List of distinct EDO scales: Difference between revisions

Each [[Equal_division_of_the_octave|EDO]] has a finite number of distinct scales, assuming that the scales are equivalent up to cyclical permutation and that they are also irreducible. By irreducible is meant a scale that is not supported by a smaller EDO (e.g. 4424442, the diatonic scale in 24-EDO, is reducible because it is also contained in 12-EDO).

Below is a table which counts every possible scale for a given EDO (columns) and number of steps/notes (rows). Note that the total number of scales for each EDO is given by {{OEIS|A059966}} and {{OEIS|A001037}}.  

==Breakdown of Scales by EDO and Number of Notes==

{| class="wikitable"

| | n-EDO

| | Number of Scales

| | 3

|-

<math>g(n) = \displaystyle \sum \limits_{m=1}^{n} \displaystyle \sum \limits_{d \mid m} \mu(m/d) (2^{m} - 1)</math>

<span style="line-height: 1.5;"> ∆ EDO (Variety = 1)</span>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

===6-EDO Scales===

 

 

411 ††

 

2121 ◊◊

 

 

 

 

===7-EDO Scales===

 

 

 

 

 

 

511 ††

 

2221 ††

 

 

4111 ††

 

21211 ††

 

 

 

 

===8-EDO Scales===

 

 

 

 

611 ††

 

 

3131 ◊◊

 

 

5111 ††

 

22121 ††

 

 

41111 ††

 

211211 ◊◊