Basal subgroup

From Xenharmonic Wiki
Jump to navigation Jump to search

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. Add n/2 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/3 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/4 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/5 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
  • 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 ed9/5 - - - BSG9/5 - - - is 9/5 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
  • 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 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 ed18/5 - - - BSG18/5 - - - is 18/5 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
  • 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 . 5/5 . 7/5 . 11/5 . 13/5 . 17/5...
  • 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 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. Add n/6 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/7 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/8 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/9 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/10 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. "elevent-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. Add n/11 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/12 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/13 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/14 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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. Add n/15 to the start of the subgroup
  6. In all instances a/b where a<b, double a
  7. 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

Retrieved from "https://en.xen.wiki/index.php?title=Basal_subgroup&oldid=164297"