List of anomalous saturated suspensions: Difference between revisions

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Below is a complete list of [http://x31eq.com/ass.htm Anomalous Saturated Suspensions] through the 23-limit. Each chord listed is either [[Otonality and utonality|ambitonal]] or has a [[Otonality and utonality|o/utonal]] inverse that is also an ASS.
Below is a complete list of [http://x31eq.com/ass.htm Anomalous Saturated Suspensions] through the 23-limit. Each chord listed is either [[Otonality and utonality|ambitonal]] or has a [[Otonality and utonality|o/utonal]] inverse that is also an '''Anomalous Saturated Suspension''' (or '''ASS''').


==Formal names==
==Formal names==
Line 8: Line 8:
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Formal name
! Formal name
! | Odd limit
! Odd limit
! | Harmonic series
! Harmonic series
! | Scale
! Scale
![[Color notation|Color Name]]
! [[Color name]]
! | Common name
! Common name
|-
|-
| | '''A'''<span style="vertical-align: sub;">{9,1a}</span>
| '''A'''<span style="vertical-align: sub;">{9,1a}</span>
| | 9
| 9
| | 3:5:9:15
| 3:5:9:15
| | 1/1 '''6/5''' 3/2 9/5
| 1/1 '''6/5''' 3/2 9/5
|g7 = y6
| g7 = y6
| | Minor 7th Chord
| Minor 7th Chord
|-
|-
| | '''A'''<span style="vertical-align: sub;">{9,2a}</span>
| '''A'''<span style="vertical-align: sub;">{9,2a}</span>
| | 9
| 9
| | 3:7:9:21
| 3:7:9:21
| | 1/1 '''7/6''' 3/2 7/4
| 1/1 '''7/6''' 3/2 7/4
|z7 = r6
| z7 = r6
| | Septimal Minor 7th Chord
| Septimal Minor 7th Chord
|-
|-
| | '''A'''<span style="vertical-align: sub;">{11,1a}</span>
| '''A'''<span style="vertical-align: sub;">{11,1a}</span>
| | 11
| 11
| | 3:9:11:33
| 3:9:11:33
| | 1/1 11/8 3/2 '''11/6'''
| 1/1 11/8 3/2 '''11/6'''
|1o7(1o4) = 1u6(1u2)
| 1o7(1o4) = 1u6(1u2)
| |  
|
|-
|-
| | '''A'''<span style="vertical-align: sub;">{13,1a}</span>
| '''A'''<span style="vertical-align: sub;">{13,1a}</span>
| | 13
| 13
| | 3:9:13:39
| 3:9:13:39
| | 1/1 '''13/12''' 3/2 13/8
| 1/1 '''13/12''' 3/2 13/8
|3o6(3o2) = 3u7(3u4)  
| 3o6(3o2) = 3u7(3u4)  
| |  
|
|-
|-
| | '''A'''<span style="vertical-align: sub;">{15,1o}</span>
| '''A'''<span style="vertical-align: sub;">{15,1o}</span>
| | 15
| 15
| | 3:7:9:15:21
| 3:7:9:15:21
| | 1/1 '''7/6''' 5/4 3/2 7/4
| 1/1 '''7/6''' 5/4 3/2 7/4
|h7,z10 = r6,ry8
| h7,z10 = r6,ry8
| | Hendrix
| Hendrix
|-
|-
| | '''A'''<span style="vertical-align: sub;">{15,1u}</span>
| '''A'''<span style="vertical-align: sub;">{15,1u}</span>
| | 15
| 15
| | 15:21:35:45:105
| 15:21:35:45:105
| | 1/1 '''7/6''' 7/5 3/2 7/4
| 1/1 '''7/6''' 7/5 3/2 7/4
|z7,zg5 = r6,g3
| z7,zg5 = r6,g3
| | Inverted Hendrix
| Inverted Hendrix
|-
|-
| | '''A'''<span style="vertical-align: sub;">{15,2o}</span>
| '''A'''<span style="vertical-align: sub;">{15,2o}</span>
| | 15
| 15
| | 3:9:11:15:33
| 3:9:11:15:33
| | 1/1 5/4 11/8 3/2 '''11/6'''
| 1/1 5/4 11/8 3/2 '''11/6'''
|y,1o7,1o11 = 1u6,1u9(1uy4)
| y,1o7,1o11 = 1u6,1u9(1uy4)
| | 11-Hendrix
| 11-Hendrix
|-
|-
| | '''A'''<span style="vertical-align: sub;">{15,2u}</span>
| '''A'''<span style="vertical-align: sub;">{15,2u}</span>
| | 15
| 15
| | 15:33:45:55:165
| 15:33:45:55:165
| | 1/1 11/10 11/8 3/2 '''11/6'''
| 1/1 11/10 11/8 3/2 '''11/6'''
|1o7(1o4)1og9 = g,1u6,1u9
| 1o7(1o4)1og9 = g,1u6,1u9
| | Inverted 11-Hendrix
| Inverted 11-Hendrix
|-
|-
| | '''A'''<span style="vertical-align: sub;">{15,3o}</span>
| '''A'''<span style="vertical-align: sub;">{15,3o}</span>
| | 15
| 15
| | 3:9:13:15:39
| 3:9:13:15:39
| | 1/1 '''13/12''' 5/4 3/2 13/8
| 1/1 '''13/12''' 5/4 3/2 13/8
|y,3o6,3o9 = 3u7(3u4)3uy9
| y,3o6,3o9 = 3u7(3u4)3uy9
| | 13-Hendrix
| 13-Hendrix
|-
|-
| | '''A'''<span style="vertical-align: sub;">{15,3u}</span>
| '''A'''<span style="vertical-align: sub;">{15,3u}</span>
| | 15
| 15
| | 15:39:45:65:195
| 15:39:45:65:195
| | 1/1 '''13/12''' 13/10 3/2 13/8
| 1/1 '''13/12''' 13/10 3/2 13/8
|3o6,3o9(3og4) = g,3u7,3u11
| 3o6,3o9(3og4) = g,3u7,3u11
| | Inverted 13-Hendrix
| Inverted 13-Hendrix
|-
|-
| | '''A'''<span style="vertical-align: sub;">{17,1o}</span>
| '''A'''<span style="vertical-align: sub;">{17,1o}</span>
| | 17
| 17
| | 3:9:15:17:51
| 3:9:15:17:51
| | 1/1 17/16 5/4 '''17/12''' 3/2
| 1/1 17/16 5/4 '''17/12''' 3/2
|y,17o9,17o12
| y,17o9,17o12
| | 17-Hendrix
| 17-Hendrix
|-
|-
| | '''A'''<span style="vertical-align: sub;">{17,1u}</span>
| '''A'''<span style="vertical-align: sub;">{17,1u}</span>
| | 17
| 17
| | 15:45:51:85:255
| 15:45:51:85:255
| | 1/1 17/16 '''17/12''' 3/2 17/10
| 1/1 17/16 '''17/12''' 3/2 17/10
|17og7(17o2)17o12
| 17og7(17o2)17o12
| | Inverted 17-Hendrix
| Inverted 17-Hendrix
|-
|-
| | '''A'''<span style="vertical-align: sub;">{19,1o}</span>
| '''A'''<span style="vertical-align: sub;">{19,1o}</span>
| | 19
| 19
| | 3:9:15:19:57
| 3:9:15:19:57
| | 1/1 19/16 5/4 3/2 '''19/12'''
| 1/1 19/16 5/4 3/2 '''19/12'''
|19o6,y3 = 19u7,19uy5
| 19o6,y3 = 19u7,19uy5
| | 19-Hendrix
| 19-Hendrix
|-
|-
| | '''A'''<span style="vertical-align: sub;">{19,1u}</span>
| '''A'''<span style="vertical-align: sub;">{19,1u}</span>
| | 19
| 19
| | 15:45:57:95:285
| 15:45:57:95:285
| | 1/1 19/16 3/2 '''19/12''' 19/10
| 1/1 19/16 3/2 '''19/12''' 19/10
|19o6,19og8 = 19u7,g3
| 19o6,19og8 = 19u7,g3
| | Inverted 19-Hendrix
| Inverted 19-Hendrix
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,1o}</span>
| '''A'''<span style="vertical-align: sub;">{21,1o}</span>
| | 21
| 21
| | 3:5:9:15:21:45
| 3:5:9:15:21:45
| |1/1 5/4 3/2 '''5/3''' 7/4 15/8
| 1/1 5/4 3/2 '''5/3''' 7/4 15/8
|y6,z7,y7 = g9,zg9
| y6,z7,y7 = g9,zg9
| |
|
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,1u}</span>
| '''A'''<span style="vertical-align: sub;">{21,1u}</span>
| | 21
| 21
| | 7:15:21:35:63:105
| 7:15:21:35:63:105
| | 1/1 15/14 9/8 '''5/4''' 3/2 15/8
| 1/1 15/14 9/8 '''5/4''' 3/2 15/8
|y9,ry8 = g7,g6,r6
| y9,ry8 = g7,g6,r6
| |
|
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,2o}</span>
| '''A'''<span style="vertical-align: sub;">{21,2o}</span>
| | 21
| 21
| | 3:7:9:15:21:63
| 3:7:9:15:21:63
| |1/1 '''7/6''' 5/4 21/16 3/2 7/4
| 1/1 '''7/6''' 5/4 21/16 3/2 7/4
|h7,z10,z11 = r6,ry8,9
| h7,z10,z11 = r6,ry8,9
| |Hendrix add11?
| Hendrix add11?
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,2u}</span>
| '''A'''<span style="vertical-align: sub;">{21,2u}</span>
| | 21
| 21
| | 5:15:21:35:45:105
| 5:15:21:35:45:105
| | 1/1 21/20 9/8 21/16 '''3/2''' '''7/4'''
| 1/1 21/20 9/8 21/16 '''3/2''' '''7/4'''
|z9(z4)zg9 = z6,4,zg5 = r6,g3,r9
| z9(z4)zg9 = z6,4,zg5 = r6,g3,r9
| |blues scale
| blues scale
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,3o}</span>
| '''A'''<span style="vertical-align: sub;">{21,3o}</span>
| | 21
| 21
| | 3:9:11:15:21:33
| 3:9:11:15:21:33
| | 1/1 5/4 11/8 3/2 7/4 '''11/6'''
| 1/1 5/4 11/8 3/2 7/4 '''11/6'''
|h7,1o7,1o11
| h7,1o7,1o11
| |  
|
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,3u}</span>
| '''A'''<span style="vertical-align: sub;">{21,3u}</span>
| | 21
| 21
| | 105:165:231:315:385:1155
| 105:165:231:315:385:1155
| | 1/1 '''12/11''' 6/5 3/2 18/11 12/7
| 1/1 '''12/11''' 6/5 3/2 18/11 12/7
|s6,1u6,1u9
| s6,1u6,1u9
| |  
|
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,4o}</span>
| '''A'''<span style="vertical-align: sub;">{21,4o}</span>
| | 21
| 21
| | 3:9:13:15:21:39
| 3:9:13:15:21:39
| | 1/1 '''13/12''' 5/4 3/2 13/8 7/4
| 1/1 '''13/12''' 5/4 3/2 13/8 7/4
|h7,3o6,3o9
| h7,3o6,3o9
| |  
|
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,4u}</span>
| '''A'''<span style="vertical-align: sub;">{21,4u}</span>
| | 21
| 21
| | 105:195:273:315:455:1365
| 105:195:273:315:455:1365
| | 1/1 6/5 18/13 3/2 12/7 '''24/13'''
| 1/1 6/5 18/13 3/2 12/7 '''24/13'''
|s6,3u4,3u7
| s6,3u4,3u7
| |
|
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,5o}</span>
| '''A'''<span style="vertical-align: sub;">{21,5o}</span>
| | 21
| 21
| | 3:9:15:17:21:51
| 3:9:15:17:21:51
| | 1/1 17/16 5/4 '''17/12''' 3/2 7/4
| 1/1 17/16 5/4 '''17/12''' 3/2 7/4
|h7,17o9,17o12
| h7,17o9,17o12
| |dom7b9#11
| dom7b9#11
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,5u}</span>
| '''A'''<span style="vertical-align: sub;">{21,5u}</span>
| | 21
| 21
| | 105:255:315:357:595:1785
| 105:255:315:357:595:1785
| | 1/1 18/17 6/5 '''24/17''' 3/2 12/7
| 1/1 18/17 6/5 '''24/17''' 3/2 12/7
|s6,17u8,17u11
| s6,17u8,17u11
| |min6b9#11
| min6b9#11
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,6o}</span>
| '''A'''<span style="vertical-align: sub;">{21,6o}</span>
| | 21
| 21
| | 3:9:15:19:21:57
| 3:9:15:19:21:57
| | 1/1 19/16 5/4 3/2 '''19/12''' 7/4
| 1/1 19/16 5/4 3/2 '''19/12''' 7/4
|19o6,y3,z7
| 19o6,y3,z7
| |dom7#9b13
| dom7#9b13
|-
|-
| | '''A'''<span style="vertical-align: sub;">{21,6u}</span>
| '''A'''<span style="vertical-align: sub;">{21,6u}</span>
| | 21
| 21
| | 105:285:315:399:665:1995
| 105:285:315:399:665:1995
| | 1/1 6/5 '''24/19''' 3/2 12/7 36/19
| 1/1 6/5 '''24/19''' 3/2 12/7 36/19
|19u7,g3,r6
| 19u7,g3,r6
| |maj7#9,13
| maj7#9,13
|-
|-
| | '''A'''<span style="vertical-align: sub;">{23,1o}</span>
| '''A'''<span style="vertical-align: sub;">{23,1o}</span>
| | 23
| 23
| | 3:9:15:21:23:69
| 3:9:15:21:23:69
| | 1/1 5/4 23/16 3/2 7/4 '''23/12'''
| 1/1 5/4 23/16 3/2 7/4 '''23/12'''
|h7,23o5,23o8
| h7,23o5,23o8
| |  
|  
|-
|-
| | '''A'''<span style="vertical-align: sub;">{23,1u}</span>
| '''A'''<span style="vertical-align: sub;">{23,1u}</span>
| | 23
| 23
| | 105:315:345:483:805:2415
| 105:315:345:483:805:2415
| | 1/1 '''24/23''' 6/5 3/2 36/23 12/7
| 1/1 '''24/23''' 6/5 3/2 36/23 12/7
|s6,23u5,23u8
| s6,23u5,23u8
| |  
|
|}
|}


[[category:todo:add definition]]
[[Category:Chords]]
[[Category:Chords]]
[[category:ass]]
[[Category:ASS]]
 
[[Category:Todo:improve synopsis]]

Revision as of 15:26, 5 December 2020

Below is a complete list of Anomalous Saturated Suspensions through the 23-limit. Each chord listed is either ambitonal or has a o/utonal inverse that is also an Anomalous Saturated Suspension (or ASS).

Formal names

For each odd limit we can list ambitonal chords in lexicographic order by harmonic series representation, along with o/utonal chord pairs according to the harmonic series representation of the otonal chord in the pair. Each chord is then designated by a capital "A" whose subscript is a tuple, where the first value is its odd limit and the second value is its index in the list for that odd limit. This is followed by an "a," "o," or "u" depending on whether the chord is ambitonal, otonal, or utonal.

Every chord has a plausible homonym. The alternate root is bolded in the scale. The Color Names column names the homonym for some of the chords. For example, if the first chord is Cg7, it has as a homonym gEby6.

Formal name Odd limit Harmonic series Scale Color name Common name
A{9,1a} 9 3:5:9:15 1/1 6/5 3/2 9/5 g7 = y6 Minor 7th Chord
A{9,2a} 9 3:7:9:21 1/1 7/6 3/2 7/4 z7 = r6 Septimal Minor 7th Chord
A{11,1a} 11 3:9:11:33 1/1 11/8 3/2 11/6 1o7(1o4) = 1u6(1u2)
A{13,1a} 13 3:9:13:39 1/1 13/12 3/2 13/8 3o6(3o2) = 3u7(3u4)
A{15,1o} 15 3:7:9:15:21 1/1 7/6 5/4 3/2 7/4 h7,z10 = r6,ry8 Hendrix
A{15,1u} 15 15:21:35:45:105 1/1 7/6 7/5 3/2 7/4 z7,zg5 = r6,g3 Inverted Hendrix
A{15,2o} 15 3:9:11:15:33 1/1 5/4 11/8 3/2 11/6 y,1o7,1o11 = 1u6,1u9(1uy4) 11-Hendrix
A{15,2u} 15 15:33:45:55:165 1/1 11/10 11/8 3/2 11/6 1o7(1o4)1og9 = g,1u6,1u9 Inverted 11-Hendrix
A{15,3o} 15 3:9:13:15:39 1/1 13/12 5/4 3/2 13/8 y,3o6,3o9 = 3u7(3u4)3uy9 13-Hendrix
A{15,3u} 15 15:39:45:65:195 1/1 13/12 13/10 3/2 13/8 3o6,3o9(3og4) = g,3u7,3u11 Inverted 13-Hendrix
A{17,1o} 17 3:9:15:17:51 1/1 17/16 5/4 17/12 3/2 y,17o9,17o12 17-Hendrix
A{17,1u} 17 15:45:51:85:255 1/1 17/16 17/12 3/2 17/10 17og7(17o2)17o12 Inverted 17-Hendrix
A{19,1o} 19 3:9:15:19:57 1/1 19/16 5/4 3/2 19/12 19o6,y3 = 19u7,19uy5 19-Hendrix
A{19,1u} 19 15:45:57:95:285 1/1 19/16 3/2 19/12 19/10 19o6,19og8 = 19u7,g3 Inverted 19-Hendrix
A{21,1o} 21 3:5:9:15:21:45 1/1 5/4 3/2 5/3 7/4 15/8 y6,z7,y7 = g9,zg9
A{21,1u} 21 7:15:21:35:63:105 1/1 15/14 9/8 5/4 3/2 15/8 y9,ry8 = g7,g6,r6
A{21,2o} 21 3:7:9:15:21:63 1/1 7/6 5/4 21/16 3/2 7/4 h7,z10,z11 = r6,ry8,9 Hendrix add11?
A{21,2u} 21 5:15:21:35:45:105 1/1 21/20 9/8 21/16 3/2 7/4 z9(z4)zg9 = z6,4,zg5 = r6,g3,r9 blues scale
A{21,3o} 21 3:9:11:15:21:33 1/1 5/4 11/8 3/2 7/4 11/6 h7,1o7,1o11
A{21,3u} 21 105:165:231:315:385:1155 1/1 12/11 6/5 3/2 18/11 12/7 s6,1u6,1u9
A{21,4o} 21 3:9:13:15:21:39 1/1 13/12 5/4 3/2 13/8 7/4 h7,3o6,3o9
A{21,4u} 21 105:195:273:315:455:1365 1/1 6/5 18/13 3/2 12/7 24/13 s6,3u4,3u7
A{21,5o} 21 3:9:15:17:21:51 1/1 17/16 5/4 17/12 3/2 7/4 h7,17o9,17o12 dom7b9#11
A{21,5u} 21 105:255:315:357:595:1785 1/1 18/17 6/5 24/17 3/2 12/7 s6,17u8,17u11 min6b9#11
A{21,6o} 21 3:9:15:19:21:57 1/1 19/16 5/4 3/2 19/12 7/4 19o6,y3,z7 dom7#9b13
A{21,6u} 21 105:285:315:399:665:1995 1/1 6/5 24/19 3/2 12/7 36/19 19u7,g3,r6 maj7#9,13
A{23,1o} 23 3:9:15:21:23:69 1/1 5/4 23/16 3/2 7/4 23/12 h7,23o5,23o8
A{23,1u} 23 105:315:345:483:805:2415 1/1 24/23 6/5 3/2 36/23 12/7 s6,23u5,23u8
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