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 ==
Line 72: Line 72:
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
== 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.


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,
=== Octochord to 4/3 -> D = 9/8 (~203.91c) ===
{| class="wikitable"
|+
!A
!B
!C
!Scale
!odd-limit of scale intervals
!Form
|-
|22/21 (~80.54c)
|126/121 (~70.10c)
|49/48 (~35.70c)
|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
|17
|-
|24/23 (~73.68c)
|529/504 (~83.81c)
|49/48 (~35.70c)
|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
|17
|-
|25/24 (~70.67c)
|128/125 (~41.06c)
|27/25 (~133.24c)
|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
|17
|-
|34/33 (~51.68c)
|297/289 (~47.27c)
|121/108 (~196.77c)
|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
|24
|-
|34/33 (~51.68c)
|363/340 (~113.32c)
|300/289 (~64.67c)
|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
|17
|-
|36/35 (~48.77c)
|175/162 (~133.63c)
|49/48 (~35.7c)
|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
|17
|-
|36/35 (~48.77c)
|1225/1188 (~53.1c)
|121/81 (~196.77c)
|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
|24
|}
 
=== Octochord to 7/5 -> D = 50/49 (~34.98c) ===
{| class="wikitable"
|+
!A
!B
!C
!Scale
!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)
|525/512 (~43.41c)
|36/35 (~48.77c)
|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
|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)
|126/121 (~166.50c)
|36/35 (~48.77c)
|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
|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
|-
|35/34 (~50.18c)
|289/280 (~54.77c)
|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
|1225
|22
|}
 
=== Octochord to 6/5 -> D = 25/18 (~568.72c) ===
{| class="wikitable"
|+
!A
!B
!C
!Scale
!odd-limit of scale intervals
!Form
|-
|36/35 (~48.77c)
|49/48 (~35.70c)
|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
|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
|-
|36/35 (~48.77c)
|49/48 (~35.70c)
|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
|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).


with a step pattern of ABACAABADABAACABA, with a mean variety of 4.625.
[[Category:Step-nested scales]]
[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]

Latest revision as of 00:34, 24 June 2025

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 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 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

Octochord to 4/3 -> D = 9/8 (~203.91c)

A B C Scale odd-limit of scale intervals
22/21 (~80.54c) 126/121 (~70.10c) 49/48 (~35.70c) 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
24/23 (~73.68c) 529/504 (~83.81c) 49/48 (~35.70c) 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
25/24 (~70.67c) 128/125 (~41.06c) 27/25 (~133.24c) 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

Octochord to 7/5 -> D = 50/49 (~34.98c)

A B C Scale odd-limit of scale intervals
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

17-form

The scales with an octachord to 4/3 follow 17-form, but with a gap for two notes between 4/3 and 3/2. In the case of the scale 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, if we wish to keep the limit at 441, a 12/11 above 4/3 and below 3/2 give 11/8 and 16/11, which are the simplest approximations of the two middle notes of 17edo in the scale's subgroup. The resulting scale,

1/1 22/21 12/11 8/7 7/6 11/9 14/11 4/3 11/8 16/11 3/2 11/7 18/11 12/7 7/4 11/6 21/11 2/1,

still remains within an odd-limit of 441, and has pattern ABACABADEDABACABA, with a mean variety of 6.

Alternatively we can keep the number of step intervals to 4 if we sacrifice the odd-limit. Of A, B, and C, the simplest intervals between 4/3 and 3/2 arise when we use C, resulting in

1/1 22/21 12/11 8/7 7/6 11/9 14/11 4/3 49/36 72/49 3/2 11/7 18/11 12/7 7/4 11/6 21/11 2/1,

which has an odd-limit of 2401, and a pattern of ABACABACDCABACABA, with mean variety of 5.

For the second scale, 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, the only way to use intervals of the scale to fill the gap whilst keeping the limit at 529 is to place A = 24/23 above 4/3 and below 3/2, resulting in

1/1 24/23 23/21 8/7 7/6 28/23 23/18 4/3 32/23 23/16 3/2 36/23 23/14 12/7 7/4 42/23 23/12 2/1,

with a step pattern of ABACABAADAABACABA, and a mean variety of 4.625.

The same true of the third scale, 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. We add A above 4/3 and below 3/2, resulting in the scale

1/1 25/24 16/15 10/9 6/5 5/4 32/25 4/3 25/18 36/25 3/2 25/16 8/5 5/3 9/5 15/8 48/25 2/1,

with an odd-limit of 625, a step pattern of ABACABAADAABACABA, and a mean variety of 4.625.

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 to approximate the scale.

Octochord to 4/3 -> D = 9/8 (~203.91c)

A B C Scale odd-limit of scale intervals Form
22/21 (~80.54c) 126/121 (~70.10c) 49/48 (~35.70c) 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 17
24/23 (~73.68c) 529/504 (~83.81c) 49/48 (~35.70c) 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 17
25/24 (~70.67c) 128/125 (~41.06c) 27/25 (~133.24c) 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 17
34/33 (~51.68c) 297/289 (~47.27c) 121/108 (~196.77c) 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 24
34/33 (~51.68c) 363/340 (~113.32c) 300/289 (~64.67c) 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 17
36/35 (~48.77c) 175/162 (~133.63c) 49/48 (~35.7c) 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 17
36/35 (~48.77c) 1225/1188 (~53.1c) 121/81 (~196.77c) 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 24

Octochord to 7/5 -> D = 50/49 (~34.98c)

A B C Scale 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) 525/512 (~43.41c) 36/35 (~48.77c) 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 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) 126/121 (~166.50c) 36/35 (~48.77c) 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 15

Octochord to 5/4 -> D = 32/25 (~427.37c)

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
35/34 (~50.18c) 289/280 (~54.77c) 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 1225 22

Octochord to 6/5 -> D = 25/18 (~568.72c)

A B C Scale odd-limit of scale intervals Form
36/35 (~48.77c) 49/48 (~35.70c) 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 1225 26/27

1225-limit ABACABADABACABA JI scales with period 3/2, with steps > 20c

Octochord to 6/5 -> D = 25/24 (~70.67c)

A B C Scale odd-limit of scale intervals
36/35 (~48.77c) 49/48 (~35.70c) 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 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).

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