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:MOS scale]]
[[Category:Tables]]
[[Category:Tables]]
{{todo|link|intro}}
{{todo|link|intro}}

Revision as of 20:55, 25 November 2024

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.

2L 3s 3L 2s 4L 1s
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 2L 3s 4 3 3 3 2
1L 3m 1s 		4 3 4 3 2
2L 2m 1s
3L 2s 3L 2s 4 4 3 4 2
3L 1m 1s
2L 5s 3L 4s 4L 3s 5L 2s 6L 1s
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 2 3 3 3 2
1L 4m 2s 		4 3 3 3 3 3 2
1L 5m 1s
2L 5s 2L 5s 4 2 3 3 3 2 2
1L 3m 3s 		4 3 2 4 3 3 2
2L 3m 2s 		4 3 2 4 3 3 2
2L 3m 2s 		4 3 3 4 3 3 2
2L 4m 1s
3L 4s 3L 4s 4 3 2 3 2 3 2
1L 3m 3s 		4 3 2 4 3 4 2
3L 2m 2s, Nicetone 		4 3 3 4 3 4 2
3L 3m 1s
4L 3s 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 5L 2s 4 4 3 4 4 4 2
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.

2L 4s 3L 3s 4L 2s 5L 1s
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 		4 3 3 3 3 2
1L 4m 1s
2L 4s 2L 4s 4 2 3 3 3 2
1L 3m 2s
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 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 4L 2s 4 4 3 4 4 2
4L 1m 1s

Since Indonesian gamelan music stems from a tradition with 9-note scales converging from max variety 7 to equal, sums of 9-note MOS make for great compromise gamelan tunings when they fail to produce max variety 3.

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
1L 1m 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 		4 3 3 3 3 3 3 3 2
1L 7m 1s
2L 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 		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 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 		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 4 3 2 4 2 4 2 4 2
4L 1m 4s 		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 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 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 7L 2s 4 4 4 3 4 4 4 4 2
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.

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