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ABACABADABACABA JI scales: Difference between revisions

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ABACABADABACABA is the (8,4,2,1) [[SNS]] pattern (the rank-4 power SNS). When covering a period of 2/1, such scales are known as Cantor-3 scales. ABACABADABACABA scales can be conceived of as two ABACABA scales separated by an interval D, akin to how ABACABA scales can be conceived of as two ABA tetrachords separated by an interval C. We will classify ABACABADABACABA scales on this page as such, grouping them by the interval subtended by ABACABA, which we will call an octochord. As [[step-nested scales]], ABACABADABACABA scales can be described of as SNS (P, P/O, O/T, A), or equivalently as SNS (P, O, T, A) etc. where P is the period, H is the interval subtended by ABACABA, the octochord, and T is the interval subtended by ABA, the tetrachord. ABACABADABACABA scales, unlike ABA and ABACABA scales, are not pairwise-well formed, and thus their mean variety is above 4. ABACABADABACABA scales have v4454654 4564544, with mean variety ~ 4.57.
ABACABADABACABA is the quaternary [[Fraenkel word]] or the rank-4 power [[SNS]], i.e., the (8,4,2,1) [[SNS]] pattern. When covering a period of 2/1, such scales are known as Cantor-3 scales. ABACABADABACABA scales can be conceived of as two equivalent [[ABACABA JI scales|ABACABA scales]] and a remaining step D, akin to how ABACABA scales can be conceived of as two equivalent ABA tetrachords and a remaining step C. We will classify ABACABADABACABA scales on this page as such, grouping them by the interval subtended by ABACABA, which we will call an octochord. As [[step-nested scales]], ABACABADABACABA scales can be described of as SNS (P, P/O, O/T, A), or equivalently as SNS (P, O, T, A) etc. where P is the period, O is the interval subtended by ABACABA, the octochord, and T is the interval subtended by ABA, the tetrachord. ABACABADABACABA scales, unlike ABA and ABACABA scales, are not [[Rank-3 scale#Pairwise well-formed scales|pairwise well-formed]], and thus their mean variety is above 4. ABACABADABACABA scales have v4454654 4564544, with mean variety ~ 4.57.


== 729-limit ABACABADABACABA JI scales with period 2/1, with steps > 20c ==
== 729-limit ABACABADABACABA JI scales with period 2/1, with steps > 20c ==
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with an odd-limit of 625, a step pattern of ABACABAADAABACABA, and a mean variety of 4.625.
with an odd-limit of 625, a step pattern of ABACABAADAABACABA, and a mean variety of 4.625.
The final scale has gaps between 28/25 and 5/4, and between 8/5 and 25/14, where a minor third and major 6th 'should' be, which we can fill with intervals of the scale while keeping the odd-limit at 625 by adding A below 5/4 and above 8/5, leading to
1/1 21/20 16/15 28/25 25/21 5/4 21/16 4/3 7/5 10/7 3/2 32/21 8/5 42/25 25/14 15/8 40/21 2/1,
with a step pattern of ABACAABADABAACABA, with a mean variety of 4.625.


== 1225-limit ABACABADABACABA JI scales with period 2/1, with steps > 20c ==
== 1225-limit ABACABADABACABA JI scales with period 2/1, with steps > 20c ==
'Form' is added to the tables below, following the above section on 17-form. The 'form' is the also the smallest [[EDO|edo]] to approximate the scale.


=== Octochord to 4/3 -> D = 9/8 (~203.91c) ===
=== Octochord to 4/3 -> D = 9/8 (~203.91c) ===
Line 88: Line 83:
!Scale
!Scale
!odd-limit of scale intervals
!odd-limit of scale intervals
!Form
|-
|-
|22/21 (~80.54c)
|22/21 (~80.54c)
Line 94: Line 90:
|1/1 22/21 12/11 8/7 7/6 11/9 14/11 4/3 3/2 11/7 18/11 12/7 7/4 11/6 21/11 2/1
|1/1 22/21 12/11 8/7 7/6 11/9 14/11 4/3 3/2 11/7 18/11 12/7 7/4 11/6 21/11 2/1
|441
|441
|17
|-
|-
|24/23 (~73.68c)
|24/23 (~73.68c)
Line 100: Line 97:
|1/1 24/23 23/21 8/7 7/6 28/23 23/18 4/3 3/2 36/23 23/14 12/7 7/4 42/23 23/12 2/1
|1/1 24/23 23/21 8/7 7/6 28/23 23/18 4/3 3/2 36/23 23/14 12/7 7/4 42/23 23/12 2/1
|529
|529
|17
|-
|-
|25/24 (~70.67c)
|25/24 (~70.67c)
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|1/1 25/24 16/15 10/9 6/5 5/4 32/25 4/3 3/2 25/16 8/5 5/3 9/5 15/8 48/25 2/1
|1/1 25/24 16/15 10/9 6/5 5/4 32/25 4/3 3/2 25/16 8/5 5/3 9/5 15/8 48/25 2/1
|625
|625
|17
|-
|-
|34/33 (~51.68c)
|34/33 (~51.68c)
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|1/1 34/33 18/17 12/11 11/9 34/27 22/17 4/3 3/2 17/11 27/17 18/11 11/6 17/9 33/17 2/1
|1/1 34/33 18/17 12/11 11/9 34/27 22/17 4/3 3/2 17/11 27/17 18/11 11/6 17/9 33/17 2/1
|1089
|1089
|24
|-
|-
|34/33 (~51.68c)  
|34/33 (~51.68c)  
Line 118: Line 118:
|1/1 34/33 11/10 17/15 20/17 40/33 22/17 4/3 3/2 17/11 33/20 17/10 30/17 20/11 33/17 2/1  
|1/1 34/33 11/10 17/15 20/17 40/33 22/17 4/3 3/2 17/11 33/20 17/10 30/17 20/11 33/17 2/1  
|1089
|1089
|17
|-
|-
|36/35 (~48.77c)
|36/35 (~48.77c)
Line 124: Line 125:
|1/1 36/35 10/9 8/7 7/6 6/5 35/27 4/3 3/2 54/35 5/3 12/7 7/4 9/5 35/18 2/1
|1/1 36/35 10/9 8/7 7/6 6/5 35/27 4/3 3/2 54/35 5/3 12/7 7/4 9/5 35/18 2/1
|1225
|1225
|17
|-
|-
|36/35 (~48.77c)
|36/35 (~48.77c)
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|1/1 36/35 35/33 12/11 11/9 44/35 35/27 4/3 3/2 54/35 35/22 18/11 11/6 66/35 35/18 2/1
|1/1 36/35 35/33 12/11 11/9 44/35 35/27 4/3 3/2 54/35 35/22 18/11 11/6 66/35 35/18 2/1
|1225
|1225
|24
|}
|}


Line 140: Line 143:
!Scale
!Scale
!odd-limit of scale intervals
!odd-limit of scale intervals
!Form
|-
|21/20 (~84.47c)
|64/63 (~27.26c)
|125/112 (~190.12c)
|1/1 21/20 16/15 28/25 5/4 21/16 4/3 7/5 10/7 3/2 32/21 8/5 25/14 15/8 40/21 2/1
|625
|19
|-
|26/25 (~67.9c)
|175/169 (~60.4c)
|125/112 (~190.12c)
|1/1 26/25 14/13 28/25 5/4 13/10 35/26 7/5 10/7 52/35 20/13 8/5 25/14 13/7 25/13 2/1
|1225
|19
|-
|28/27 (~62.96c)
|729/700 (~70.28c)
|125/112 (~190.12c)
|1/1 28/27 27/25 28/25 5/4 35/27 27/20 7/5 10/7 40/27 54/35 8/5 25/14 50/27 27/14 2/1
|1225
|19
|-
|-
|16/15 (~111.73c)
|16/15 (~111.73c)
|525/512 (~43.41c)
|525/512 (~43.41c)
|36/35 (~48.77c)
|36/35 (~48.77c)
|16/15 35/32 7/6 6/5 32/25 21/16 7/5 10/7 32/21 25/16 5/3 12/7 64/35 15/8 2/1
|1/1 16/15 35/32 7/6 6/5 32/25 21/16 7/5 10/7 32/21 25/16 5/3 12/7 64/35 15/8 2/1
|1225
|1225
|15
|-
|31/30 (~56.77c)
|1050/961 (~153.34c)
|36/35 (~48.77c)
|1/1 31/30 35/31 7/6 6/5 31/25 42/31 7/5 10/7 31/21 50/31 5/3 12/7 62/35 40/31 2/1
|1225
|15
|-
|-
|35/34 (~50.18c)
|35/34 (~50.18c)
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|1/1 35/34 17/15 7/6 6/5 21/17 34/25 7/5 10/7 25/17 34/21 5/3 12/7 30/17 68/35 2/1
|1/1 35/34 17/15 7/6 6/5 21/17 34/25 7/5 10/7 25/17 34/21 5/3 12/7 30/17 68/35 2/1
|1225
|1225
|15
|}
=== Octochord to 5/4 -> D = 32/25 (~427.37c) ===
{| class="wikitable"
|+
!A
!B
!C
!Scale
!odd-limit of scale intervals
!Form
|-
|33/32 (~53.27c)
|1120/1089 (48.59c)
|256/245 (~76.03c)
|1/1 33/32 35/33 35/32 8/7 33/28 40/33 5/4 8/5 33/20 48/33 7/4 64/35 66/35 64/33 2/1
|1225
|22
|-
|-
|31/30 (~56.77c)
|35/34 (~50.18c)
|1050/961 (~153.34c)
|289/280 (~54.77c)
|36/35 (~48.77c)
|256/245 (~76.03c)
|1/1 31/30 35/31 7/6 6/5 31/25 42/31 7/5 10/7 31/21 50/31 5/3 12/7 62/35 40/31 2/1
|1/1 35/34 17/16 35/32 8/7 20/17 17/14 5/4 8/5 28/17 17/10 7/4 64/35 32/17 35/17 2/1
|1225
|1225
|22
|}
=== Octochord to 6/5 -> D = 25/18 (~568.72c) ===
{| class="wikitable"
|+
!A
!B
!C
!Scale
!odd-limit of scale intervals
!Form
|-
|-
|28/27 (~62.96c)
|36/35 (~48.77c)
|729/700 (~70.28c)
|49/48 (~35.70c)
|125/112 (190.12c)
|250/243 (~49.17c)
|1/1 28/27 27/25 28/25 5/4 35/27 27/20 7/5 10/7 40/27 54/35 8/5 25/14 50/27 27/14 2/1
|1/1 36/35 21/20 27/25 10/9 8/7 7/6 6/5 5/3 12/7 7/4 9/5 50/27 40/21 35/18 2/1
|1225
|1225
|26/27
|}
== 1225-limit ABACABADABACABA JI scales with period 3/2, with steps > 20c ==
=== Octochord to 6/5 -> D = 25/24 (~70.67c) ===
{| class="wikitable"
|+
!A
!B
!C
!Scale
!odd-limit of scale intervals
|-
|-
|26/25 (~67.9c)
|36/35 (~48.77c)
|175/169 (~60.4c)
|49/48 (~35.70c)
|125/112 (190.12c)
|250/243 (~49.17c)
|1/1 26/25 14/13 28/25 5/4 13/10 35/26 7/5 10/7 52/35 20/13 8/5 25/14 13/7 25/13 2/1
|1/1 36/35 21/20 27/25 10/9 8/7 7/6 6/5 5/4 9/7 21/16 27/20 25/18 10/7 35/24 3/2
|1225
|1225
|}
|}
Noticing that A and C are almost exactly the same size, we temper them together without much loss of accuracy, tempering out [[4375/4374]], the [[Ragisma]]. The tempered scale then has a scale pattern of ABAAABACABAAABA (relabeling so the most frequent step is A and the least frequent is C).
[[Category:Step-nested scales]]
[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]
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