13 . 47/13...
* The basal subgroup of '''[[ed29/13]]''' - - - BSG29/13 - - - is 29/13 . 31/13 . 37/13 . 41/13 . 43/13 . 47/13...
''and so on...''
''and so on...''


=== ed17/14, ed19/14, ed23/14, ed29/14... ===
=== ed17/14, ed19/14, ed23/14, ed29/14... ===
A.k.a. "[[half-prime subgroup|fourteenth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|fourteenth-prime subgroups]]".


To find BSGn/14:
To find BSGn/14:
Line 258: Line 247:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed17/14]]''' (BSG17/14) is 17/14 . 19/14 . 23/14 . 29/14 . 31/14 . 37/14...
* The basal subgroup of '''[[ed17/14]]''' - - - BSG17/14 - - - is 17/14 . 19/14 . 23/14 . 29/14 . 31/14 . 37/14...
* The basal subgroup of '''[[ed19/14]]''' (BSG19/14) is 19/14 . 23/14 . 29/14 . 31/14 . 37/14 . 41/14...
* The basal subgroup of '''[[ed19/14]]''' - - - BSG19/14 - - - is 19/14 . 23/14 . 29/14 . 31/14 . 37/14 . 41/14...
* The basal subgroup of '''[[ed23/14]]''' (BSG23/14) is 23/14 . 29/14 . 31/14 . 37/14 . 41/14 . 43/14...
* The basal subgroup of '''[[ed23/14]]''' - - - BSG23/14 - - - is 23/14 . 29/14 . 31/14 . 37/14 . 41/14 . 43/14...
* The basal subgroup of '''[[ed29/14]]''' (BSG29/14) is 29/14 . 31/14 . 37/14 . 41/14 . 43/14 . 47/14...
* The basal subgroup of '''[[ed29/14]]''' - - - BSG29/14 - - - is 29/14 . 31/14 . 37/14 . 41/14 . 43/14 . 47/14...
''and so on...''
''and so on...''


=== ed17/15, ed19/15, ed23/15, ed29/15... ===
=== ed17/15, ed19/15, ed23/15, ed29/15... ===
A.k.a. "[[half-prime subgroup|fifteenth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|fifteenth-prime subgroups]]".


To find BSGn/15:
To find BSGn/15:
Line 274: Line 262:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed17/15]]''' (BSG17/15) is 17/15 . 19/15 . 23/15 . 29/15 . 31/15 . 37/15...
* The basal subgroup of '''[[ed17/15]]''' - - - BSG17/15 - - - is 17/15 . 19/15 . 23/15 . 29/15 . 31/15 . 37/15...
* The basal subgroup of '''[[ed19/15]]''' (BSG19/15) is 19/15 . 23/15 . 29/15 . 31/15 . 37/15 . 41/15...
* The basal subgroup of '''[[ed19/15]]''' - - - BSG19/15 - - - is 19/15 . 23/15 . 29/15 . 31/15 . 37/15 . 41/15...
* The basal subgroup of '''[[ed23/15]]''' (BSG23/15) is 23/15 . 29/15 . 31/15 . 37/15 . 41/15 . 43/15...
* The basal subgroup of '''[[ed23/15]]''' - - - BSG23/15 - - - is 23/15 . 29/15 . 31/15 . 37/15 . 41/15 . 43/15...
* The basal subgroup of '''[[ed29/15]]''' (BSG29/15) is 29/15 . 31/15 . 37/15 . 41/15 . 43/15 . 47/15...
* The basal subgroup of '''[[ed29/15]]''' - - - BSG29/15 - - - is 29/15 . 31/15 . 37/15 . 41/15 . 43/15 . 47/15...
''and so on...''
''and so on...''


Line 290: Line 278:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed4|ed4/1]]''' (BSG4/1) is 4/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed4|ed4/1]]''' - - - BSG4/1 - - - is 4/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed6|ed6/1]]''' (BSG6/1) is 6/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed6|ed6/1]]''' - - - BSG6/1 - - - is 6/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed8|ed8/1]]''' (BSG8/1) is 8/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed8|ed8/1]]''' - - - BSG8/1 - - - is 8/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed9|ed9/1]]''' (BSG9/1) is 9/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed9|ed9/1]]''' - - - BSG9/1 - - - is 9/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed10|ed10/1]]''' (BSG10/1) is 10/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
* The basal subgroup of '''[[ed10|ed10/1]]''' - - - BSG10/1 - - - is 10/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
* The basal subgroup of '''[[ed12|ed12/1]]''' (BSG12/1) is 12/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed12|ed12/1]]''' - - - BSG12/1 - - - is 12/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed14|ed14/1]]''' (BSG14/1) is 14/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
* The basal subgroup of '''[[ed14|ed14/1]]''' - - - BSG14/1 - - - is 14/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
* The basal subgroup of '''[[ed15|ed15/1]]''' (BSG15/1) is 15/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
* The basal subgroup of '''[[ed15|ed15/1]]''' - - - BSG15/1 - - - is 15/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
* The basal subgroup of '''[[ed16|ed16/1]]''' (BSG16/1) is 16/1 . 3/1 . 5/1 . 7/1 . 11/1 . 13/1...
* The basal subgroup of '''[[ed16|ed16/1]]''' - - - BSG16/1 - - - is 16/1 . 3/1 . 5/1 . 7/1 . 11/1 . 13/1...
* The basal subgroup of '''[[ed18|ed18/1]]''' (BSG18/1) is 18/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed18|ed18/1]]''' - - - BSG18/1 - - - is 18/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed20|ed20/1]]''' (BSG20/1) is 20/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
* The basal subgroup of '''[[ed20|ed20/1]]''' - - - BSG20/1 - - - is 20/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
* The basal subgroup of '''[[ed21|ed21/1]]''' (BSG21/1) is 21/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
* The basal subgroup of '''[[ed21|ed21/1]]''' - - - BSG21/1 - - - is 21/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
* The basal subgroup of '''[[ed22|ed22/1]]''' (BSG22/1) is 22/1 . 13/1 . 17/1 . 19/1 . 23/1 . 29/1...
* The basal subgroup of '''[[ed22|ed22/1]]''' - - - BSG22/1 - - - is 22/1 . 13/1 . 17/1 . 19/1 . 23/1 . 29/1...
* The basal subgroup of '''[[ed24|ed24/1]]''' (BSG24/1) is 24/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed24|ed24/1]]''' - - - BSG24/1 - - - is 24/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed25|ed25/1]]''' (BSG25/1) is 25/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
* The basal subgroup of '''[[ed25|ed25/1]]''' - - - BSG25/1 - - - is 25/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
* The basal subgroup of '''[[ed26|ed26/1]]''' (BSG26/1) is 26/1 . 17/1 . 19/1 . 23/1 . 29/1 . 31/1...
* The basal subgroup of '''[[ed26|ed26/1]]''' - - - BSG26/1 - - - is 26/1 . 17/1 . 19/1 . 23/1 . 29/1 . 31/1...
* The basal subgroup of '''[[ed27|ed27/1]]''' (BSG27/1) is 27/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed27|ed27/1]]''' - - - BSG27/1 - - - is 27/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
* The basal subgroup of '''[[ed28|ed28/1]]''' (BSG28/1) is 28/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
* The basal subgroup of '''[[ed28|ed28/1]]''' - - - BSG28/1 - - - is 28/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
* The basal subgroup of '''[[ed30|ed30/1]]''' (BSG30/1) is 30/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1 ...
* The basal subgroup of '''[[ed30|ed30/1]]''' - - - BSG30/1 - - - is 30/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1 ...
''and so on...''
''and so on...''


Line 320: Line 308:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/2
# Add n/2 to the start of the subgroup
# Add n/2 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 325: Line 314:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed9/2]]''' (BSG9/2) is 9/2 . 5/2 . 7/2 . 11/2 . 13/2 . 17/2...
* The basal subgroup of '''[[ed9/2]]''' - - - BSG9/2 - - - is 9/2 . 5/2 . 7/2 . 11/2 . 13/2 . 17/2...
* The basal subgroup of '''[[ed15/2]]''' (BSG15/2) is 15/2 . 7/2 . 11/2 . 13/2 . 17/2 . 19/2...
* The basal subgroup of '''[[ed15/2]]''' - - - BSG15/2 - - - is 15/2 . 7/2 . 11/2 . 13/2 . 17/2 . 19/2...
* The basal subgroup of '''[[ed21/2]]''' (BSG21/2) is 21/2 . 11/2 . 13/2 . 17/2 . 19/2 . 23/2...
* The basal subgroup of '''[[ed21/2]]''' - - - BSG21/2 - - - is 21/2 . 11/2 . 13/2 . 17/2 . 19/2 . 23/2...
* The basal subgroup of '''[[ed25/2]]''' (BSG25/2) is 25/2 . 7/2 . 11/2 . 13/2 . 17/2 . 19/2...
* The basal subgroup of '''[[ed25/2]]''' - - - BSG25/2 - - - is 25/2 . 7/2 . 11/2 . 13/2 . 17/2 . 19/2...
* The basal subgroup of '''[[ed27/2]]''' (BSG27/2) is 27/2 . 5/2 . 7/2 . 11/2 . 13/2 . 17/2...
* The basal subgroup of '''[[ed27/2]]''' - - - BSG27/2 - - - is 27/2 . 5/2 . 7/2 . 11/2 . 13/2 . 17/2...
''and so on...''
''and so on...''


=== ed4/3, ed8/3, ed10/3, ed14/3... ===
=== ed4/3, ed8/3, ed10/3, ed14/3... ===
A.k.a. "[[half-prime subgroup|third-prime subgroups]]".
A.k.a. "[[half-prime subgroup|third-prime subgroups]]".


To find BSGn/3:
To find BSGn/3:
Line 341: Line 329:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/3
# Add n/3 to the start of the subgroup
# Add n/3 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 346: Line 335:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed4/3]]''' (BSG4/3) is 4/3 . 5/3 . 7/3 . 11/3 . 13/3 . 17/3...
* The basal subgroup of '''[[ed4/3]]''' - - - BSG4/3 - - - is 4/3 . 5/3 . 7/3 . 11/3 . 13/3 . 17/3...
* The basal subgroup of '''[[ed8/3]]''' (BSG8/3) is 8/3 . 5/3 . 7/3 . 11/3 . 13/3 . 17/3...
* The basal subgroup of '''[[ed8/3]]''' - - - BSG8/3 - - - is 8/3 . 5/3 . 7/3 . 11/3 . 13/3 . 17/3...
* The basal subgroup of '''[[ed10/3]]''' (BSG10/3) is 10/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
* The basal subgroup of '''[[ed10/3]]''' - - - BSG10/3 - - - is 10/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
* The basal subgroup of '''[[ed14/3]]''' (BSG14/3) is 14/3 . 11/3 . 13/3 . 17/3 . 19/3 . 23/3...
* The basal subgroup of '''[[ed14/3]]''' - - - BSG14/3 - - - is 14/3 . 11/3 . 13/3 . 17/3 . 19/3 . 23/3...
* The basal subgroup of '''[[ed16/3]]''' (BSG16/3) is 16/3 . 3/3 . 5/3 . 7/3 . 11/3 . 13/3...
* The basal subgroup of '''[[ed16/3]]''' - - - BSG16/3 - - - is 16/3 . 3/3 . 5/3 . 7/3 . 11/3 . 13/3...
* The basal subgroup of '''[[ed18/3]]''' (BSG18/3) is 18/3 . 5/3 . 7/3 . 11/3 . 13/3 . 17/3...
* The basal subgroup of '''[[ed18/3]]''' - - - BSG18/3 - - - is 18/3 . 5/3 . 7/3 . 11/3 . 13/3 . 17/3...
* The basal subgroup of '''[[ed20/3]]''' (BSG20/3) is 20/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
* The basal subgroup of '''[[ed20/3]]''' - - - BSG20/3 - - - is 20/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
* The basal subgroup of '''[[ed22/3]]''' (BSG22/3) is 22/3 . 13/3 . 17/3 . 19/3 . 23/3 . 29/3...
* The basal subgroup of '''[[ed22/3]]''' - - - BSG22/3 - - - is 22/3 . 13/3 . 17/3 . 19/3 . 23/3 . 29/3...
* The basal subgroup of '''[[ed25/3]]''' (BSG25/3) is 25/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
* The basal subgroup of '''[[ed25/3]]''' - - - BSG25/3 - - - is 25/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
* The basal subgroup of '''[[ed26/3]]''' (BSG26/3) is 26/3 . 17/3 . 19/3 . 23/3 . 29/3 . 31/3...
* The basal subgroup of '''[[ed26/3]]''' - - - BSG26/3 - - - is 26/3 . 17/3 . 19/3 . 23/3 . 29/3 . 31/3...
* The basal subgroup of '''[[ed28/3]]''' (BSG28/3) is 28/3 . 11/3 . 13/3 . 17/3 . 19/3 . 23/3...
* The basal subgroup of '''[[ed28/3]]''' - - - BSG28/3 - - - is 28/3 . 11/3 . 13/3 . 17/3 . 19/3 . 23/3...
''and so on...''
''and so on...''


=== ed9/4, ed15/4, ed21/4, ed25/4... ===
=== ed9/4, ed15/4, ed21/4, ed25/4... ===
A.k.a. "[[half-prime subgroup|quarter-prime subgroups]]".
A.k.a. "[[half-prime subgroup|quarter-prime subgroups]]".


To find BSGn/4:
To find BSGn/4:
Line 368: Line 356:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/4
# Add n/4 to the start of the subgroup
# Add n/4 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 373: Line 362:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed9/4]]''' (BSG9/4) is 9/4 . 5/4 . 7/4 . 11/4 . 13/4 . 17/4...
* The basal subgroup of '''[[ed9/4]]''' - - - BSG9/4 - - - is 9/4 . 5/4 . 7/4 . 11/4 . 13/4 . 17/4...
* The basal subgroup of '''[[ed15/4]]''' (BSG15/4) is 15/4 . 7/4 . 11/4 . 13/4 . 17/4 . 19/4...
* The basal subgroup of '''[[ed15/4]]''' - - - BSG15/4 - - - is 15/4 . 7/4 . 11/4 . 13/4 . 17/4 . 19/4...
* The basal subgroup of '''[[ed21/4]]''' (BSG21/4) is 21/4 . 11/4 . 13/4 . 17/4 . 19/4 . 23/4...
* The basal subgroup of '''[[ed21/4]]''' - - - BSG21/4 - - - is 21/4 . 11/4 . 13/4 . 17/4 . 19/4 . 23/4...
* The basal subgroup of '''[[ed25/4]]''' (BSG25/4) is 25/4 . 7/4 . 11/4 . 13/4 . 17/4 . 19/4...
* The basal subgroup of '''[[ed25/4]]''' - - - BSG25/4 - - - is 25/4 . 7/4 . 11/4 . 13/4 . 17/4 . 19/4...
* The basal subgroup of '''[[ed27/4]]''' (BSG27/4) is 27/4 . 5/4 . 7/4 . 11/4 . 13/4 . 17/4...
* The basal subgroup of '''[[ed27/4]]''' - - - BSG27/4 - - - is 27/4 . 5/4 . 7/4 . 11/4 . 13/4 . 17/4...
''and so on...''
''and so on...''


=== ed6/5, ed8/5, ed9/5, ed12/5... ===
=== ed6/5, ed8/5, ed9/5, ed12/5... ===
A.k.a. "[[half-prime subgroup|fifth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|fifth-prime subgroups]]".


To find BSGn/5:
To find BSGn/5:
Line 389: Line 377:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/5
# Add n/5 to the start of the subgroup
# Add n/5 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 394: Line 383:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed6/5]]''' (BSG6/5) is 6/5 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
* The basal subgroup of '''[[ed6/5]]''' - - - BSG6/5 - - - is 6/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
* The basal subgroup of '''[[ed8/5]]''' (BSG8/5) is 8/5 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
* The basal subgroup of '''[[ed8/5]]''' - - - BSG8/5 - - - is 8/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
* The basal subgroup of '''[[ed9/5]]''' (BSG9/5) is 9/5 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
* The basal subgroup of '''[[ed9/5]]''' - - - BSG9/5 - - - is 9/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
* The basal subgroup of '''[[ed12/5]]''' (BSG12/5) is 12/5 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
* The basal subgroup of '''[[ed12/5]]''' - - - BSG12/5 - - - is 12/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
* The basal subgroup of '''[[ed14/5]]''' (BSG14/5) is 14/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
* The basal subgroup of '''[[ed14/5]]''' - - - BSG14/5 - - - is 14/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
* The basal subgroup of '''[[ed16/5]]''' (BSG16/5) is 16/5 . 3/5 . 5/5 . 7/5 . 11/5 . 13/5...
* The basal subgroup of '''[[ed16/5]]''' - - - BSG16/5 - - - is 16/5 . 3/5 . 7/5 . 11/5 . 13/5 . 17/5...
* The basal subgroup of '''[[ed18/5]]''' (BSG18/5) is 18/5 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
* The basal subgroup of '''[[ed18/5]]''' - - - BSG18/5 - - - is 18/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
* The basal subgroup of '''[[ed21/5]]''' (BSG21/5) is 21/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
* The basal subgroup of '''[[ed21/5]]''' - - - BSG21/5 - - - is 21/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
* The basal subgroup of '''[[ed22/5]]''' (BSG22/5) is 22/5 . 13/5 . 17/5 . 19/5 . 23/5 . 29/5...
* The basal subgroup of '''[[ed22/5]]''' - - - BSG22/5 - - - is 22/5 . 13/5 . 17/5 . 19/5 . 23/5 . 29/5...
* The basal subgroup of '''[[ed24/5]]''' (BSG24/5) is 24/5 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
* The basal subgroup of '''[[ed24/5]]''' - - - BSG24/5 - - - is 24/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
* The basal subgroup of '''[[ed26/5]]''' (BSG26/5) is 26/5 . 17/5 . 19/5 . 23/5 . 29/5 . 31/5...
* The basal subgroup of '''[[ed26/5]]''' - - - BSG26/5 - - - is 26/5 . 17/5 . 19/5 . 23/5 . 29/5 . 31/5...
* The basal subgroup of '''[[ed27/5]]''' (BSG27/5) is 27/5 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
* The basal subgroup of '''[[ed27/5]]''' - - - BSG27/5 - - - is 27/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
* The basal subgroup of '''[[ed28/5]]''' (BSG28/5) is 28/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
* The basal subgroup of '''[[ed28/5]]''' - - - BSG28/5 - - - is 28/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
''and so on...''
''and so on...''


=== ed35/6, ed55/6, ed65/6, ed77/6... ===
=== ed35/6, ed55/6, ed65/6, ed77/6... ===
A.k.a. "[[half-prime subgroup|sixth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|sixth-prime subgroups]]".


To find BSGn/6:
To find BSGn/6:
Line 418: Line 406:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/6
# Add n/6 to the start of the subgroup
# Add n/6 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 423: Line 412:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed35/6]]''' (BSG35/6) is 35/6 . 11/6 . 13/6 . 17/6 . 19/6 . 23/6...
* The basal subgroup of '''[[ed35/6]]''' - - - BSG35/6 - - - is 35/6 . 11/6 . 13/6 . 17/6 . 19/6 . 23/6...
* The basal subgroup of '''[[ed55/6]]''' (BSG55/6) is 55/6 . 13/6 . 17/6 . 19/6 . 23/6 . 29/6...
* The basal subgroup of '''[[ed55/6]]''' - - - BSG55/6 - - - is 55/6 . 13/6 . 17/6 . 19/6 . 23/6 . 29/6...
* The basal subgroup of '''[[ed65/6]]''' (BSG65/6) is 65/6 . 17/6 . 19/6 . 23/6 . 29/6 . 31/6...
* The basal subgroup of '''[[ed65/6]]''' - - - BSG65/6 - - - is 65/6 . 17/6 . 19/6 . 23/6 . 29/6 . 31/6...
* The basal subgroup of '''[[ed77/6]]''' (BSG77/6) is 77/6 . 13/6 . 17/6 . 19/6 . 23/6 . 29/6...
* The basal subgroup of '''[[ed77/6]]''' - - - BSG77/6 - - - is 77/6 . 13/6 . 17/6 . 19/6 . 23/6 . 29/6...
''and so on...''
''and so on...''


=== ed8/7, ed9/7, ed10/7, ed12/7... ===
=== ed8/7, ed9/7, ed10/7, ed12/7... ===
A.k.a. "[[half-prime subgroup|seventh-prime subgroups]]".
A.k.a. "[[half-prime subgroup|seventh-prime subgroups]]".


To find BSGn/7:
To find BSGn/7:
Line 438: Line 426:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/7
# Add n/7 to the start of the subgroup
# Add n/7 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 443: Line 432:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed8/7]]''' (BSG8/7) is 8/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed8/7]]''' - - - BSG8/7 - - - is 8/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed9/7]]''' (BSG9/7) is 9/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed9/7]]''' - - - BSG9/7 - - - is 9/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed10/7]]''' (BSG10/7) is 10/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
* The basal subgroup of '''[[ed10/7]]''' - - - BSG10/7 - - - is 10/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
* The basal subgroup of '''[[ed12/7]]''' (BSG12/7) is 12/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed12/7]]''' - - - BSG12/7 - - - is 12/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed15/7]]''' (BSG15/7) is 15/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
* The basal subgroup of '''[[ed15/7]]''' - - - BSG15/7 - - - is 15/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
* The basal subgroup of '''[[ed16/7]]''' (BSG16/7) is 16/7 . 12/7 . 10/7 . 11/7 . 13/7 . 17/7...
* The basal subgroup of '''[[ed16/7]]''' - - - BSG16/7 - - - is 16/7 . 12/7 . 10/7 . 11/7 . 13/7 . 17/7...
* The basal subgroup of '''[[ed18/7]]''' (BSG18/7) is 18/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed18/7]]''' - - - BSG18/7 - - - is 18/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed20/7]]''' (BSG20/7) is 20/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
* The basal subgroup of '''[[ed20/7]]''' - - - BSG20/7 - - - is 20/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
* The basal subgroup of '''[[ed22/7]]''' (BSG22/7) is 22/7 . 13/7 . 17/7 . 19/7 . 23/7 . 29/7...
* The basal subgroup of '''[[ed22/7]]''' - - - BSG22/7 - - - is 22/7 . 13/7 . 17/7 . 19/7 . 23/7 . 29/7...
* The basal subgroup of '''[[ed24/7]]''' (BSG24/7) is 24/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed24/7]]''' - - - BSG24/7 - - - is 24/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed25/7]]''' (BSG25/7) is 25/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
* The basal subgroup of '''[[ed25/7]]''' - - - BSG25/7 - - - is 25/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
* The basal subgroup of '''[[ed26/7]]''' (BSG26/7) is 26/7 . 17/7 . 19/7 . 23/7 . 29/7 . 31/7...
* The basal subgroup of '''[[ed26/7]]''' - - - BSG26/7 - - - is 26/7 . 17/7 . 19/7 . 23/7 . 29/7 . 31/7...
* The basal subgroup of '''[[ed27/7]]''' (BSG27/7) is 27/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed27/7]]''' - - - BSG27/7 - - - is 27/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
* The basal subgroup of '''[[ed30/7]]''' (BSG30/7) is 30/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
* The basal subgroup of '''[[ed30/7]]''' - - - BSG30/7 - - - is 30/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
''and so on...''
''and so on...''


=== ed9/8, ed15/8, ed21/8, ed25/8... ===
=== ed9/8, ed15/8, ed21/8, ed25/8... ===
A.k.a. "[[half-prime subgroup|eighth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|eighth-prime subgroups]]".


To find BSGn/8:
To find BSGn/8:
Line 468: Line 456:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/8
# Add n/8 to the start of the subgroup
# Add n/8 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 473: Line 462:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed9/8]]''' (BSG9/8) is 9/8 . 10/8 (ie 5/4) . 7/8 . 11/8 . 13/8 . 17/8...
* The basal subgroup of '''[[ed9/8]]''' - - - BSG9/8 - - - is 9/8 . 10/8 (ie 5/4) . 7/8 . 11/8 . 13/8 . 17/8...
* The basal subgroup of '''[[ed15/8]]''' (BSG15/8) is 15/8 . 14/8 (ie 7/4) . 11/8 . 13/8 . 17/8 . 19/8...
* The basal subgroup of '''[[ed15/8]]''' - - - BSG15/8 - - - is 15/8 . 14/8 (ie 7/4) . 11/8 . 13/8 . 17/8 . 19/8...
* The basal subgroup of '''[[ed21/8]]''' (BSG21/8) is 21/8 . 11/8 . 13/8 . 17/8 . 19/8 . 23/8..
* The basal subgroup of '''[[ed21/8]]''' - - - BSG21/8 - - - is 21/8 . 11/8 . 13/8 . 17/8 . 19/8 . 23/8..
* The basal subgroup of '''[[ed25/8]]''' (BSG25/8) is 25/8 . 14/8 (ie 7/4) . 11/8 . 13/8 . 17/8 . 19/8...
* The basal subgroup of '''[[ed25/8]]''' - - - BSG25/8 - - - is 25/8 . 14/8 (ie 7/4) . 11/8 . 13/8 . 17/8 . 19/8...
* The basal subgroup of '''[[ed27/8]]''' (BSG27/8) is 27/8 . 10/8 (ie 5/4) . 7/8 . 11/8 . 13/8 . 17/8...
* The basal subgroup of '''[[ed27/8]]''' - - - BSG27/8 - - - is 27/8 . 10/8 (ie 5/4) . 7/8 . 11/8 . 13/8 . 17/8...
''and so on...''
''and so on...''


=== ed10/9, ed14/9, ed16/9, ed20/9... ===
=== ed10/9, ed14/9, ed16/9, ed20/9... ===
A.k.a. "[[half-prime subgroup|ninth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|ninth-prime subgroups]]".


To find BSGn/9:
To find BSGn/9:
Line 489: Line 477:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/9
# Add n/9 to the start of the subgroup
# Add n/9 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 494: Line 483:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed10/9]]''' (BSG10/9) is 10/9 . 14/9 . 11/9 . 13/9 . 17/9 . 19/9...
* The basal subgroup of '''[[ed10/9]]''' - - - BSG10/9 - - - is 10/9 . 14/9 . 11/9 . 13/9 . 17/9 . 19/9...
* The basal subgroup of '''[[ed14/9]]''' (BSG14/9) is 14/9 . 11/9 . 13/9 . 17/9 . 19/9 . 23/9...
* The basal subgroup of '''[[ed14/9]]''' - - - BSG14/9 - - - is 14/9 . 11/9 . 13/9 . 17/9 . 19/9 . 23/9...
* The basal subgroup of '''[[ed16/9]]''' (BSG16/9) is 16/9 . 10/9 . 7/9 . 11/9 . 13/9 . 17/9...
* The basal subgroup of '''[[ed16/9]]''' - - - BSG16/9 - - - is 16/9 . 10/9 . 7/9 . 11/9 . 13/9 . 17/9...
* The basal subgroup of '''[[ed20/9]]''' (BSG20/9) is 20/9 . 14/9 . 11/9 . 13/9 . 17/9 . 19/9...
* The basal subgroup of '''[[ed20/9]]''' - - - BSG20/9 - - - is 20/9 . 14/9 . 11/9 . 13/9 . 17/9 . 19/9...
* The basal subgroup of '''[[ed22/9]]''' (BSG22/9) is 22/9 . 13/9 . 17/9 . 19/9 . 23/9 . 29/9...
* The basal subgroup of '''[[ed22/9]]''' - - - BSG22/9 - - - is 22/9 . 13/9 . 17/9 . 19/9 . 23/9 . 29/9...
* The basal subgroup of '''[[ed25/9]]''' (BSG25/9) is 25/9 . 14/9 . 11/9 . 13/9 . 17/9 . 19/9...
* The basal subgroup of '''[[ed25/9]]''' - - - BSG25/9 - - - is 25/9 . 14/9 . 11/9 . 13/9 . 17/9 . 19/9...
* The basal subgroup of '''[[ed26/9]]''' (BSG26/9) is 26/9 . 17/9 . 19/9 . 23/9 . 29/9 . 31/9...
* The basal subgroup of '''[[ed26/9]]''' - - - BSG26/9 - - - is 26/9 . 17/9 . 19/9 . 23/9 . 29/9 . 31/9...
* The basal subgroup of '''[[ed28/9]]''' (BSG28/9) is 28/9 . 11/9 . 13/9 . 17/9 . 19/9 . 23/9...
* The basal subgroup of '''[[ed28/9]]''' - - - BSG28/9 - - - is 28/9 . 11/9 . 13/9 . 17/9 . 19/9 . 23/9...
''and so on...''
''and so on...''


=== ed21/10, ed27/10, ed33/10, ed39/10... ===
=== ed21/10, ed27/10, ed33/10, ed39/10... ===
A.k.a. "[[half-prime subgroup|tenth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|tenth-prime subgroups]]".


To find BSGn/10:
To find BSGn/10:
Line 513: Line 501:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/10
# Add n/10 to the start of the subgroup
# Add n/10 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 518: Line 507:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed21/10]]''' (BSG21/10) is 21/10 . 11/10 . 13/10 . 17/10 . 19/10 . 23/10...
* The basal subgroup of '''[[ed21/10]]''' - - - BSG21/10 - - - is 21/10 . 11/10 . 13/10 . 17/10 . 19/10 . 23/10...
* The basal subgroup of '''[[ed27/10]]''' (BSG27/10) is 27/10 . 14/10 (ie 7/5) . 11/10 . 13/10 . 17/10 . 19/10...
* The basal subgroup of '''[[ed27/10]]''' - - - BSG27/10 - - - is 27/10 . 14/10 (ie 7/5) . 11/10 . 13/10 . 17/10 . 19/10...
* The basal subgroup of '''[[ed33/10]]''' (BSG33/10) is 33/10 . 13/10 . 17/10 . 19/10 . 23/10 . 29/10...
* The basal subgroup of '''[[ed33/10]]''' - - - BSG33/10 - - - is 33/10 . 13/10 . 17/10 . 19/10 . 23/10 . 29/10...
* The basal subgroup of '''[[ed39/10]]''' (BSG39/10) is 39/10 . 17/10 . 19/10 . 23/10 . 29/10 . 31/10...
* The basal subgroup of '''[[ed39/10]]''' - - - BSG39/10 - - - is 39/10 . 17/10 . 19/10 . 23/10 . 29/10 . 31/10...
''and so on...''
''and so on...''


=== ed12/11, ed14/11, ed15/11, ed16/11... ===
=== ed12/11, ed14/11, ed15/11, ed16/11... ===
A.k.a. "[[half-prime subgroup|elevent-prime subgroups]]".
A.k.a. "[[half-prime subgroup|eleventh-prime subgroups]]".
 


To find BSGn/11:
To find BSGn/11:
Line 533: Line 521:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/11
# Add n/11 to the start of the subgroup
# Add n/11 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 538: Line 527:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed12/11]]''' (BSG12/11) is 12/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
* The basal subgroup of '''[[ed12/11]]''' - - - BSG12/11 - - - is 12/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
* The basal subgroup of '''[[ed14/11]]''' (BSG14/11) is 14/11 . 13/11 . 17/11 . 19/11 . 23/11 . 29/11...
* The basal subgroup of '''[[ed14/11]]''' - - - BSG14/11 - - - is 14/11 . 13/11 . 17/11 . 19/11 . 23/11 . 29/11...
* The basal subgroup of '''[[ed15/11]]''' (BSG15/11) is 15/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
* The basal subgroup of '''[[ed15/11]]''' - - - BSG15/11 - - - is 15/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
* The basal subgroup of '''[[ed16/11]]''' (BSG16/11) is 16/11 . 12/11 . 20/11 . 14/11 . 13/11 . 17/11...
* The basal subgroup of '''[[ed16/11]]''' - - - BSG16/11 - - - is 16/11 . 12/11 . 20/11 . 14/11 . 13/11 . 17/11...
* The basal subgroup of '''[[ed18/11]]''' (BSG18/11) is 18/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
* The basal subgroup of '''[[ed18/11]]''' - - - BSG18/11 - - - is 18/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
* The basal subgroup of '''[[ed20/11]]''' (BSG20/11) is 20/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
* The basal subgroup of '''[[ed20/11]]''' - - - BSG20/11 - - - is 20/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
* The basal subgroup of '''[[ed21/11]]''' (BSG21/11) is 21/11 . 13/11 . 17/11 . 19/11 . 23/11 . 29/11...
* The basal subgroup of '''[[ed21/11]]''' - - - BSG21/11 - - - is 21/11 . 13/11 . 17/11 . 19/11 . 23/11 . 29/11...
* The basal subgroup of '''[[ed24/11]]''' (BSG24/11) is 24/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
* The basal subgroup of '''[[ed24/11]]''' - - - BSG24/11 - - - is 24/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
* The basal subgroup of '''[[ed25/11]]''' (BSG25/11) is 25/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
* The basal subgroup of '''[[ed25/11]]''' - - - BSG25/11 - - - is 25/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
* The basal subgroup of '''[[ed26/11]]''' (BSG26/11) is 26/11 . 17/11 . 19/11 . 23/11 . 29/11 . 31/11...
* The basal subgroup of '''[[ed26/11]]''' - - - BSG26/11 - - - is 26/11 . 17/11 . 19/11 . 23/11 . 29/11 . 31/11...
* The basal subgroup of '''[[ed27/11]]''' (BSG27/11) is 27/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
* The basal subgroup of '''[[ed27/11]]''' - - - BSG27/11 - - - is 27/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
* The basal subgroup of '''[[ed28/11]]''' (BSG28/11) is 28/11 . 13/11 . 17/11 . 19/11 . 23/11 . 29/11...
* The basal subgroup of '''[[ed28/11]]''' - - - BSG28/11 - - - is 28/11 . 13/11 . 17/11 . 19/11 . 23/11 . 29/11...
* The basal subgroup of '''[[ed30/11]]''' (BSG30/11) is 30/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
* The basal subgroup of '''[[ed30/11]]''' - - - BSG30/11 - - - is 30/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
''and so on...''
''and so on...''


=== ed35/12, ed55/12, ed65/12, ed77/12... ===
=== ed35/12, ed55/12, ed65/12, ed77/12... ===
A.k.a. "[[half-prime subgroup|twelfth-prime subgroups]]".
A.k.a. "[[half-prime subgroup|twelfth-prime subgroups]]".


To find BSGn/12:
To find BSGn/12:
Line 562: Line 550:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/12
# Add n/12 to the start of the subgroup
# Add n/12 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 567: Line 556:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed35/12]]''' (BSG35/12) is 35/12 . 22/12 (ie 11/6) . 13/12 . 17/12 . 19/12 . 23/12...
* The basal subgroup of '''[[ed35/12]]''' - - - BSG35/12 - - - is 35/12 . 22/12 (ie 11/6) . 13/12 . 17/12 . 19/12 . 23/12...
* The basal subgroup of '''[[ed55/12]]''' (BSG55/12) is 55/12 . 13/12 . 17/12 . 19/12 . 23/12 . 29/12...
* The basal subgroup of '''[[ed55/12]]''' - - - BSG55/12 - - - is 55/12 . 13/12 . 17/12 . 19/12 . 23/12 . 29/12...
* The basal subgroup of '''[[ed65/12]]''' (BSG65/12) is 65/12 . 17/12 . 19/12 . 23/12 . 29/12 . 31/12...
* The basal subgroup of '''[[ed65/12]]''' - - - BSG65/12 - - - is 65/12 . 17/12 . 19/12 . 23/12 . 29/12 . 31/12...
* The basal subgroup of '''[[ed77/12]]''' (BSG77/12) is 77/12 . 13/12 . 17/12 . 19/12 . 23/12 . 29/12...
* The basal subgroup of '''[[ed77/12]]''' - - - BSG77/12 - - - is 77/12 . 13/12 . 17/12 . 19/12 . 23/12 . 29/12...
''and so on...''
''and so on...''


Line 579: Line 568:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/13
# Add n/13 to the start of the subgroup
# Add n/13 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 584: Line 574:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed14/13]]''' (BSG14/13) is 14/13 . 22/13 . 17/13 . 19/13 . 23/13 . 29/13...
* The basal subgroup of '''[[ed14/13]]''' - - - BSG14/13 - - - is 14/13 . 22/13 . 17/13 . 19/13 . 23/13 . 29/13...
* The basal subgroup of '''[[ed15/13]]''' (BSG15/13) is 15/13 . 14/13 . 22/13 . 13/13 . 17/13 . 19/13...
* The basal subgroup of '''[[ed15/13]]''' - - - BSG15/13 - - - is 15/13 . 14/13 . 22/13 . 13/13 . 17/13 . 19/13...
* The basal subgroup of '''[[ed16/13]]''' (BSG16/13) is 16/13 . 24/13 . 20/13 . 14/13 . 22/13 . 17/13...
* The basal subgroup of '''[[ed16/13]]''' - - - BSG16/13 - - - is 16/13 . 24/13 . 20/13 . 14/13 . 22/13 . 17/13...
* The basal subgroup of '''[[ed18/13]]''' (BSG18/13) is 18/13 . 20/13 . 14/13 . 22/13 . 17/13 . 19/13...
* The basal subgroup of '''[[ed18/13]]''' - - - BSG18/13 - - - is 18/13 . 20/13 . 14/13 . 22/13 . 17/13 . 19/13...
* The basal subgroup of '''[[ed20/13]]''' (BSG20/13) is 20/13 . 14/13 . 22/13 . 17/13 . 19/13 . 23/13...
* The basal subgroup of '''[[ed20/13]]''' - - - BSG20/13 - - - is 20/13 . 14/13 . 22/13 . 17/13 . 19/13 . 23/13...
* The basal subgroup of '''[[ed21/13]]''' (BSG21/13) is 21/13 . 22/13 . 17/13 . 19/13 . 23/13 . 29/13...
* The basal subgroup of '''[[ed21/13]]''' - - - BSG21/13 - - - is 21/13 . 22/13 . 17/13 . 19/13 . 23/13 . 29/13...
* The basal subgroup of '''[[ed22/13]]''' (BSG22/13) is 22/13 . 17/13 . 19/13 . 23/13 . 29/13 . 31/13...
* The basal subgroup of '''[[ed22/13]]''' - - - BSG22/13 - - - is 22/13 . 17/13 . 19/13 . 23/13 . 29/13 . 31/13...
* The basal subgroup of '''[[ed24/13]]''' (BSG24/13) is 24/13 . 20/13 . 14/13 . 22/13 . 17/13 . 19/13...
* The basal subgroup of '''[[ed24/13]]''' - - - BSG24/13 - - - is 24/13 . 20/13 . 14/13 . 22/13 . 17/13 . 19/13...
* The basal subgroup of '''[[ed25/13]]''' (BSG25/13) is 25/13 . 14/13 . 22/13 . 17/13 . 19/13 . 23/13...
* The basal subgroup of '''[[ed25/13]]''' - - - BSG25/13 - - - is 25/13 . 14/13 . 22/13 . 17/13 . 19/13 . 23/13...
* The basal subgroup of '''[[ed27/13]]''' (BSG27/13) is 27/13 . 20/13 . 14/13 . 22/13 . 17/13 . 19/13...
* The basal subgroup of '''[[ed27/13]]''' - - - BSG27/13 - - - is 27/13 . 20/13 . 14/13 . 22/13 . 17/13 . 19/13...
* The basal subgroup of '''[[ed28/13]]''' (BSG28/13) is 28/13 . 22/13 . 17/13 . 19/13 . 23/13 . 29/13...
* The basal subgroup of '''[[ed28/13]]''' - - - BSG28/13 - - - is 28/13 . 22/13 . 17/13 . 19/13 . 23/13 . 29/13...
* The basal subgroup of '''[[ed30/13]]''' (BSG30/13) is 30/13 . 14/13 . 22/13 . 17/13 . 19/13 . 23/13...
* The basal subgroup of '''[[ed30/13]]''' - - - BSG30/13 - - - is 30/13 . 14/13 . 22/13 . 17/13 . 19/13 . 23/13...
''and so on...''
''and so on...''


Line 604: Line 594:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/14
# Add n/14 to the start of the subgroup
# Add n/14 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 609: Line 600:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed15/14]]''' (BSG15/14) is 15/14 . 22/14 (ie 11/7) . 26/14 (ie 13/7) . 17/14 . 19/14 . 23/14...
* The basal subgroup of '''[[ed15/14]]''' - - - BSG15/14 - - - is 15/14 . 22/14 (ie 11/7) . 26/14 (ie 13/7) . 17/14 . 19/14 . 23/14...
* The basal subgroup of '''[[ed25/14]]''' (BSG25/14) is 25/14 . 11/14 . 26/14 (ie 13/7) . 17/14 . 19/14 . 23/14...
* The basal subgroup of '''[[ed25/14]]''' - - - BSG25/14 - - - is 25/14 . 11/14 . 26/14 (ie 13/7) . 17/14 . 19/14 . 23/14...
* The basal subgroup of '''[[ed27/14]]''' (BSG27/14) is 27/14 . 20/14 (ie 10/7) . 11/14 . 13/14 . 17/14 . 19/14...
* The basal subgroup of '''[[ed27/14]]''' - - - BSG27/14 - - - is 27/14 . 20/14 (ie 10/7) . 11/14 . 13/14 . 17/14 . 19/14...
* The basal subgroup of '''[[ed33/14]]''' (BSG33/14) is 33/14 . 26/14 (ie 13/7) . 17/14 . 19/14 . 23/14 . 29/14...
* The basal subgroup of '''[[ed33/14]]''' - - - BSG33/14 - - - is 33/14 . 26/14 (ie 13/7) . 17/14 . 19/14 . 23/14 . 29/14...
''and so on...''
''and so on...''


Line 621: Line 612:
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is a prime factor of n
# Remove all instances of m/1 where m is less than n's largest prime factor
# Remove all instances of m/1 where m is less than n's largest prime factor
# Replace all instances of m/1 with m/15
# Add n/15 to the start of the subgroup
# Add n/15 to the start of the subgroup
# In all instances a/b where a<b, double a
# In all instances a/b where a<b, double a
Line 626: Line 618:


'''List:'''
'''List:'''
* The basal subgroup of '''[[ed16/15]]''' (BSG16/15) is 16/15 . 28/15 . 22/15 . 26/15...
* The basal subgroup of '''[[ed16/15]]''' - - - BSG16/15 - - - is 16/15 . 28/15 . 22/15 . 26/15...
* The basal subgroup of '''[[ed22/15]]''' (BSG22/15) is 22/15 . 26/15 . 17/15 . 19/15 . 23/15 . 29/15...
* The basal subgroup of '''[[ed22/15]]''' - - - BSG22/15 - - - is 22/15 . 26/15 . 17/15 . 19/15 . 23/15 . 29/15...
* The basal subgroup of '''[[ed26/15]]''' (BSG26/15) is 26/15 . 17/15 . 19/15 . 23/15 . 29/15 . 31/15...
* The basal subgroup of '''[[ed26/15]]''' - - - BSG26/15 - - - is 26/15 . 17/15 . 19/15 . 23/15 . 29/15 . 31/15...
* The basal subgroup of '''[[ed28/15]]''' (BSG28/15) is 28/15 . 22/15 . 26/15 . 17/15 . 19/15 . 23/15...
* The basal subgroup of '''[[ed28/15]]''' - - - BSG28/15 - - - is 28/15 . 22/15 . 26/15 . 17/15 . 19/15 . 23/15...
''and so on...''
''and so on...''


Latest revision as of 02:56, 7 May 2025

An basal subgroup[idiosyncratic term] is a subgroup which has a unique affiliation with one specific family of equal tunings.

It is built on top of the concept of a half-prime subgroup, so it is recommended to study half-prime subgroups first before exploring basal subgroups.

The basal subgroup for equal divisions of n/m is named BSGn/m[idiosyncratic term]. This makes it easy to refer to a specific BSG.


Example:

The basal subgroup of ed2/1 - - - BSG2/1 - - - is 2.3.5.7.11.13... a.k.a. 2/1 . 3/1 . 5/1 . 7/1 . 11/1 . 13/1...

Prime basal subgroups

ed3, ed5, ed7, ed11...

To find BSGn/1:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n

List:

  • The basal subgroup of ed3/1 - - - BSG3/1 - - - is 3/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
  • The basal subgroup of ed5/1 - - - BSG5/1 - - - is 5/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
  • The basal subgroup of ed7/1 - - - BSG7/1 - - - is 7/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
  • The basal subgroup of ed11/1 - - - BSG11/1 - - - is 11/1 . 13/1 . 17/1 . 19/1 . 23/1 . 29/1...
  • The basal subgroup of ed13/1 - - - BSG13/1 - - - is 13/1 . 17/1 . 19/1 . 23/1 . 29/1 . 31/1...
  • The basal subgroup of ed17/1 - - - BSG17/1 - - - is 17/1 . 19/1 . 23/1 . 29/1 . 31/1 . 37/1...
  • The basal subgroup of ed19/1 - - - BSG19/1 - - - is 19/1 . 23/1 . 29/1 . 31/1 . 37/1 . 41/1...
  • The basal subgroup of ed23/1 - - - BSG23/1 - - - is 23/1 . 29/1 . 31/1 . 37/1 . 41/1 . 43/1...
  • The basal subgroup of ed29/1 - - - BSG29/1 - - - is 29/1 . 31/1 . 37/1 . 41/1 . 43/1 . 47/1...

and so on...

ed3/2, ed5/2, ed7/2, ed11/2...

A.k.a. "half-prime subgroups".

To find BSGn/2:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/2

List:

  • The basal subgroup of ed3/2 - - - BSG3/2 - - - is 3/2 . 5/2 . 7/2 . 11/2 . 13/2 . 17/2...
  • The basal subgroup of ed5/2 - - - BSG5/2 - - - is 5/2 . 7/2 . 11/2 . 13/2 . 17/2 . 19/2...
  • The basal subgroup of ed7/2 - - - BSG7/2 - - - is 7/2 . 11/2 . 13/2 . 17/2 . 19/2 . 23/2...
  • The basal subgroup of ed11/2 - - - BSG11/2 - - - is 11/2 . 13/2 . 17/2 . 19/2 . 23/2 . 29/2...
  • The basal subgroup of ed13/2 - - - BSG13/2 - - - is 13/2 . 17/2 . 19/2 . 23/2 . 29/2 . 31/2...
  • The basal subgroup of ed17/2 - - - BSG17/2 - - - is 17/2 . 19/2 . 23/2 . 29/2 . 31/2 . 37/2...
  • The basal subgroup of ed19/2 - - - BSG19/2 - - - is 19/2 . 23/2 . 29/2 . 31/2 . 37/2 . 41/2...
  • The basal subgroup of ed23/2 - - - BSG23/2 - - - is 23/2 . 29/2 . 31/2 . 37/2 . 41/2 . 43/2...
  • The basal subgroup of ed29/2 - - - BSG29/2 - - - is 29/2 . 31/2 . 37/2 . 41/2 . 43/2 . 47/2...

and so on...

ed5/3, ed7/3, ed11/3, ed17/3...

A.k.a. "third-prime subgroups".

To find BSGn/3:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/3

List:

  • The basal subgroup of ed5/3 - - - BSG5/3 - - - is 5/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
  • The basal subgroup of ed7/3 - - - BSG7/3 - - - is 7/3 . 11/3 . 13/3 . 17/3 . 19/3 . 23/3...
  • The basal subgroup of ed11/3 - - - BSG11/3 - - - is 11/3 . 13/3 . 17/3 . 19/3 . 23/3 . 29.3...
  • The basal subgroup of ed13/3 - - - BSG13/3 - - - is 13/3 . 17/3 . 19/3 . 23/3 . 29/3 . 31/3...
  • The basal subgroup of ed17/3 - - - BSG17/3 - - - is 17/3 . 19/3 . 23/3 . 29/3 . 31/3 . 37/3...
  • The basal subgroup of ed19/3 - - - BSG19/3 - - - is 19/3 . 23/3 . 29/3 . 31/3 . 37/3 . 41/3...
  • The basal subgroup of ed23/3 - - - BSG23/3 - - - is 23/3 . 29/3 . 31/3 . 37/3 . 41/3 . 43/3...
  • The basal subgroup of ed29/3 - - - BSG29/3 - - - is 29/3 . 31/3 . 37/3 . 41/3 . 43/3 . 47/3...

and so on...

ed5/4, ed7/4, ed11/4, ed17/4...

A.k.a. "quarter-prime subgroups".

To find BSGn/4:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/4

List:

  • The basal subgroup of ed5/4 - - - BSG5/4 - - - is 5/4 . 7/4 . 11/4 . 13/4 . 17/4 . 19/4...
  • The basal subgroup of ed7/4 - - - BSG7/4 - - - is 7/4 . 11/4 . 13/4 . 17/4 . 19/4 . 23/4...
  • The basal subgroup of ed11/4 - - - BSG11/4 - - - is 11/4 . 13/4 . 17/4 . 19/4 . 23/4 . 29/4...
  • The basal subgroup of ed13/4 - - - BSG13/4 - - - is 13/4 . 17/4 . 19/4 . 23/4 . 29/4 . 31/4...
  • The basal subgroup of ed17/4 - - - BSG17/4 - - - is 17/4 . 19/4 . 23/4 . 29/4 . 31/4 . 37/4...
  • The basal subgroup of ed19/4 - - - BSG19/4 - - - is 19/4 . 23/4 . 29/4 . 31/4 . 37/4 . 41/4...
  • The basal subgroup of ed23/4 - - - BSG23/4 - - - is 23/4 . 29/4 . 31/4 . 37/4 . 41/4 . 43/4...
  • The basal subgroup of ed29/4 - - - BSG29/4 - - - is 29/4 . 31/4 . 37/4 . 41/4 . 43/4 . 47/4...

and so on...

ed7/5, ed11/5, ed17/5, ed19/5...

A.k.a. "fifth-prime subgroups".

To find BSGn/5:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/5

List:

  • The basal subgroup of ed7/5 - - - BSG7/5 - - - is 7/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
  • The basal subgroup of ed11/5 - - - BSG11/5 - - - is 11/5 . 13/5 . 17/5 . 19/5 . 23/5 . 29/5...
  • The basal subgroup of ed13/5 - - - BSG13/5 - - - is 13/5 . 17/5 . 19/5 . 23/5 . 29/5 . 31/5...
  • The basal subgroup of ed17/5 - - - BSG17/5 - - - is 17/5 . 19/5 . 23/5 . 29/5 . 31/5 . 37/5...
  • The basal subgroup of ed19/5 - - - BSG19/5 - - - is 19/5 . 23/5 . 29/5 . 31/5 . 37/5 . 41/5...
  • The basal subgroup of ed23/5 - - - BSG23/5 - - - is 23/5 . 29/5 . 31/5 . 37/5 . 41/5 . 43/5...
  • The basal subgroup of ed29/5 - - - BSG29/5 - - - is 29/5 . 31/5 . 37/5 . 41/5 . 43/5 . 47/5...

and so on...

ed7/6, ed11/6, ed17/6, ed19/6...

A.k.a. "sixth-prime subgroups".

To find BSGn/6:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/6

List:

  • The basal subgroup of ed7/6 - - - BSG7/6 - - - is 7/6 . 11/6 . 13/6 . 17/6 . 19/6 . 23/6...
  • The basal subgroup of ed11/6 - - - BSG11/6 - - - is 11/6 . 13/6 . 17/6 . 19/6 . 23/6 . 29/6...
  • The basal subgroup of ed13/6 - - - BSG13/6 - - - is 13/6 . 17/6 . 19/6 . 23/6 . 29/6 . 31/6...
  • The basal subgroup of ed17/6 - - - BSG17/6 - - - is 17/6 . 19/6 . 23/6 . 29/6 . 31/6 . 37/6...
  • The basal subgroup of ed19/6 - - - BSG19/6 - - - is 19/6 . 23/6 . 29/6 . 31/6 . 37/6 . 41/6...
  • The basal subgroup of ed23/6 - - - BSG23/6 - - - is 23/6 . 29/6 . 31/6 . 37/6 . 41/6 . 43/6...
  • The basal subgroup of ed29/6 - - - BSG29/6 - - - is 29/6 . 31/6 . 37/6 . 41/6 . 43/6 . 47/6...

and so on...

ed11/7, ed17/7, ed19/7, ed23/7...

A.k.a. "seventh-prime subgroups".

To find BSGn/7:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/7

List:

  • The basal subgroup of ed11/7 - - - BSG11/7 - - - is 11/7 . 13/7 . 17/7 . 19/7 . 23/7 . 29/7...
  • The basal subgroup of ed13/7 - - - BSG13/7 - - - is 13/7 . 17/7 . 19/7 . 23/7 . 29/7 . 31/7...
  • The basal subgroup of ed17/7 - - - BSG17/7 - - - is 17/7 . 19/7 . 23/7 . 29/7 . 31/7 . 37/7...
  • The basal subgroup of ed19/7 - - - BSG19/7 - - - is 19/7 . 23/7 . 29/7 . 31/7 . 37/7 . 41/7...
  • The basal subgroup of ed23/7 - - - BSG23/7 - - - is 23/7 . 29/7 . 31/7 . 37/7 . 41/7 . 43/7...
  • The basal subgroup of ed29/7 - - - BSG29/7 - - - is 29/7 . 31/7 . 37/7 . 41/7 . 43/7 . 47/7...

and so on...

ed11/8, ed17/8, ed19/8, ed23/8...

A.k.a. "eighth-prime subgroups".

To find BSGn/8:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/8

List:

  • The basal subgroup of ed11/8 - - - BSG11/8 - - - is 11/8 . 13/8 . 17/8 . 19/8 . 23/8 . 29/8...
  • The basal subgroup of ed13/8 - - - BSG13/8 - - - is 13/8 . 17/8 . 19/8 . 23/8 . 29/8 . 31/8...
  • The basal subgroup of ed17/8 - - - BSG17/8 - - - is 17/8 . 19/8 . 23/8 . 29/8 . 31/8 . 37/8...
  • The basal subgroup of ed19/8 - - - BSG19/8 - - - is 19/8 . 23/8 . 29/8 . 31/8 . 37/8 . 41/8...
  • The basal subgroup of ed23/8 - - - BSG23/8 - - - is 23/8 . 29/8 . 31/8 . 37/8 . 41/8 . 43/8...
  • The basal subgroup of ed29/8 - - - BSG29/8 - - - is 29/8 . 31/8 . 37/8 . 41/8 . 43/8 . 47/8...

and so on...

ed11/9, ed17/9, ed19/9, ed23/9...

A.k.a. "ninth-prime subgroups".

To find BSGn/9:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/9

List:

  • The basal subgroup of ed11/9 - - - BSG11/9 - - - is 11/9 . 13/9 . 17/9 . 19/9 . 23/9 . 29/9...
  • The basal subgroup of ed13/9 - - - BSG13/9 - - - is 13/9 . 17/9 . 19/9 . 23/9 . 29/9 . 31/9...
  • The basal subgroup of ed17/9 - - - BSG17/9 - - - is 17/9 . 19/9 . 23/9 . 29/9 . 31/9 . 37/9...
  • The basal subgroup of ed19/9 - - - BSG19/9 - - - is 19/9 . 23/9 . 29/9 . 31/9 . 37/9 . 41/9...
  • The basal subgroup of ed23/9 - - - BSG23/9 - - - is 23/9 . 29/9 . 31/9 . 37/9 . 41/9 . 43/9...
  • The basal subgroup of ed29/9 - - - BSG29/9 - - - is 29/9 . 31/9 . 37/9 . 41/9 . 43/9 . 47/9...

and so on...

ed11/10, ed17/10, ed19/10, ed23/10...

A.k.a. "tenth-prime subgroups".

To find BSGn/10:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/10

List:

  • The basal subgroup of ed11/10 - - - BSG11/10 - - - is 11/10 . 13/10 . 17/10 . 19/10 . 23/10 . 29/10...
  • The basal subgroup of ed13/10 - - - BSG13/10 - - - is 13/10 . 17/10 . 19/10 . 23/10 . 29/10 . 31/10...
  • The basal subgroup of ed17/10 - - - BSG17/10 - - - is 17/10 . 19/10 . 23/10 . 29/10 . 31/10 . 37/10...
  • The basal subgroup of ed19/10 - - - BSG19/10 - - - is 19/10 . 23/10 . 29/10 . 31/10 . 37/10 . 41/10...
  • The basal subgroup of ed23/10 - - - BSG23/10 - - - is 23/10 . 29/10 . 31/10 . 37/10 . 41/10 . 43/10...
  • The basal subgroup of ed29/10 - - - BSG29/10 - - - is 29/10 . 31/10 . 37/10 . 41/10 . 43/10 . 47/10...

and so on...

ed13/11, ed17/11, ed19/11, ed23/11...

A.k.a. "eleventh-prime subgroups".

To find BSGn/11:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/11

List:

  • The basal subgroup of ed13/11 - - - BSG13/11 - - - is 13/11 . 17/11 . 19/11 . 23/11 . 29/11 . 31/11...
  • The basal subgroup of ed17/11 - - - BSG17/11 - - - is 17/11 . 19/11 . 23/11 . 29/11 . 31/11 . 37/11...
  • The basal subgroup of ed19/11 - - - BSG19/11 - - - is 19/11 . 23/11 . 29/11 . 31/11 . 37/11 . 41/11...
  • The basal subgroup of ed23/11 - - - BSG23/11 - - - is 23/11 . 29/11 . 31/11 . 37/11 . 41/11 . 43/11...
  • The basal subgroup of ed29/11 - - - BSG29/11 - - - is 29/11 . 31/11 . 37/11 . 41/11 . 43/11 . 47/11...

and so on...

ed13/12, ed17/12, ed29/12, ed23/12...

A.k.a. "twelfth-prime subgroups".

To find BSGn/12:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/12

List:

  • The basal subgroup of ed13/12 - - - BSG13/12 - - - is 13/12 . 17/12 . 19/12 . 23/12 . 29/12 . 31/12...
  • The basal subgroup of ed17/12 - - - BSG17/12 - - - is 17/12 . 19/12 . 23/12 . 29/12 . 31/12 . 37/12...
  • The basal subgroup of ed19/12 - - - BSG19/12 - - - is 19/12 . 23/12 . 29/12 . 31/12 . 37/12 . 41/12...
  • The basal subgroup of ed23/12 - - - BSG23/12 - - - is 23/12 . 29/12 . 31/12 . 37/12 . 41/12 . 43/12...
  • The basal subgroup of ed29/12 - - - BSG29/12 - - - is 29/12 . 31/12 . 37/12 . 41/12 . 43/12 . 47/12...

and so on...

ed17/13, ed19/13, ed23/13, ed29/13...

A.k.a. "thirteenth-prime subgroups".

To find BSGn/13:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/13

List:

  • The basal subgroup of ed17/13 - - - BSG17/13 - - - is 17/13 . 19/13 . 23/13 . 29/13 . 31/13 . 37/13...
  • The basal subgroup of ed19/13 - - - BSG19/13 - - - is 19/13 . 23/13 . 29/13 . 31/13 . 37/13 . 41/13...
  • The basal subgroup of ed23/13 - - - BSG23/13 - - - is 23/13 . 29/13 . 31/13 . 37/13 . 41/13 . 43/13...
  • The basal subgroup of ed29/13 - - - BSG29/13 - - - is 29/13 . 31/13 . 37/13 . 41/13 . 43/13 . 47/13...

and so on...

ed17/14, ed19/14, ed23/14, ed29/14...

A.k.a. "fourteenth-prime subgroups".

To find BSGn/14:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/14

List:

  • The basal subgroup of ed17/14 - - - BSG17/14 - - - is 17/14 . 19/14 . 23/14 . 29/14 . 31/14 . 37/14...
  • The basal subgroup of ed19/14 - - - BSG19/14 - - - is 19/14 . 23/14 . 29/14 . 31/14 . 37/14 . 41/14...
  • The basal subgroup of ed23/14 - - - BSG23/14 - - - is 23/14 . 29/14 . 31/14 . 37/14 . 41/14 . 43/14...
  • The basal subgroup of ed29/14 - - - BSG29/14 - - - is 29/14 . 31/14 . 37/14 . 41/14 . 43/14 . 47/14...

and so on...

ed17/15, ed19/15, ed23/15, ed29/15...

A.k.a. "fifteenth-prime subgroups".

To find BSGn/15:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m<n
  3. Replace all instances of m/1 with m/15

List:

  • The basal subgroup of ed17/15 - - - BSG17/15 - - - is 17/15 . 19/15 . 23/15 . 29/15 . 31/15 . 37/15...
  • The basal subgroup of ed19/15 - - - BSG19/15 - - - is 19/15 . 23/15 . 29/15 . 31/15 . 37/15 . 41/15...
  • The basal subgroup of ed23/15 - - - BSG23/15 - - - is 23/15 . 29/15 . 31/15 . 37/15 . 41/15 . 43/15...
  • The basal subgroup of ed29/15 - - - BSG29/15 - - - is 29/15 . 31/15 . 37/15 . 41/15 . 43/15 . 47/15...

and so on...

Composite basal subgroups

ed4, ed6, ed8, ed9...

To find BSGn/1:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m is a prime factor of n
  3. Remove all instances of m/1 where m is less than n's largest prime factor
  4. Add n/1 to the start of the subgroup

List:

  • The basal subgroup of ed4/1 - - - BSG4/1 - - - is 4/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
  • The basal subgroup of ed6/1 - - - BSG6/1 - - - is 6/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
  • The basal subgroup of ed8/1 - - - BSG8/1 - - - is 8/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
  • The basal subgroup of ed9/1 - - - BSG9/1 - - - is 9/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
  • The basal subgroup of ed10/1 - - - BSG10/1 - - - is 10/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
  • The basal subgroup of ed12/1 - - - BSG12/1 - - - is 12/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
  • The basal subgroup of ed14/1 - - - BSG14/1 - - - is 14/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
  • The basal subgroup of ed15/1 - - - BSG15/1 - - - is 15/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
  • The basal subgroup of ed16/1 - - - BSG16/1 - - - is 16/1 . 3/1 . 5/1 . 7/1 . 11/1 . 13/1...
  • The basal subgroup of ed18/1 - - - BSG18/1 - - - is 18/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
  • The basal subgroup of ed20/1 - - - BSG20/1 - - - is 20/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
  • The basal subgroup of ed21/1 - - - BSG21/1 - - - is 21/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
  • The basal subgroup of ed22/1 - - - BSG22/1 - - - is 22/1 . 13/1 . 17/1 . 19/1 . 23/1 . 29/1...
  • The basal subgroup of ed24/1 - - - BSG24/1 - - - is 24/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
  • The basal subgroup of ed25/1 - - - BSG25/1 - - - is 25/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1...
  • The basal subgroup of ed26/1 - - - BSG26/1 - - - is 26/1 . 17/1 . 19/1 . 23/1 . 29/1 . 31/1...
  • The basal subgroup of ed27/1 - - - BSG27/1 - - - is 27/1 . 5/1 . 7/1 . 11/1 . 13/1 . 17/1...
  • The basal subgroup of ed28/1 - - - BSG28/1 - - - is 28/1 . 11/1 . 13/1 . 17/1 . 19/1 . 23/1...
  • The basal subgroup of ed30/1 - - - BSG30/1 - - - is 30/1 . 7/1 . 11/1 . 13/1 . 17/1 . 19/1 ...

and so on...

ed9/2, ed15/2, ed21/2, ed25/2...

A.k.a. "half-prime subgroups".


To find BSGn/2:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 2
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/2
  6. Add n/2 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed9/2 - - - BSG9/2 - - - is 9/2 . 5/2 . 7/2 . 11/2 . 13/2 . 17/2...
  • The basal subgroup of ed15/2 - - - BSG15/2 - - - is 15/2 . 7/2 . 11/2 . 13/2 . 17/2 . 19/2...
  • The basal subgroup of ed21/2 - - - BSG21/2 - - - is 21/2 . 11/2 . 13/2 . 17/2 . 19/2 . 23/2...
  • The basal subgroup of ed25/2 - - - BSG25/2 - - - is 25/2 . 7/2 . 11/2 . 13/2 . 17/2 . 19/2...
  • The basal subgroup of ed27/2 - - - BSG27/2 - - - is 27/2 . 5/2 . 7/2 . 11/2 . 13/2 . 17/2...

and so on...

ed4/3, ed8/3, ed10/3, ed14/3...

A.k.a. "third-prime subgroups".

To find BSGn/3:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 3
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/3
  6. Add n/3 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed4/3 - - - BSG4/3 - - - is 4/3 . 5/3 . 7/3 . 11/3 . 13/3 . 17/3...
  • The basal subgroup of ed8/3 - - - BSG8/3 - - - is 8/3 . 5/3 . 7/3 . 11/3 . 13/3 . 17/3...
  • The basal subgroup of ed10/3 - - - BSG10/3 - - - is 10/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
  • The basal subgroup of ed14/3 - - - BSG14/3 - - - is 14/3 . 11/3 . 13/3 . 17/3 . 19/3 . 23/3...
  • The basal subgroup of ed16/3 - - - BSG16/3 - - - is 16/3 . 3/3 . 5/3 . 7/3 . 11/3 . 13/3...
  • The basal subgroup of ed18/3 - - - BSG18/3 - - - is 18/3 . 5/3 . 7/3 . 11/3 . 13/3 . 17/3...
  • The basal subgroup of ed20/3 - - - BSG20/3 - - - is 20/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
  • The basal subgroup of ed22/3 - - - BSG22/3 - - - is 22/3 . 13/3 . 17/3 . 19/3 . 23/3 . 29/3...
  • The basal subgroup of ed25/3 - - - BSG25/3 - - - is 25/3 . 7/3 . 11/3 . 13/3 . 17/3 . 19/3...
  • The basal subgroup of ed26/3 - - - BSG26/3 - - - is 26/3 . 17/3 . 19/3 . 23/3 . 29/3 . 31/3...
  • The basal subgroup of ed28/3 - - - BSG28/3 - - - is 28/3 . 11/3 . 13/3 . 17/3 . 19/3 . 23/3...

and so on...

ed9/4, ed15/4, ed21/4, ed25/4...

A.k.a. "quarter-prime subgroups".

To find BSGn/4:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 4
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/4
  6. Add n/4 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed9/4 - - - BSG9/4 - - - is 9/4 . 5/4 . 7/4 . 11/4 . 13/4 . 17/4...
  • The basal subgroup of ed15/4 - - - BSG15/4 - - - is 15/4 . 7/4 . 11/4 . 13/4 . 17/4 . 19/4...
  • The basal subgroup of ed21/4 - - - BSG21/4 - - - is 21/4 . 11/4 . 13/4 . 17/4 . 19/4 . 23/4...
  • The basal subgroup of ed25/4 - - - BSG25/4 - - - is 25/4 . 7/4 . 11/4 . 13/4 . 17/4 . 19/4...
  • The basal subgroup of ed27/4 - - - BSG27/4 - - - is 27/4 . 5/4 . 7/4 . 11/4 . 13/4 . 17/4...

and so on...

ed6/5, ed8/5, ed9/5, ed12/5...

A.k.a. "fifth-prime subgroups".

To find BSGn/5:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 5
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/5
  6. Add n/5 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed6/5 - - - BSG6/5 - - - is 6/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
  • The basal subgroup of ed8/5 - - - BSG8/5 - - - is 8/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
  • The basal subgroup of ed9/5 - - - BSG9/5 - - - is 9/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
  • The basal subgroup of ed12/5 - - - BSG12/5 - - - is 12/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
  • The basal subgroup of ed14/5 - - - BSG14/5 - - - is 14/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
  • The basal subgroup of ed16/5 - - - BSG16/5 - - - is 16/5 . 3/5 . 7/5 . 11/5 . 13/5 . 17/5...
  • The basal subgroup of ed18/5 - - - BSG18/5 - - - is 18/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
  • The basal subgroup of ed21/5 - - - BSG21/5 - - - is 21/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...
  • The basal subgroup of ed22/5 - - - BSG22/5 - - - is 22/5 . 13/5 . 17/5 . 19/5 . 23/5 . 29/5...
  • The basal subgroup of ed24/5 - - - BSG24/5 - - - is 24/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
  • The basal subgroup of ed26/5 - - - BSG26/5 - - - is 26/5 . 17/5 . 19/5 . 23/5 . 29/5 . 31/5...
  • The basal subgroup of ed27/5 - - - BSG27/5 - - - is 27/5 . 7/5 . 11/5 . 13/5 . 17/5 . 19/5...
  • The basal subgroup of ed28/5 - - - BSG28/5 - - - is 28/5 . 11/5 . 13/5 . 17/5 . 19/5 . 23/5...

and so on...

ed35/6, ed55/6, ed65/6, ed77/6...

A.k.a. "sixth-prime subgroups".

To find BSGn/6:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 6
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/6
  6. Add n/6 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed35/6 - - - BSG35/6 - - - is 35/6 . 11/6 . 13/6 . 17/6 . 19/6 . 23/6...
  • The basal subgroup of ed55/6 - - - BSG55/6 - - - is 55/6 . 13/6 . 17/6 . 19/6 . 23/6 . 29/6...
  • The basal subgroup of ed65/6 - - - BSG65/6 - - - is 65/6 . 17/6 . 19/6 . 23/6 . 29/6 . 31/6...
  • The basal subgroup of ed77/6 - - - BSG77/6 - - - is 77/6 . 13/6 . 17/6 . 19/6 . 23/6 . 29/6...

and so on...

ed8/7, ed9/7, ed10/7, ed12/7...

A.k.a. "seventh-prime subgroups".

To find BSGn/7:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 7
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/7
  6. Add n/7 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed8/7 - - - BSG8/7 - - - is 8/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
  • The basal subgroup of ed9/7 - - - BSG9/7 - - - is 9/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
  • The basal subgroup of ed10/7 - - - BSG10/7 - - - is 10/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
  • The basal subgroup of ed12/7 - - - BSG12/7 - - - is 12/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
  • The basal subgroup of ed15/7 - - - BSG15/7 - - - is 15/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
  • The basal subgroup of ed16/7 - - - BSG16/7 - - - is 16/7 . 12/7 . 10/7 . 11/7 . 13/7 . 17/7...
  • The basal subgroup of ed18/7 - - - BSG18/7 - - - is 18/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
  • The basal subgroup of ed20/7 - - - BSG20/7 - - - is 20/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
  • The basal subgroup of ed22/7 - - - BSG22/7 - - - is 22/7 . 13/7 . 17/7 . 19/7 . 23/7 . 29/7...
  • The basal subgroup of ed24/7 - - - BSG24/7 - - - is 24/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
  • The basal subgroup of ed25/7 - - - BSG25/7 - - - is 25/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...
  • The basal subgroup of ed26/7 - - - BSG26/7 - - - is 26/7 . 17/7 . 19/7 . 23/7 . 29/7 . 31/7...
  • The basal subgroup of ed27/7 - - - BSG27/7 - - - is 27/7 . 10/7 . 11/7 . 13/7 . 17/7 . 19/7...
  • The basal subgroup of ed30/7 - - - BSG30/7 - - - is 30/7 . 11/7 . 13/7 . 17/7 . 19/7 . 23/7...

and so on...

ed9/8, ed15/8, ed21/8, ed25/8...

A.k.a. "eighth-prime subgroups".

To find BSGn/8:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 8
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/8
  6. Add n/8 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed9/8 - - - BSG9/8 - - - is 9/8 . 10/8 (ie 5/4) . 7/8 . 11/8 . 13/8 . 17/8...
  • The basal subgroup of ed15/8 - - - BSG15/8 - - - is 15/8 . 14/8 (ie 7/4) . 11/8 . 13/8 . 17/8 . 19/8...
  • The basal subgroup of ed21/8 - - - BSG21/8 - - - is 21/8 . 11/8 . 13/8 . 17/8 . 19/8 . 23/8..
  • The basal subgroup of ed25/8 - - - BSG25/8 - - - is 25/8 . 14/8 (ie 7/4) . 11/8 . 13/8 . 17/8 . 19/8...
  • The basal subgroup of ed27/8 - - - BSG27/8 - - - is 27/8 . 10/8 (ie 5/4) . 7/8 . 11/8 . 13/8 . 17/8...

and so on...

ed10/9, ed14/9, ed16/9, ed20/9...

A.k.a. "ninth-prime subgroups".

To find BSGn/9:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 9
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/9
  6. Add n/9 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed10/9 - - - BSG10/9 - - - is 10/9 . 14/9 . 11/9 . 13/9 . 17/9 . 19/9...
  • The basal subgroup of ed14/9 - - - BSG14/9 - - - is 14/9 . 11/9 . 13/9 . 17/9 . 19/9 . 23/9...
  • The basal subgroup of ed16/9 - - - BSG16/9 - - - is 16/9 . 10/9 . 7/9 . 11/9 . 13/9 . 17/9...
  • The basal subgroup of ed20/9 - - - BSG20/9 - - - is 20/9 . 14/9 . 11/9 . 13/9 . 17/9 . 19/9...
  • The basal subgroup of ed22/9 - - - BSG22/9 - - - is 22/9 . 13/9 . 17/9 . 19/9 . 23/9 . 29/9...
  • The basal subgroup of ed25/9 - - - BSG25/9 - - - is 25/9 . 14/9 . 11/9 . 13/9 . 17/9 . 19/9...
  • The basal subgroup of ed26/9 - - - BSG26/9 - - - is 26/9 . 17/9 . 19/9 . 23/9 . 29/9 . 31/9...
  • The basal subgroup of ed28/9 - - - BSG28/9 - - - is 28/9 . 11/9 . 13/9 . 17/9 . 19/9 . 23/9...

and so on...

ed21/10, ed27/10, ed33/10, ed39/10...

A.k.a. "tenth-prime subgroups".

To find BSGn/10:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 10
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/10
  6. Add n/10 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed21/10 - - - BSG21/10 - - - is 21/10 . 11/10 . 13/10 . 17/10 . 19/10 . 23/10...
  • The basal subgroup of ed27/10 - - - BSG27/10 - - - is 27/10 . 14/10 (ie 7/5) . 11/10 . 13/10 . 17/10 . 19/10...
  • The basal subgroup of ed33/10 - - - BSG33/10 - - - is 33/10 . 13/10 . 17/10 . 19/10 . 23/10 . 29/10...
  • The basal subgroup of ed39/10 - - - BSG39/10 - - - is 39/10 . 17/10 . 19/10 . 23/10 . 29/10 . 31/10...

and so on...

ed12/11, ed14/11, ed15/11, ed16/11...

A.k.a. "eleventh-prime subgroups".

To find BSGn/11:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 11
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/11
  6. Add n/11 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed12/11 - - - BSG12/11 - - - is 12/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
  • The basal subgroup of ed14/11 - - - BSG14/11 - - - is 14/11 . 13/11 . 17/11 . 19/11 . 23/11 . 29/11...
  • The basal subgroup of ed15/11 - - - BSG15/11 - - - is 15/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
  • The basal subgroup of ed16/11 - - - BSG16/11 - - - is 16/11 . 12/11 . 20/11 . 14/11 . 13/11 . 17/11...
  • The basal subgroup of ed18/11 - - - BSG18/11 - - - is 18/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
  • The basal subgroup of ed20/11 - - - BSG20/11 - - - is 20/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
  • The basal subgroup of ed21/11 - - - BSG21/11 - - - is 21/11 . 13/11 . 17/11 . 19/11 . 23/11 . 29/11...
  • The basal subgroup of ed24/11 - - - BSG24/11 - - - is 24/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
  • The basal subgroup of ed25/11 - - - BSG25/11 - - - is 25/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...
  • The basal subgroup of ed26/11 - - - BSG26/11 - - - is 26/11 . 17/11 . 19/11 . 23/11 . 29/11 . 31/11...
  • The basal subgroup of ed27/11 - - - BSG27/11 - - - is 27/11 . 20/11 . 14/11 . 13/11 . 17/11 . 19/11...
  • The basal subgroup of ed28/11 - - - BSG28/11 - - - is 28/11 . 13/11 . 17/11 . 19/11 . 23/11 . 29/11...
  • The basal subgroup of ed30/11 - - - BSG30/11 - - - is 30/11 . 14/11 . 13/11 . 17/11 . 19/11 . 23/11...

and so on...

ed35/12, ed55/12, ed65/12, ed77/12...

A.k.a. "twelfth-prime subgroups".

To find BSGn/12:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 12
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/12
  6. Add n/12 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed35/12 - - - BSG35/12 - - - is 35/12 . 22/12 (ie 11/6) . 13/12 . 17/12 . 19/12 . 23/12...
  • The basal subgroup of ed55/12 - - - BSG55/12 - - - is 55/12 . 13/12 . 17/12 . 19/12 . 23/12 . 29/12...
  • The basal subgroup of ed65/12 - - - BSG65/12 - - - is 65/12 . 17/12 . 19/12 . 23/12 . 29/12 . 31/12...
  • The basal subgroup of ed77/12 - - - BSG77/12 - - - is 77/12 . 13/12 . 17/12 . 19/12 . 23/12 . 29/12...

and so on...

ed14/13, ed15/13, ed16/13, ed18/13...

To find BSGn/13:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 13
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/13
  6. Add n/13 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed14/13 - - - BSG14/13 - - - is 14/13 . 22/13 . 17/13 . 19/13 . 23/13 . 29/13...
  • The basal subgroup of ed15/13 - - - BSG15/13 - - - is 15/13 . 14/13 . 22/13 . 13/13 . 17/13 . 19/13...
  • The basal subgroup of ed16/13 - - - BSG16/13 - - - is 16/13 . 24/13 . 20/13 . 14/13 . 22/13 . 17/13...
  • The basal subgroup of ed18/13 - - - BSG18/13 - - - is 18/13 . 20/13 . 14/13 . 22/13 . 17/13 . 19/13...
  • The basal subgroup of ed20/13 - - - BSG20/13 - - - is 20/13 . 14/13 . 22/13 . 17/13 . 19/13 . 23/13...
  • The basal subgroup of ed21/13 - - - BSG21/13 - - - is 21/13 . 22/13 . 17/13 . 19/13 . 23/13 . 29/13...
  • The basal subgroup of ed22/13 - - - BSG22/13 - - - is 22/13 . 17/13 . 19/13 . 23/13 . 29/13 . 31/13...
  • The basal subgroup of ed24/13 - - - BSG24/13 - - - is 24/13 . 20/13 . 14/13 . 22/13 . 17/13 . 19/13...
  • The basal subgroup of ed25/13 - - - BSG25/13 - - - is 25/13 . 14/13 . 22/13 . 17/13 . 19/13 . 23/13...
  • The basal subgroup of ed27/13 - - - BSG27/13 - - - is 27/13 . 20/13 . 14/13 . 22/13 . 17/13 . 19/13...
  • The basal subgroup of ed28/13 - - - BSG28/13 - - - is 28/13 . 22/13 . 17/13 . 19/13 . 23/13 . 29/13...
  • The basal subgroup of ed30/13 - - - BSG30/13 - - - is 30/13 . 14/13 . 22/13 . 17/13 . 19/13 . 23/13...

and so on...

ed15/14, ed25/14, ed27/14, ed33/14...

To find BSGn/14:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 14
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/14
  6. Add n/14 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed15/14 - - - BSG15/14 - - - is 15/14 . 22/14 (ie 11/7) . 26/14 (ie 13/7) . 17/14 . 19/14 . 23/14...
  • The basal subgroup of ed25/14 - - - BSG25/14 - - - is 25/14 . 11/14 . 26/14 (ie 13/7) . 17/14 . 19/14 . 23/14...
  • The basal subgroup of ed27/14 - - - BSG27/14 - - - is 27/14 . 20/14 (ie 10/7) . 11/14 . 13/14 . 17/14 . 19/14...
  • The basal subgroup of ed33/14 - - - BSG33/14 - - - is 33/14 . 26/14 (ie 13/7) . 17/14 . 19/14 . 23/14 . 29/14...

and so on...

ed16/15, ed22/15, ed26/15, ed28/15...

To find BSGn/15:

  1. Start with BSG2/1
  2. Remove all instances of m/1 where m shares a prime factor with 15
  3. Remove all instances of m/1 where m is a prime factor of n
  4. Remove all instances of m/1 where m is less than n's largest prime factor
  5. Replace all instances of m/1 with m/15
  6. Add n/15 to the start of the subgroup
  7. In all instances a/b where a<b, double a
  8. If a is still not bigger than b, keep doubling a until it becomes bigger than b

List:

  • The basal subgroup of ed16/15 - - - BSG16/15 - - - is 16/15 . 28/15 . 22/15 . 26/15...
  • The basal subgroup of ed22/15 - - - BSG22/15 - - - is 22/15 . 26/15 . 17/15 . 19/15 . 23/15 . 29/15...
  • The basal subgroup of ed26/15 - - - BSG26/15 - - - is 26/15 . 17/15 . 19/15 . 23/15 . 29/15 . 31/15...
  • The basal subgroup of ed28/15 - - - BSG28/15 - - - is 28/15 . 22/15 . 26/15 . 17/15 . 19/15 . 23/15...

and so on...

Use cases

Basal subgroups are most useful as starting points when beginning to explore an equal tuning. They give you a solid fundamental skeleton to start with, which you can then tweak and tinker with by adding and removing elements from the subgroup, or swapping them out for other ones.

Basal subgroups can also be used as a rough way to measure the "unusualness" of a subgroup in a given equal tuning.

If the subgroup is quite similar to the basal subgroup, then it is a to-be-expected subgroup. For example subgroups like 2.3.7.11 and 2.3.5.11 are very similar to BSG2/1 (2.3.5.7.11...) with slight modifications, so it is not unexpected to see them get used often in ed2/1s (edos).

However if the subgroup is quite different to the basal subgroup, then that means it is highly unusual (not better or worse necessarily, just more unexpected). So for example it would be quite strange to see a 2.53/10.111/43 subgroup get used in an ed2, because that is extremely far removed from BSG2/1.

Related concepts

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