Diaslen: Difference between revisions

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'''{{PAGENAME}}''' (also denoted 4s in groundfault's aberrismic theory) is a family of quasi-diatonic ~~ternary~~ scale ~~patterns~~ with [[~~scale~~ signature]] ~~5L2m4s~~, so named by groundfault who considers [[33edo]]'s L:m:s = 5:2:1 a good tuning for it. The name ''~~diamech~~'' is a blend of ''dia-'' and ~~the~~ [[~~TAMNAMS]] prefix ''mech-'' for [[machinoid~~]]. [[23edo]] is the only edo without a diatonic fifth that has a ~~5L2mks~~ ternary scale pattern, namely ~~5L2m4s~~ with L:m:s = 3:2:1.

'''{{PAGENAME}}''' (also denoted 4s in [[groundfault]]'s [[aberrismic theory]]) is a family of quasi-[[diatonic]] [[scale pattern]]s with [[step signature]] 5L 2m 4s, so named by groundfault who considers [[33edo]]'s L:m:s = 5:2:1 a good tuning for it. The name ''diaslen'' is a blend of ''dia-'' and [[slendric]]. [[23edo]] is the only [[edo]] without a diatonic fifth that has a 5L 2m ks [[ternary]] scale pattern, namely 5L 2m 4s with L:m:s = 3:2:1.

The three ~~canonical diamech~~ scale patterns, denoted 4sL (LsLsLmLsLsm), 4sC (LmLsLsLmLss), and 4sR (LsLmLsLsLms), may be constructed via [[MOS substitution]], giving [[generator sequence]]s for the three scales:

== Structure ==

The three standard diaslen scale patterns, denoted 4sL (LsLsLmLsLsm), 4sC (LmLsLsLmLss), and 4sR (LsLmLsLsLms), may be constructed via [[MOS substitution]], giving [[generator sequence]]s for the three scales:

{| class="wikitable"

|+ ~~5L2m4s~~ as ~~MOS_subst(5, 2, 4; m, s; ''k'')~~

|+ diaslen scales as MOS substitution scales

|-

!rowspan=2| ~~''k''~~

!rowspan=2| chirality

!rowspan=2| filling MOS

!rowspan=2| [[UDP]] for filling MOS

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|| 2-steps

|-

| ~~2 (~~4sC) || <code>mssmss</code> || 4|0(2)

| 4sC || <code>mssmss</code> || 4|0(2)

|colspan=2 style="text-align:right;"| <code>LmLsLsLmLss</code>

|colspan=2| GS('''L'''+'''m''', '''L'''+'''s''', '''L'''+'''s''')

|-

| ~~1 (~~4sR) || <code>smssms</code> || 2|2(2)

| 4sR || <code>smssms</code> || 2|2(2)

|colspan=2 style="text-align:right;"| <code>LsLmLsLsLms</code>

|colspan=2| GS('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s''')

|-

| ~~0 (~~4sL) || <code>ssmssm</code> || 0|4(2)

| 4sL || <code>ssmssm</code> || 0|4(2)

|colspan=2 style="text-align:right;"| <code>LsLsLmLsLsm</code>

|colspan=2| GS('''L'''+'''s''', '''L'''+'''s''', '''L'''+'''m''')

|}

== Tunings and interpretations ==

=== Untempered 2.3.7 ===

The standard untempered 2.3.7 interpretation for diaslen has L = [[9/8]], m = [[49/48]], s = [[64/63]].

=== Buzzard (2.3.7.13/5[640/637, 676/675]) ===

[[Buzzard]] is an important temperament for diaslen offering an elegant combination of interordinals and making 7/4 sharper in systems exaggerating [[49/48]] and splitting it into two equal parts. Assuming the 2.3.7 interpretation above and the constraint m:s = 2:1 makes L + m = 147/128 exactly half of 2L + m + 2s = 4/3, making it natural to set 147/128 equal to 15/13, tempering out [[640/637]] and [[676/675]]. [[53edo]] (9:2:1) and [[58edo]] (10:2:1) offer particularly good tunings for this interpretation; while the 58edo tuning's s step is smaller than that of the 53edo tuning, 58edo has better 7/6, 9/7, 15/13, and 13/10 as well as giving you a free 11:13:15.

== External links ==

* [https://sw3.lumipakkanen.com/scale/LboZJ9YLp 4sC in 53edo tuning]

* [https://sw3.lumipakkanen.com/scale/Lbo60GpgU 4sR in 53edo tuning]

* [https://sw3.lumipakkanen.com/scale/LboMdyjGJ 4sL in 53edo tuning]

[[Category:Rank-3 scales]]

[[Category:Aberrismic theory]]

@@ Line 1: / Line 1: @@
-'''{{PAGENAME}}''' (also denoted 4s in groundfault's aberrismic theory) is a family of quasi-diatonic ternary scale patterns with [[scale signature]] 5L2m4s, so named by groundfault who considers [[33edo]]'s L:m:s = 5:2:1 a good tuning for it. The name ''diamech'' is a blend of ''dia-'' and the [[TAMNAMS]] prefix ''mech-'' for [[machinoid]]. [[23edo]] is the only edo without a diatonic fifth that has a 5L2mks ternary scale pattern, namely 5L2m4s with L:m:s = 3:2:1.
+'''{{PAGENAME}}''' (also denoted 4s in [[groundfault]]'s [[aberrismic theory]]) is a family of quasi-[[diatonic]] [[scale pattern]]s with [[step signature]] 5L 2m 4s, so named by groundfault who considers [[33edo]]'s L:m:s = 5:2:1 a good tuning for it. The name ''diaslen'' is a blend of ''dia-'' and [[slendric]]. [[23edo]] is the only [[edo]] without a diatonic fifth that has a 5L 2m ks [[ternary]] scale pattern, namely 5L 2m 4s with L:m:s = 3:2:1.
-The three canonical diamech scale patterns, denoted 4sL (LsLsLmLsLsm), 4sC (LmLsLsLmLss), and 4sR (LsLmLsLsLms), may be constructed via [[MOS substitution]], giving [[generator sequence]]s for the three scales:
+== Structure ==
+The three standard diaslen scale patterns, denoted 4sL (LsLsLmLsLsm), 4sC (LmLsLsLmLss), and 4sR (LsLmLsLsLms), may be constructed via [[MOS substitution]], giving [[generator sequence]]s for the three scales:
 {| class="wikitable"
-|+ 5L2m4s as MOS_subst(5, 2, 4; m, s; ''k'')
+|+ diaslen scales as MOS substitution scales
 |-
-!rowspan=2| ''k''
+!rowspan=2| chirality
 !rowspan=2| filling MOS
 !rowspan=2| [[UDP]] for filling MOS
@@ Line 17: / Line 18: @@
 || 2-steps
 |-
-| 2 (4sC) || <code>mssmss</code> || 4&#124;0(2)
+| 4sC || <code>mssmss</code> || 4&#124;0(2)
 |colspan=2 style="text-align:right;"| <code>LmLsLsLmLss</code>
 |colspan=2| GS('''L'''+'''m''', '''L'''+'''s''', '''L'''+'''s''')
 |-
-| 1 (4sR) || <code>smssms</code> || 2&#124;2(2)
+| 4sR || <code>smssms</code> || 2&#124;2(2)
 |colspan=2 style="text-align:right;"| <code>LsLmLsLsLms</code>
 |colspan=2| GS('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s''')
 |-
-| 0 (4sL) || <code>ssmssm</code> || 0&#124;4(2)
+| 4sL || <code>ssmssm</code> || 0&#124;4(2)
 |colspan=2 style="text-align:right;"| <code>LsLsLmLsLsm</code>
 |colspan=2| GS('''L'''+'''s''', '''L'''+'''s''', '''L'''+'''m''')
 |}
+== Tunings and interpretations ==
+=== Untempered 2.3.7 ===
+The standard untempered 2.3.7 interpretation for diaslen has L = [[9/8]], m = [[49/48]], s = [[64/63]].
+=== Buzzard (2.3.7.13/5[640/637, 676/675]) ===
+[[Buzzard]] is an important temperament for diaslen offering an elegant combination of interordinals and making 7/4 sharper in systems exaggerating [[49/48]] and splitting it into two equal parts. Assuming the 2.3.7 interpretation above and the constraint m:s = 2:1 makes L + m  = 147/128 exactly half of 2L + m + 2s = 4/3, making it natural to set 147/128 equal to 15/13, tempering out [[640/637]] and [[676/675]]. [[53edo]] (9:2:1) and [[58edo]] (10:2:1) offer particularly good tunings for this interpretation; while the 58edo tuning's s step is smaller than that of the 53edo tuning, 58edo has better 7/6, 9/7, 15/13, and 13/10 as well as giving you a free 11:13:15.
+== External links ==
+* [https://sw3.lumipakkanen.com/scale/LboZJ9YLp 4sC in 53edo tuning]
+* [https://sw3.lumipakkanen.com/scale/Lbo60GpgU 4sR in 53edo tuning]
+* [https://sw3.lumipakkanen.com/scale/LboMdyjGJ 4sL in 53edo tuning]
 [[Category:Rank-3 scales]]
+[[Category:Aberrismic theory]]