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 equivalent 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.

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 ==

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|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

|~~1125~~

|1225

|22

|-

Line 209:

|256/245 (~76.03c)

|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

|~~1125~~

|1225

|22

|}

Line 227:

|250/243 (~49.17c)

|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

|~~1125~~

|1225

|26/27

|}

Line 245:

|250/243 (~49.17c)

|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

|~~1125~~

|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]]

@@ Line 1: / Line 1: @@
-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 equivalent 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.
+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 ==
@@ Line 202: / Line 202: @@
 |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
-|1125
+|1225
 |22
 |-
@@ Line 209: / Line 209: @@
 |256/245 (~76.03c)
 |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
-|1125
+|1225
 |22
 |}
@@ Line 227: / Line 227: @@
 |250/243 (~49.17c)
 |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
-|1125
+|1225
 |26/27
 |}
@@ Line 245: / Line 245: @@
 |250/243 (~49.17c)
 |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
-|1125
+|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]]