2L 9s: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox MOS
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Name =  
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-02-11 18:48:58 UTC</tt>.<br>
| Periods = 1
: The original revision id was <tt>540678354</tt>.<br>
| nLargeSteps = 2
: The revision comment was: <tt></tt><br>
| nSmallSteps = 9
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Equalized = 5
<h4>Original Wikitext content:</h4>
| Collapsed = 1
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">This MOS, with a generator between 5/11edo (545 5/11 cents) and 1/2edo (600), reaches a harmonic entropy minimum when the generator is 7/5. It represents temperaments like Tritonic, Triton, Heinz and Liese.
| Pattern = LssssLsssss
|| 5/11 ||  ||  ||  ||  || 545 5/11 ||
}}
||  ||  ||  ||  || 21/46 || 547.826087 ||
{{MOS intro}}
||  ||  ||  || 16/35 ||  || 548 4/7 ||
||  ||  ||  ||  || 27/59 || 549.152542 ||
||  ||  || 11/24 ||  ||  || 550 ||
||  ||  ||  ||  || 28/61 || 550.819672 ||
||  ||  ||  || 17/37 ||  || 551 13/37 ||
||  ||  ||  ||  || 23/50 || 552 ||
||  || 6/13 ||  ||  ||  || 553 11/13 ||
||  ||  ||  ||  || 19/41 || 556 4/41 ||
||  ||  ||  || 13/28 ||  || 557 1/7 ||
||  ||  ||  ||  || 20/43 || 558.139535 ||
||  ||  ||  ||  ||  || 559.852266 ||
||  ||  || 7/15 ||  ||  || 560 ||
||  ||  ||  ||  ||  || 560.148007 ||
||  ||  ||  ||  || 15/32 || 562.5 ||
||  ||  ||  || 8/17 ||  || 564.705882 ||
||  ||  ||  ||  || 9/19 || 568.421053 ||
|| 1/2 ||  ||  ||  ||  || 600 ||</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;2L 9s&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This MOS, with a generator between 5/11edo (545 5/11 cents) and 1/2edo (600), reaches a harmonic entropy minimum when the generator is 7/5. It represents temperaments like Tritonic, Triton, Heinz and Liese.&lt;br /&gt;


This MOS reaches a [[harmonic entropy]] minimum when the generator is [[7/5]]. It represents temperaments like [[Tritonic]], [[Triton]], [[Heinz]] and [[Liese]].


&lt;table class="wiki_table"&gt;
== Scale properties ==
    &lt;tr&gt;
{{TAMNAMS use}}
        &lt;td&gt;5/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;545 5/11&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/46&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;547.826087&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/35&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;548 4/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/59&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;549.152542&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;550&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;28/61&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;550.819672&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;17/37&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;551 13/37&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;23/50&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;552&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;553 11/13&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;19/41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;556 4/41&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;557 1/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20/43&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;558.139535&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;559.852266&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;560&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;560.148007&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;562.5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;564.705882&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/19&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;568.421053&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;600&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Intervals ===
{{MOS intervals}}
 
=== Generator chain ===
{{MOS genchain}}
 
=== Modes ===
{{MOS mode degrees}}
 
== Scale tree ==
{{MOS tuning spectrum
| 6/1 = [[Tritonic]]&nbsp;↓
| 5/1 = [[Triton]] / [[liese]]
| 5/4 = [[Heinz]]
}}
 
{{todo|expand}}
 
[[Category:11-tone scales]]

Latest revision as of 18:57, 3 March 2025

↖ 1L 8s ↑ 2L 8s 3L 8s ↗
← 1L 9s 2L 9s 3L 9s →
↙ 1L 10s ↓ 2L 10s 3L 10s ↘ 
┌╥┬┬┬┬╥┬┬┬┬┬┐
│║││││║││││││
│││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LssssLsssss
sssssLssssL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 5\11 to 1\2 (545.5 ¢ to 600.0 ¢)
Dark 1\2 to 6\11 (600.0 ¢ to 654.5 ¢)
TAMNAMS information
Related to 2L 7s (balzano)
With tunings 2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent 2L 7s
Sister 9L 2s
Daughters 11L 2s, 2L 11s
Neutralized 4L 7s
2-Flought 13L 9s, 2L 20s
Equal tunings
Equalized (L:s = 1:1) 5\11 (545.5 ¢)
Supersoft (L:s = 4:3) 16\35 (548.6 ¢)
Soft (L:s = 3:2) 11\24 (550.0 ¢)
Semisoft (L:s = 5:3) 17\37 (551.4 ¢)
Basic (L:s = 2:1) 6\13 (553.8 ¢)
Semihard (L:s = 5:2) 13\28 (557.1 ¢)
Hard (L:s = 3:1) 7\15 (560.0 ¢)
Superhard (L:s = 4:1) 8\17 (564.7 ¢)
Collapsed (L:s = 1:0) 1\2 (600.0 ¢)

2L 9s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 2 large steps and 9 small steps, repeating every octave. 2L 9s is a child scale of 2L 7s, expanding it by 2 tones. Generators that produce this scale range from 545.5 ¢ to 600 ¢, or from 600 ¢ to 654.5 ¢.

This MOS reaches a harmonic entropy minimum when the generator is 7/5. It represents temperaments like Tritonic, Triton, Heinz and Liese.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 2L 9s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Minor 1-mosstep m1ms s 0.0 ¢ to 109.1 ¢
Major 1-mosstep M1ms L 109.1 ¢ to 600.0 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 218.2 ¢
Major 2-mosstep M2ms L + s 218.2 ¢ to 600.0 ¢
3-mosstep Minor 3-mosstep m3ms 3s 0.0 ¢ to 327.3 ¢
Major 3-mosstep M3ms L + 2s 327.3 ¢ to 600.0 ¢
4-mosstep Minor 4-mosstep m4ms 4s 0.0 ¢ to 436.4 ¢
Major 4-mosstep M4ms L + 3s 436.4 ¢ to 600.0 ¢
5-mosstep Diminished 5-mosstep d5ms 5s 0.0 ¢ to 545.5 ¢
Perfect 5-mosstep P5ms L + 4s 545.5 ¢ to 600.0 ¢
6-mosstep Perfect 6-mosstep P6ms L + 5s 600.0 ¢ to 654.5 ¢
Augmented 6-mosstep A6ms 2L + 4s 654.5 ¢ to 1200.0 ¢
7-mosstep Minor 7-mosstep m7ms L + 6s 600.0 ¢ to 763.6 ¢
Major 7-mosstep M7ms 2L + 5s 763.6 ¢ to 1200.0 ¢
8-mosstep Minor 8-mosstep m8ms L + 7s 600.0 ¢ to 872.7 ¢
Major 8-mosstep M8ms 2L + 6s 872.7 ¢ to 1200.0 ¢
9-mosstep Minor 9-mosstep m9ms L + 8s 600.0 ¢ to 981.8 ¢
Major 9-mosstep M9ms 2L + 7s 981.8 ¢ to 1200.0 ¢
10-mosstep Minor 10-mosstep m10ms L + 9s 600.0 ¢ to 1090.9 ¢
Major 10-mosstep M10ms 2L + 8s 1090.9 ¢ to 1200.0 ¢
11-mosstep Perfect 11-mosstep P11ms 2L + 9s 1200.0 ¢

Generator chain

Generator chain of 2L 9s
Bright gens Scale degree Abbrev.
12 Augmented 5-mosdegree A5md
11 Augmented 0-mosdegree A0md
10 Augmented 6-mosdegree A6md
9 Major 1-mosdegree M1md
8 Major 7-mosdegree M7md
7 Major 2-mosdegree M2md
6 Major 8-mosdegree M8md
5 Major 3-mosdegree M3md
4 Major 9-mosdegree M9md
3 Major 4-mosdegree M4md
2 Major 10-mosdegree M10md
1 Perfect 5-mosdegree P5md
0 Perfect 0-mosdegree
Perfect 11-mosdegree		 P0md
P11md
−1 Perfect 6-mosdegree P6md
−2 Minor 1-mosdegree m1md
−3 Minor 7-mosdegree m7md
−4 Minor 2-mosdegree m2md
−5 Minor 8-mosdegree m8md
−6 Minor 3-mosdegree m3md
−7 Minor 9-mosdegree m9md
−8 Minor 4-mosdegree m4md
−9 Minor 10-mosdegree m10md
−10 Diminished 5-mosdegree d5md
−11 Diminished 11-mosdegree d11md
−12 Diminished 6-mosdegree d6md

Modes

Scale degrees of the modes of 2L 9s
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11
10|0 1 LssssLsssss Perf. Maj. Maj. Maj. Maj. Perf. Aug. Maj. Maj. Maj. Maj. Perf.
9|1 6 LsssssLssss Perf. Maj. Maj. Maj. Maj. Perf. Perf. Maj. Maj. Maj. Maj. Perf.
8|2 11 sLssssLssss Perf. Min. Maj. Maj. Maj. Perf. Perf. Maj. Maj. Maj. Maj. Perf.
7|3 5 sLsssssLsss Perf. Min. Maj. Maj. Maj. Perf. Perf. Min. Maj. Maj. Maj. Perf.
6|4 10 ssLssssLsss Perf. Min. Min. Maj. Maj. Perf. Perf. Min. Maj. Maj. Maj. Perf.
5|5 4 ssLsssssLss Perf. Min. Min. Maj. Maj. Perf. Perf. Min. Min. Maj. Maj. Perf.
4|6 9 sssLssssLss Perf. Min. Min. Min. Maj. Perf. Perf. Min. Min. Maj. Maj. Perf.
3|7 3 sssLsssssLs Perf. Min. Min. Min. Maj. Perf. Perf. Min. Min. Min. Maj. Perf.
2|8 8 ssssLssssLs Perf. Min. Min. Min. Min. Perf. Perf. Min. Min. Min. Maj. Perf.
1|9 2 ssssLsssssL Perf. Min. Min. Min. Min. Perf. Perf. Min. Min. Min. Min. Perf.
0|10 7 sssssLssssL Perf. Min. Min. Min. Min. Dim. Perf. Min. Min. Min. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 2L 9s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
5\11 545.455 654.545 1:1 1.000 Equalized 2L 9s
26\57 547.368 652.632 6:5 1.200
21\46 547.826 652.174 5:4 1.250 Heinz
37\81 548.148 651.852 9:7 1.286
16\35 548.571 651.429 4:3 1.333 Supersoft 2L 9s
43\94 548.936 651.064 11:8 1.375
27\59 549.153 650.847 7:5 1.400
38\83 549.398 650.602 10:7 1.429
11\24 550.000 650.000 3:2 1.500 Soft 2L 9s
39\85 550.588 649.412 11:7 1.571
28\61 550.820 649.180 8:5 1.600
45\98 551.020 648.980 13:8 1.625
17\37 551.351 648.649 5:3 1.667 Semisoft 2L 9s
40\87 551.724 648.276 12:7 1.714
23\50 552.000 648.000 7:4 1.750
29\63 552.381 647.619 9:5 1.800
6\13 553.846 646.154 2:1 2.000 Basic 2L 9s
Scales with tunings softer than this are proper
25\54 555.556 644.444 9:4 2.250
19\41 556.098 643.902 7:3 2.333
32\69 556.522 643.478 12:5 2.400
13\28 557.143 642.857 5:2 2.500 Semihard 2L 9s
33\71 557.746 642.254 13:5 2.600
20\43 558.140 641.860 8:3 2.667
27\58 558.621 641.379 11:4 2.750
7\15 560.000 640.000 3:1 3.000 Hard 2L 9s
22\47 561.702 638.298 10:3 3.333
15\32 562.500 637.500 7:2 3.500
23\49 563.265 636.735 11:3 3.667
8\17 564.706 635.294 4:1 4.000 Superhard 2L 9s
17\36 566.667 633.333 9:2 4.500
9\19 568.421 631.579 5:1 5.000 Triton / liese
10\21 571.429 628.571 6:1 6.000 Tritonic ↓
1\2 600.000 600.000 1:0 → ∞ Collapsed 2L 9s
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