Xenharmonic Wiki

ABACABA JI scales: Difference between revisions

← Older editNewer edit →
VisualWikitext
Lhearne (talk | contribs)
autopatrolled, emailconfirmed
1,636 edits
m small edit to include existence of ABACABA scales with other periods
Lhearne (talk | contribs)
autopatrolled, emailconfirmed
1,636 edits
No edit summary
Line 1: Line 1:
ABACABA is the singular pairwise well-formed generalised step pattern, and the (4, 2, 1) [[SN scale|SNS]] pattern. Scales with this step pattern are known as Cantor-2 scales. Such scales can be thought of as mirror-symmetrical tetrachordal scales. As [[step-nested scales]], all ABACABA scales can be described as SNS (P, P/T, A), or equivalently as SNS (P, T, A), where P is the period, and T = ABA, the outer interval of the tetrachord.
ABACABA is the singular pairwise well-formed generalised step pattern, and the (4, 2, 1) [[SN scale|SNS]] pattern. Scales with this step pattern are known as Cantor-2 scales. Such scales can be thought of as mirror-symmetrical tetrachordal scales. As [[step-nested scales]], all ABACABA scales can be described as SNS (P, P/T, A), or equivalently as SNS (P, T, A), where P is the period, and T = ABA, the outer interval of the tetrachord.
== 225-limit ABACABA scales with period 2/1, with steps > 20c ==
== 225-limit ABACABA scales with period 2/1, with steps > 20c ==
225 is chosen as the limit so that the list includes all ABACABA scales with complexity up to that of the [[5-limit]] double harmonic major scale — [[SNS (2/1, 3/2, 5/4)-7|16/15 5/4 4/3 3/2 8/5 15/8 2/1]] — and a lower limit of 20c for step sizes is chosen so that there are no steps smaller than [[81/80]].   
225 is chosen as the [[odd-limit]] so that the list includes all ABACABA scales with complexity up to that of the [[5-limit]] double harmonic major scale — [[SNS (2/1, 3/2, 5/4)-7|16/15 5/4 4/3 3/2 8/5 15/8 2/1]] — and a lower limit of 20c for step sizes is chosen so that there are no steps smaller than [[81/80]]. For ABACABA scales, 225-odd-limit implies 13-limit.   


=== Tetrachord to 4/3 -> C = 9/8 (~203.91c) ===
=== Tetrachord to 4/3 -> C = 9/8 (~203.91c) ===
Line 243: Line 243:


== 729-limit ABACABA scales with period 2/1, with steps > 20c ==
== 729-limit ABACABA scales with period 2/1, with steps > 20c ==
One scale under such constraints is a degenerate case, wherein A = C: the [[Pythagorean]] diatonic scale, where A = C = [[9/8]], and B = [[256/243]], with a rank-2 form ABABABA. This scale is well-formed, and (5, 2) SNS. Specifically it is SNS (2/1, 3/2)[7]. The most complex interval in this scale is the Pythagorean augmented fourth — [[729/512]] — and it's inversion, the Pythagorean diminished fifth — [[1024/729]]. Accordingly, the scale is 729-limit. 729 is chosen as the limit so that the list includes all ABACABA scales with complexity up to that of the Pythagorean diatonic scale (with steps > 20c). As [[step-nested scales]], all other ABACABA scales with period 2/1 can be best described as SNS (2/1, 2/T, A), or equivalently as SNS (2/1, T, A), where T = ABA, the outer interval of the tetrachord.  
One scale under such constraints is a degenerate case, wherein A = C: the [[Pythagorean]] diatonic scale, where A = C = [[9/8]], and B = [[256/243]], with a rank-2 form ABABABA. This scale is well-formed, and (5, 2) SNS. Specifically it is SNS (2/1, 3/2)[7]. The most complex interval in this scale is the Pythagorean augmented fourth — [[729/512]] — and it's inversion, the Pythagorean diminished fifth — [[1024/729]]. Accordingly, the scale is 729-limit. 729 is chosen as the limit so that the list includes all ABACABA scales with complexity up to that of the Pythagorean diatonic scale (with steps > 20c). As [[step-nested scales]], all other ABACABA scales with period 2/1 can be best described as SNS (2/1, 2/T, A), or equivalently as SNS (2/1, T, A), where T = ABA, the outer interval of the tetrachord. For ABACABA scales, 729-odd-limit implies 23-limit, and a 27-odd limit for A.  


=== Tetrachord to 4/3 -> C = 9/8 ===
=== Tetrachord to 4/3 -> C = 9/8 ===
Line 1,186: Line 1,186:
|}
|}


=== Tetrachord to 5/4 -> C = 32/25 ===
=== Tetrachord to 23/20 -> C = 800/529 ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,629: Line 1,629:
|405/392
|405/392
|1/1 28/27 15/14 10/9 9/5 28/15 27/14 2/1
|1/1 28/27 15/14 10/9 9/5 28/15 27/14 2/1
|729
|}
== 729-limit ABACABA scales with period 3/2, with steps > 20c ==
Given the scales repeat at 3/2, factors of 3 in the odd-limit vary with transposition by a period. Accordingly the odd-limit listed is the odd-limit for intervals in a single period of the scale.
=== Tetrachord to 6/5 -> C = 25/24 (~70.67c) ===
{| class="wikitable"
|+
!A
!B
!Scale
!no-3 odd-limit of scale intervals
|-
|21/20 (~84.47c)
|160/147 (~146.71c)
|1/1 21/20 8/7 6/5 5/4 21/16 10/7 3/2
|147
|-
|24/23 (~73.68c)
|529/480 (~168.28c)
|1/1 24/23 23/20 6/5 5/4 30/23 23/16 3/2
|529
|-
|27/25 (~133.24c)
|250/243 (~49.17c)
|1/1 27/25 10/9 6/5 5/4 27/20 25/18 3/2
|625
|-
|16/15 (~111.73c)
|135/128 (~92.18c)
|1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2
|675
|-
|18/17 (~98.95c)
|289/270 (~117.73c)
|1/1 18/17 17/15 6/5 5/4 45/34 17/12 3/2
|675
|}
=== Tetrachord to 7/6 -> C = 54/49 (~168.21c) ===
{| class="wikitable"
|+
!A
!B
!Scale
!odd-limit of scale intervals
|-
|21/20 (~84.47c)
|200/189 (~97.94c)
|1/1 21/20 10/9 7/6 9/7 27/20 10/7 3/2
|243
|-
|19/18 (~93.60c)
|378/361 (~79.65c)
|1/1 19/18 21/19 7/6 9/7 19/14 27/19 3/2
|361
|-
|28/27 (~62.96c)
|243/224 (~140.95c)
|1/1 28/27 9/8 7/6 9/7 16/9 27/14 3/2
|729
|729
|}
|}
Retrieved from "https://en.xen.wiki/w/ABACABA_JI_scales"