Diaslen: Difference between revisions
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'''{{PAGENAME}}''' (also denoted 4s in groundfault's aberrismic theory) is a family of quasi-diatonic | '''{{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 | == 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" | {| class="wikitable" | ||
|+ | |+ diaslen scales as MOS substitution scales | ||
|- | |- | ||
!rowspan=2| | !rowspan=2| chirality | ||
!rowspan=2| filling MOS | !rowspan=2| filling MOS | ||
!rowspan=2| [[UDP]] for filling MOS | !rowspan=2| [[UDP]] for filling MOS | ||
!colspan=2| step pattern | !colspan=2| step pattern | ||
!colspan=2| generator sequence | !colspan=2| [[generator sequence]] | ||
|- | |- | ||
!| template MOS: | !| template MOS: | ||
|| <code>LXLXLXLXLXX</code> | || <code>LXLXLXLXLXX</code> | ||
!| intvl. class of gen.: | !| intvl. class of gen.: | ||
|| 2-steps | || 2-steps | ||
|- | |- | ||
| | | 4sC || <code>mssmss</code> || 4|0(2) | ||
|colspan=2 style="text-align:right;"| <code>LmLsLsLmLss</code> | |colspan=2 style="text-align:right;"| <code>LmLsLsLmLss</code> | ||
|colspan=2| GS('''L'''+'''m''', '''L'''+'''s''', '''L'''+'''s''') | |colspan=2| GS('''L'''+'''m''', '''L'''+'''s''', '''L'''+'''s''') | ||
|- | |- | ||
| | | 4sR || <code>smssms</code> || 2|2(2) | ||
|colspan=2 style="text-align:right;"| <code>LsLmLsLsLms</code> | |colspan=2 style="text-align:right;"| <code>LsLmLsLsLms</code> | ||
|colspan=2| GS('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s''') | |colspan=2| GS('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s''') | ||
|- | |- | ||
| | | 4sL || <code>ssmssm</code> || 0|4(2) | ||
|colspan=2 style="text-align:right;"| <code>LsLsLmLsLsm</code> | |colspan=2 style="text-align:right;"| <code>LsLsLmLsLsm</code> | ||
|colspan=2| GS('''L'''+'''s''', '''L'''+'''s''', '''L'''+'''m''') | |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:Rank-3 scales]] | ||
[[Category:Aberrismic theory]] | |||
Latest revision as of 23:05, 6 March 2025
Diaslen (also denoted 4s in groundfault's aberrismic theory) is a family of quasi-diatonic scale patterns 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.
Structure
The three standard diaslen scale patterns, denoted 4sL (LsLsLmLsLsm), 4sC (LmLsLsLmLss), and 4sR (LsLmLsLsLms), may be constructed via MOS substitution, giving generator sequences for the three scales:
| chirality | filling MOS | UDP for filling MOS | step pattern | generator sequence | ||
|---|---|---|---|---|---|---|
| template MOS: | LXLXLXLXLXX
|
intvl. class of gen.: | 2-steps | |||
| 4sC | mssmss |
4|0(2) | LmLsLsLmLss
|
GS(L+m, L+s, L+s) | ||
| 4sR | smssms |
2|2(2) | LsLmLsLsLms
|
GS(L+s, L+m, L+s) | ||
| 4sL | ssmssm |
0|4(2) | LsLsLmLsLsm
|
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.