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If the number of notes is even, the max-variety-3 scale consists of two chains of 3/2 of equal length, each of which contains half the notes. If the number of notes is odd, the two 3/2-chains differ by 1 in length.
If the number of notes is even, the max-variety-3 scale consists of two chains of 3/2 of equal length, each of which contains half the notes. If the number of notes is odd, the two 3/2-chains differ by 1 in length.


Any scale at all with only 3 notes has max variety 3, so let's begin with the 4-note scales. A 4-note scale consists of two parallel 3/2 intervals separated by some other 5-limit interval. Although strictly speaking this could be any other 5-limit interval at all, if the two chains are separated from each other there will be no potential for 5-limit harmony. For these 4-note scales it turns out that all the configurations work, so we can easily list all the possible scales: ...{1/1,10/9,3/2,5/3}, {1/1,5/4,3/2,5/3}, {1/1,5/4,3/2,15/8}, {1/1,45/32,3/2,15/18}... Only the ones where the two chains are lined up closely have good potential for harmony.
Any scale at all with only 3 notes has max variety 3, so let's begin with the 4-note scales. A 4-note scale consists of two parallel 3/2 intervals separated by some other 5-limit interval. Although strictly speaking this could be any other 5-limit interval at all, if the two chains are separated from each other there will be no potential for 5-limit harmony. For these 4-note scales it turns out that all the configurations work, so we can easily list all the possible scales: ...{1/1,10/9,3/2,5/3}, {1/1,5/4,3/2,5/3}, {1/1,5/4,3/2,15/8}, {1/1,45/32,3/2,125/64}... Only the ones where the two chains are lined up closely have good potential for harmony.


The 5-note scales will consist of a chain of 3 notes and a parallel chain of 2 notes. The 3-note chain has the same pattern as {1/1, 9/8, 3/2}, and in order for it to make an actual max-variety-3 scale, the other 2 notes must fall in between 9/8 and 3/2, and in between 3/2 and 2/1. If either of the notes falls between 1/1 and 9/8, the scale will not be max-variety-3. Let's look at all the scales we get as we move the 2-note chain past the 3-note chain in the 5-limit lattice.
The 5-note scales will consist of a chain of 3 notes and a parallel chain of 2 notes. The 3-note chain has the same pattern as {1/1, 9/8, 3/2}, and in order for it to make an actual max-variety-3 scale, the other 2 notes must fall in between 9/8 and 3/2, and in between 3/2 and 2/1. If either of the notes falls between 1/1 and 9/8, the scale will not be max-variety-3. Let's look at all the scales we get as we move the 2-note chain past the 3-note chain in the 5-limit lattice.
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1/1 9/8 1215/1024 3/2 405/256 ... max variety 3
1/1 9/8 1215/1024 3/2 405/256 ... max variety 3


TODO: Finish example
Scales with three step sizes are also derivable by summing MOS scales together. All 5-note and 7-note scales derived this way will have max variety 3.
{| class="wikitable"
|+
!
!1L 4s
!2L 3s
!3L 2s
!4L 1s
|-
!1L 4s
!1L 4s
|4 2 3 2 2
1L 1m 3s
|4 3 3 2 2
1L 2m 2s
|4 3 3 3 2
1L 3m 1s
|-
!2L 3s
|4 2 3 2 2
1L 1m 3s
!2L 3s
|4 3 3 3 2
1L 3m 1s
|4 3 4 3 2
2L 2m 1s
|-
!3L 2s
|4 3 3 2 2
1L 2m 2s
|4 3 3 3 2
1L 3m 1s
!3L 2s
|4 4 3 4 2
3L 1m 1s
|-
!4L 1s
|4 3 3 3 2
1L 3m 1s
|4 3 4 3 2
2L 2m 1s
|4 4 3 4 2
3L 1m 1s
!4L 1s
|}
{| class="wikitable"
|+
!
!1L 6s
!2L 5s
!3L 4s
!4L 3s
!5L 2s
!6L 1s
|-
!1L 6s
!1L 6s
|4 2 2 3 2 2 2
1L 1m 5s
|4 2 2 3 2 3 2
1L 2m 4s
|4 3 2 3 2 3 2
1L 3m 3s
|4 3 3 3 3 3 2
1L 5m 1s
|7ed
|-
!2L 5s
|4 2 2 3 2 2 2
1L 1m 5s
!2L 5s
|4 2 3 3 3 2 2
[[5L 2s MODMOSes|1L 3m 3s]]
|4 3 2 4 3 3 2
2L 3m 3s
|7ed
|4 3 3 4 3 3 2
2L 4m 1s
|-
!3L 4s
|4 2 2 3 2 3 2
1L 2m 4s
|4 2 3 3 3 2 2
[[5L 2s MODMOSes|1L 3m 3s]]
!3L 4s
|7ed
|4 3 2 4 3 4 2
3L 2m 2s, [[29edo#Nicetone|Nicetone]]
|4 3 3 4 3 4 2
3L 3m 1s
|-
!4L 3s
|4 3 2 3 2 3 2
1L 3m 3s
|4 3 2 4 3 3 2
2L 3m 3s
|7ed
!4L 3s
|4 4 2 4 3 4 2
4L 2m 1s
|4 4 3 4 2 4 2
4L 2m 1s, [[Mohajira]]
|-
!5L 2s
|4 3 3 3 3 3 2
1L 5m 1s
|7ed
|4 3 2 4 3 4 2
3L 2m 2s, [[29edo#Nicetone|Nicetone]]
|4 4 2 4 3 4 2
4L 2m 1s
!5L 2s
|4 4 3 4 4 4 2
5L 1m 1s
|-
!6L 1s
|7ed
|4 3 3 4 3 3 2
2L 4m 1s
|4 3 3 4 3 4 2
3L 3m 1s
|4 4 3 4 2 4 2
4L 2m 1s, [[Mohajira]]
|4 4 3 4 4 4 2
5L 1m 1s
!6L 1s
|}
However, if a scale derived this way has any other number of notes, its three step sizes will not always produce max variety 3.
{| class="wikitable"
!
!1L 5s
!2L 4s
!3L 3s
!4L 2s
!5L 1s
|-
!1L 5s
!1L 5s
|4 2 2 3 2 2
1L 1m 4s
|4 2 3 2 3 2
1L 2m 3s
 
fails, max variety 4
|4 3 2 3 2 2
1L 2m 3s
 
fails, max variety 4
|6ed
|-
!2L 4s
|4 2 2 3 2 2
1L 1m 4s
!2L 4s
|4 2 3 3 3 2
1L 3m 2s
 
fails, max variety 4
|6ed
|4 3 3 4 3 2
2L 3m 1s
 
fails, max variety 4
|-
!3L 3s
|4 2 3 2 3 2
1L 2m 3s
 
fails, max variety 4
|4 2 3 3 3 3
1L 3m 2s
 
fails, max variety 4
|6ed
|4 3 3 4 3 2
2L 3m 1s
 
fails, max variety 4
|4 3 4 3 4 2
3L 2m 1s
 
fails, max variety 4
|-
!4L 2s
|4 3 2 3 2 2
1L 2m 3s
 
fails, max variety 4
|6ed
|4 3 3 4 3 2
2L 3m 1s
 
fails, max variety 4
!4L 2s
|4 4 3 4 4 2
4L 1m 1s
|-
!5L 1s
|6ed
|4 3 3 4 3 2
2L 3m 1s
 
fails, max variety 4
|4 3 4 3 4 2
3L 2m 1s
 
fails, max variety 4
|4 4 3 4 4 2
4L 1m 1s
!5L 1s
|}
Since Indonesian gamelan music stems from a tradition with 9-note scales converging from [http://www.neuroscience-of-music.se/Gamelan%20pelog%20A.htm max variety ''7''] to equal, sums of 9-note MOS make for great compromise gamelan tunings when they fail to produce max variety 3.
{| class="wikitable"
!
!1L 8s
!2L 7s
!3L 6s
!4L 5s
!5L 4s
!6L 3s
!7L 2s
!8L 1s
|-
!1L 8s
!1L 8s
|4 2 2 2 3 2 2 2 2
1L 1s 7s
|4 2 2 3 2 2 3 2 2
1L 2m 6s
 
fails, max variety 4
|4 2 2 3 2 3 2 3 2
1L 3m 5s
 
fails, max variety 4
|4 3 2 3 2 3 2 3 2
1L 4m 4s
|4 4 2 3 2 2 3 2 2
2L 2m 5s
|4 3 3 2 3 3 3 3 2
1L 6m 2s
 
fails, max variety 4
|9ed
|-
!2L 7s
|4 2 2 2 3 2 2 2 2
1L 1s 7s
!2L 7s
|4 2 2 3 3 2 3 2 2
1L 3m 5s
 
fails, max variety 4
|4 2 2 3 3 3 2 3 2
1L 4m 4s
|4 3 2 2 4 3 2 2 2
 
2L 2m 5s
|4 3 2 3 3 2 3 3 2
1L 5m 3s
 
fails, max variety 4
|9ed
|4 3 3 3 3 3 3 3 2
1L 7m 1s
|-
!3L 6s
|4 2 2 3 2 2 3 2 2
1L 2m 6s
 
fails, max variety 4
|4 2 2 3 3 2 3 2 2
1L 3m 5s
 
fails, max variety 4
!3L 6s
|4 2 2 4 3 2 3 3 2
2L 3m 4s
 
fails, max variety 4
|4 3 2 4 2 3 3 3 2
2L 5m 2s
|9ed
|4 3 3 3 3 3 3 4 2
2L 6m 1s
 
fails, max variety 4
|4 3 3 4 3 3 4 3 2
1L 5m 3s
 
fails, max variety 4
|-
!4L 5s
|4 2 2 3 2 3 2 3 2
1L 4m 4s
|4 2 2 3 3 3 2 3 2
1L 4m 4s
|4 2 2 4 3 2 3 3 2
2L 3m 4s
 
fails, max variety 4
!4L 5s
|9ed
|4 3 2 3 3 2 3 4 2
2L 4m 3s
 
fails, max variety 4
|4 4 3 3 3 4 3 4 2
3L 5m 1s
 
fails, max variety 4
|4 3 3 4 3 4 3 4 2
4L 4m 1s
|-
!5L 4s
|4 3 2 3 2 3 2 3 2
1L 4m 4s
|4 3 2 2 4 3 2 2 2
 
2L 2m 5s
|4 3 2 4 2 3 3 3 2
2L 5m 2s
|9ed
!5L 4s
|4 4 2 4 3 3 3 4 4 2
5L 2m 2s, [[diasem]]
|4 4 3 3 3 4 3 4 2
4L 4m 1s
|4 4 3 4 3 4 3 4 2
5L 3m 1s
 
fails, max variety 4
|-
!6L 3s
|4 4 2 3 2 2 3 2 2
2L 2m 5s
|4 3 2 3 3 2 3 3 2
1L 5m 3s
 
fails, max variety 4
|9ed
|4 3 2 3 3 2 3 4 2
2L 4m 3s
 
fails, max variety 4
|4 4 2 4 3 3 3 4 4 2
5L 2m 2s, [[diasem]]
!6L 3s
|4 4 3 4 3 3 4 4 2
5L 3m 1s
 
fails, max variety 4
|4 4 3 4 4 3 4 4 2
6L 2m 1s
 
fails, max variety 4
|-
!7L 2s
|4 3 3 2 3 3 3 3 2
1L 6m 2s
 
fails, max variety 4
|9ed
|4 3 3 3 3 3 4 3 2
2L 6m 1s
 
fails, max variety 4
|4 4 3 3 3 4 3 4 2
1L 5m 3s
 
fails, max variety 4
|4 4 3 3 3 4 3 4 2
4L 4m 1s
|4 4 3 4 3 3 4 4 2
5L 3m 1s
 
fails, max variety 4
!7L 2s
|4 4 4 3 4 4 4 4 2
7L 1m 1s
|-
!8L 1s
|9ed
|4 3 3 3 3 3 3 3 2
1L 7m 1s
|4 3 3 4 3 3 4 3 2
1L 5m 3s
 
fails, max variety 4
|4 3 3 4 3 4 3 4 2
4L 4m 1s
|4 4 3 4 3 4 3 4 2
5L 3m 1s
 
fails, max variety 4
|4 4 3 4 4 3 4 4 2
6L 2m 1s
 
fails, max variety 4
|4 4 4 3 4 4 4 4 2
7L 1m 1s
!8L 1s
|}
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