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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 separ
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.
 
ated 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 a heptachord. As [[step-nested scales]], ABACABADABACABA scales can be described of as SNS (P, P/H, H/T, A), or equivalently as SNS (P, H, T, A) etc. where P is the period, H is the interval subtended by ABACABA, the heptachord, and T is the interval subtended by ABA, the tetrachord.


== 729-limit ABACABADABACABA JI scales with period 2/1, with steps > 20c ==
== 729-limit ABACABADABACABA JI scales with period 2/1, with steps > 20c ==


=== Heptachord to 4/3 -> D = 9/8 (~203.91c) ===
=== Octochord to 4/3 -> D = 9/8 (~203.91c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 33: Line 31:
|}
|}


=== Heptachord to 7/5 -> D = 50/49 (~34.98c) ===
=== Octochord to 7/5 -> D = 50/49 (~34.98c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 46: Line 44:
|125/112 (~190.12c)
|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
|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|edo]] to approximate the scale.
=== 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
|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).
[[Category:Step-nested scales]]
[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]
Retrieved from "https://en.xen.wiki/w/ABACABADABACABA_JI_scales"