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Interval variety/Maximum variety MOS summing: Difference between revisions

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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.
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"
{| class="wikitable"
|+
!
!2L 3s
!3L 2s
!4L 1s
|-
|-
!1L 4s
!
|4 2 3 2 2
! 2L 3s
1L 1m 3s
! 3L 2s
|4 3 3 2 2
! 4L 1s
1L 2m 2s
|-
|4 3 3 3 2
! 1L 4s
1L 3m 1s
| 4 2 3 2 2<br />1L 1m 3s
| 4 3 3 2 2<br />1L 2m 2s
| 4 3 3 3 2<br />1L 3m 1s
|-
|-
!2L 3s
! 2L 3s
!2L 3s
! 2L 3s
|4 3 3 3 2
| 4 3 3 3 2<br />1L 3m 1s
1L 3m 1s
| 4 3 4 3 2<br />2L 2m 1s
|4 3 4 3 2
2L 2m 1s
|-
|-
!3L 2s
! 3L 2s
|
|  
!3L 2s
! 3L 2s
|4 4 3 4 2
| 4 4 3 4 2<br />3L 1m 1s
3L 1m 1s
|}
|}
{| class="wikitable"
{| class="wikitable"
|+
!
!2L 5s
!3L 4s
!4L 3s
!5L 2s
!6L 1s
|-
|-
!1L 6s
!
|4 2 2 3 2 2 2
! 2L 5s
1L 1m 5s  
! 3L 4s
|4 2 2 3 2 3 2
! 4L 3s
1L 2m 4s
! 5L 2s
|4 3 2 3 2 3 2
! 6L 1s
1L 3m 3s
|-
|4 3 2 3 3 3 2
! 1L 6s
1L 4m 2s
| 4 2 2 3 2 2 2<br />1L 1m 5s  
|4 3 3 3 3 3 2
| 4 2 2 3 2 3 2<br />1L 2m 4s
1L 5m 1s
| 4 3 2 3 2 3 2<br />1L 3m 3s
| 4 3 2 3 3 3 2<br />1L 4m 2s
| 4 3 3 3 3 3 2<br />1L 5m 1s
|-
|-
!2L 5s
! 2L 5s
!2L 5s
! 2L 5s
|4 2 3 3 3 2 2
| 4 2 3 3 3 2 2<br />[[5L 2s MODMOSes| 1L 3m 3s]]
[[5L 2s MODMOSes|1L 3m 3s]]
| 4 3 2 4 3 3 2<br />2L 3m 2s
|4 3 2 4 3 3 2
| 4 3 2 4 3 3 2<br />2L 3m 2s
2L 3m 2s
| 4 3 3 4 3 3 2<br />2L 4m 1s
|4 3 2 4 3 3 2
2L 3m 2s
|4 3 3 4 3 3 2
2L 4m 1s
|-
|-
!3L 4s
! 3L 4s
|
|  
!3L 4s
! 3L 4s
|4 3 2 3 2 3 2
| 4 3 2 3 2 3 2<br />[[5L 2s MODMOSes| 1L 3m 3s]]
[[5L 2s MODMOSes|1L 3m 3s]]
| 4 3 2 4 3 4 2<br />3L 2m 2s, [[29edo#Nicetone| Nicetone]]
|4 3 2 4 3 4 2
| 4 3 3 4 3 4 2<br />3L 3m 1s
3L 2m 2s, [[29edo#Nicetone|Nicetone]]
|4 3 3 4 3 4 2
3L 3m 1s
|-
|-
!4L 3s
! 4L 3s
|
|  
|
|  
!4L 3s
! 4L 3s
|4 4 2 4 3 4 2
| 4 4 2 4 3 4 2<br />4L 2m 1s
4L 2m 1s
| 4 4 3 4 2 4 2<br />4L 2m 1s, [[Mohajira]]
|4 4 3 4 2 4 2
4L 2m 1s, [[Mohajira]]
|-
|-
!5L 2s
! 5L 2s
|
|  
|
|  
|
|  
!5L 2s
! 5L 2s
|4 4 3 4 4 4 2
| 4 4 3 4 4 4 2<br />5L 1m 1s
5L 1m 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.
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"
{| class="wikitable"
!
!2L 4s
!3L 3s
!4L 2s
!5L 1s
|-
|-
!1L 5s
!
|4 2 2 3 2 2
! 2L 4s
1L 1m 4s
! 3L 3s
|4 2 3 2 3 2
! 4L 2s
1L 2m 3s
! 5L 1s
 
|-
fails, max variety 4
! 1L 5s
|4 3 2 3 2 2
| 4 2 2 3 2 2<br />1L 1m 4s
1L 2m 3s
| 4 2 3 2 3 2<br />1L 2m 3s<br />fails, max variety 4
 
| 4 3 2 3 2 2<br />1L 2m 3s<br />fails, max variety 4
fails, max variety 4
| 4 3 3 3 3 2<br />1L 4m 1s
|4 3 3 3 3 2
1L 4m 1s
|-
|-
!2L 4s
! 2L 4s
!2L 4s
! 2L 4s
|4 2 3 3 3 2
| 4 2 3 3 3 2<br />1L 3m 2s<br />fails, max variety 4
1L 3m 2s
| 4 3 2 4 3 2<br />2L 2m 2s
 
| 4 3 3 4 3 2<br />2L 3m 1s<br />fails, max variety 4
fails, max variety 4
|4 3 2 4 3 2
2L 2m 2s
|4 3 3 4 3 2
2L 3m 1s
 
fails, max variety 4
|-
|-
!3L 3s
! 3L 3s
|
|  
!3L 3s
! 3L 3s
|4 3 3 4 3 2
| 4 3 3 4 3 2<br />2L 3m 1s<br />fails, max variety 4
2L 3m 1s
| 4 3 4 3 4 2<br />3L 2m 1s<br />fails, max variety 4
 
fails, max variety 4
|4 3 4 3 4 2
3L 2m 1s
 
fails, max variety 4
|-
|-
!4L 2s
! 4L 2s
|
|  
|
|  
!4L 2s
! 4L 2s
|4 4 3 4 4 2
| 4 4 3 4 4 2<br />4L 1m 1s
4L 1m 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.
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"
{| class="wikitable"
!
!2L 7s
!3L 6s
!4L 5s
!5L 4s
!6L 3s
!7L 2s
!8L 1s
|-
|-
!1L 8s
!
|4 2 2 2 3 2 2 2 2
! 2L 7s
1L 1m 7s
! 3L 6s
|4 2 2 3 2 2 3 2 2
! 4L 5s
1L 2m 6s
! 5L 4s
 
! 6L 3s
fails, max variety 4
! 7L 2s
|4 2 2 3 2 3 2 3 2
! 8L 1s
1L 3m 5s
|-
 
! 1L 8s
fails, max variety 4
| 4 2 2 2 3 2 2 2 2<br />1L 1m 7s
|4 3 2 3 2 3 2 3 2
| 4 2 2 3 2 2 3 2 2<br />1L 2m 6s<br />fails, max variety 4
1L 4m 4s
| 4 2 2 3 2 3 2 3 2<br />1L 3m 5s<br />fails, max variety 4
|4 4 2 3 2 2 3 2 2
| 4 3 2 3 2 3 2 3 2<br />1L 4m 4s
2L 2m 5s
| 4 4 2 3 2 2 3 2 2<br />2L 2m 5s
|4 3 3 2 3 3 3 3 2
| 4 3 3 2 3 3 3 3 2<br />1L 6m 2s<br />fails, max variety 4
1L 6m 2s
| 4 3 3 3 3 3 3 3 2<br />1L 7m 1s
 
fails, max variety 4
|4 3 3 3 3 3 3 3 2
1L 7m 1s
|-
|-
!2L 7s
! 2L 7s
!2L 7s
! 2L 7s
|4 2 2 3 3 2 3 2 2
| 4 2 2 3 3 2 3 2 2<br />1L 3m 5s<br />fails, max variety 4
1L 3m 5s
| 4 2 2 3 3 3 2 3 2<br />1L 4m 4s
 
| 4 3 2 2 4 3 2 2 2<br />2L 2m 5s
fails, max variety 4
| 4 3 2 3 3 2 3 3 2<br />1L 5m 3s<br />fails, max variety 4
|4 2 2 3 3 3 2 3 2
| 4 3 3 2 4 3 3 3 2<br />2L 5m 2s
1L 4m 4s
| 4 3 3 4 4 3 3 3 2 <br />1L 6m 2s<br />fails, max variety 4
|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
|4 3 3 2 4 3 3 3 2
2L 5m 2s
|4 3 3 4 4 3 3 3 2  
1L 6m 2s
 
fails, max variety 4
|-
|-
!3L 6s
! 3L 6s
|
|  
!3L 6s
! 3L 6s
|4 2 2 4 3 2 3 3 2
| 4 2 2 4 3 2 3 3 2<br />2L 3m 4s<br />fails, max variety 4
2L 3m 4s
| 4 3 2 4 2 3 3 3 2<br />2L 5m 2s
 
| 4 3 2 4 3 2 4 3 2<br />3L 3m 3s
fails, max variety 4
| 4 3 3 3 3 3 3 4 2<br />2L 6m 1s<br />fails, max variety 4
|4 3 2 4 2 3 3 3 2
| 4 3 3 4 3 3 4 3 2<br />1L 5m 3s<br />fails, max variety 4
2L 5m 2s
|4 3 2 4 3 2 4 3 2
3L 3m 3s
|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
! 4L 5s
|
|  
|
|  
!4L 5s
! 4L 5s
|4 3 2 4 2 4 2 4 2
| 4 3 2 4 2 4 2 4 2<br />4L 1m 4s
4L 1m 4s
| 4 3 2 3 3 2 3 4 2<br />2L 4m 3s <br />fails, max variety 4
|4 3 2 3 3 2 3 4 2
| 4 4 3 3 3 4 3 4 2<br />3L 5m 1s<br />fails, max variety 4
2L 4m 3s  
| 4 3 3 4 3 4 3 4 2<br />4L 4m 1s
 
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
! 5L 4s
|
|  
|
|  
|
|  
!5L 4s
! 5L 4s
|4 4 2 4 3 3 3 4 4 2
| 4 4 2 4 3 3 3 4 4 2<br />5L 2m 2s, [[diasem]]
5L 2m 2s, [[diasem]]
| 4 4 3 3 3 4 3 4 2<br />4L 4m 1s
|4 4 3 3 3 4 3 4 2
| 4 4 3 4 3 4 3 4 2<br />5L 3m 1s<br />fails, max variety 4
4L 4m 1s
|4 4 3 4 3 4 3 4 2
5L 3m 1s
 
fails, max variety 4
|-
|-
!6L 3s
! 6L 3s
|
|  
|
|  
|
|  
|
|  
!6L 3s
! 6L 3s
|4 4 3 4 3 3 4 4 2
| 4 4 3 4 3 3 4 4 2<br />5L 3m 1s<br />fails, max variety 4
5L 3m 1s
| 4 4 3 4 4 3 4 4 2<br />6L 2m 1s<br />fails, max variety 4
 
fails, max variety 4
|4 4 3 4 4 3 4 4 2
6L 2m 1s
 
fails, max variety 4
|-
|-
!7L 2s
! 7L 2s
|
|  
|
|  
|
|  
|
|  
|
|  
!7L 2s
! 7L 2s
|4 4 4 3 4 4 4 4 2
| 4 4 4 3 4 4 4 4 2<br />7L 1m 1s
7L 1m 1s
|}
|}
As the 6-note and 9-note addition tables prove, the requirement of having the same number of steps of different sizes makes max variety 3 scales with three step sizes rare.
As the 6-note and 9-note addition tables prove, the requirement of having the same number of steps of different sizes makes max variety 3 scales with three step sizes rare.
[[Category:MOS scale]]
[[Category:Tables]]
{{todo|link|intro}}
Retrieved from "https://en.xen.wiki/w/Interval_variety/Maximum_variety_MOS_summing"