12L 5s: Difference between revisions

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The [[leapday]]/[[leapweek]] version is proper, but the Pythagorean/schismic version is improper (it does not become proper until you add 12 more notes to form the schismic 29-note scale).
The [[leapday]]/[[leapweek]] version is proper, but the Pythagorean/schismic version is improper (it does not become proper until you add 12 more notes to form the schismic 29-note scale).
==Intervals==
 
== Scale properties ==
{{TAMNAMS use}}
 
=== Intervals ===
{{MOS intervals}}
{{MOS intervals}}
== Modes ==
 
=== Generator chain ===
{{MOS genchain}}
 
=== Modes ===
{{MOS mode degrees}}
{{MOS mode degrees}}
=== Proposed tuning-specific names ===
=== Proposed tuning-specific names ===
[[Declan Paul Boushy]] has proposed names for these modes corresponding to step ratios [[Subaru scale|3:1]] and [[Tanegashima scale|4:1]].
[[Declan Paul Boushy]] has proposed names for these modes corresponding to step ratios [[Subaru scale|3:1]] and [[Tanegashima scale|4:1]].
{{todo|add etymology|add Template:MOS modes and annotate each row using Boushy’s names}}


== Scales ==
== Scales ==
Line 21: Line 31:


== Scale tree ==
== Scale tree ==
{{Scale tree|Comments=
{{MOS tuning spectrum
4/3: [[Leapfrog]];
| 4/3 = [[Leapfrog]]  
7/5: [[Leapweek]];
| 7/5 = [[Leapweek]]  
11/7: [[Leapday]] / [[polypyth]];
| 3/2 = [[Leapday]]  
13/8: Golden neogothic (495.9044¢);
| 11/7 = [[Polypyth]]  
7/3: [[Undecental]];
| 13/8 = Golden neogothic (495.904{{c}})  
12/5: Argent tuning (497.0563¢);
| 7/3 = [[Undecental]]  
13/5: Unnamed golden tuning (497.2540¢)
| 12/5 = Argent tuning (497.056{{c}})  
11/4: [[Kwai]];
| 13/5 = Unnamed golden tuning (497.254{{c}})  
3/1: [[Garibaldi]] / [[andromeda]];
| 11/4 = [[Kwai]]  
7/2: Garibaldi / [[cassandra]];
| 3/1 = [[Garibaldi]] / [[andromeda]]  
4/1: Garibaldi / [[helenus]], Pythagorean tuning (498.0450¢);
| 7/2 = Garibaldi / [[cassandra]]  
9/2: [[Pontiac]];
| 4/1 = Garibaldi / [[helenus]], Pythagorean tuning (498.045{{c}})  
5/1: [[Photia]];
| 9/2 = [[Pontiac]]  
6/1: ↓ [[Grackle]]
| 5/1 = [[Photia]]  
| 6/1 = ↓ [[Grackle]], ↓↓ [[gracecordial]]  
}}
}}


[[Category:17-tone scales]]
[[Category:17-tone scales]]
[[Category:Mega chromatic scales]]
[[Category:Mega chromatic scales]]

Latest revision as of 18:42, 3 March 2025

↖ 11L 4s ↑ 12L 4s 13L 4s ↗
← 11L 5s 12L 5s 13L 5s →
↙ 11L 6s ↓ 12L 6s 13L 6s ↘ 
┌╥╥╥┬╥╥┬╥╥╥┬╥╥┬╥╥┬┐
│║║║│║║│║║║│║║│║║││
│││││││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLsLLLsLLsLLs
sLLsLLsLLLsLLsLLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 7\17 to 5\12 (494.1 ¢ to 500.0 ¢)
Dark 7\12 to 10\17 (700.0 ¢ to 705.9 ¢)
TAMNAMS information
Related to 5L 2s (diatonic)
With tunings 2:1 to 3:1 (hypohard)
Related MOS scales
Parent 5L 7s
Sister 5L 12s
Daughters 17L 12s, 12L 17s
Neutralized 7L 10s
2-Flought 29L 5s, 12L 22s
Equal tunings
Equalized (L:s = 1:1) 7\17 (494.1 ¢)
Supersoft (L:s = 4:3) 26\63 (495.2 ¢)
Soft (L:s = 3:2) 19\46 (495.7 ¢)
Semisoft (L:s = 5:3) 31\75 (496.0 ¢)
Basic (L:s = 2:1) 12\29 (496.6 ¢)
Semihard (L:s = 5:2) 29\70 (497.1 ¢)
Hard (L:s = 3:1) 17\41 (497.6 ¢)
Superhard (L:s = 4:1) 22\53 (498.1 ¢)
Collapsed (L:s = 1:0) 5\12 (500.0 ¢)

12L 5s, also called p-enharmonic, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 12 large steps and 5 small steps, repeating every octave. 12L 5s is a grandchild scale of 5L 2s, expanding it by 10 tones. Generators that produce this scale range from 494.1 ¢ to 500 ¢, or from 700 ¢ to 705.9 ¢. Temperaments supported by this scale include those under the Pythagorean and schismic families, characterized by a diesis (the difference between a large step and two small steps) that is smaller than the chroma.

The leapday/leapweek version is proper, but the Pythagorean/schismic version is improper (it does not become proper until you add 12 more notes to form the schismic 29-note scale).

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 12L 5s
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 70.6 ¢
Major 1-mosstep M1ms L 70.6 ¢ to 100.0 ¢
2-mosstep Minor 2-mosstep m2ms L + s 100.0 ¢ to 141.2 ¢
Major 2-mosstep M2ms 2L 141.2 ¢ to 200.0 ¢
3-mosstep Minor 3-mosstep m3ms 2L + s 200.0 ¢ to 211.8 ¢
Major 3-mosstep M3ms 3L 211.8 ¢ to 300.0 ¢
4-mosstep Minor 4-mosstep m4ms 2L + 2s 200.0 ¢ to 282.4 ¢
Major 4-mosstep M4ms 3L + s 282.4 ¢ to 300.0 ¢
5-mosstep Minor 5-mosstep m5ms 3L + 2s 300.0 ¢ to 352.9 ¢
Major 5-mosstep M5ms 4L + s 352.9 ¢ to 400.0 ¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 400.0 ¢ to 423.5 ¢
Major 6-mosstep M6ms 5L + s 423.5 ¢ to 500.0 ¢
7-mosstep Diminished 7-mosstep d7ms 4L + 3s 400.0 ¢ to 494.1 ¢
Perfect 7-mosstep P7ms 5L + 2s 494.1 ¢ to 500.0 ¢
8-mosstep Minor 8-mosstep m8ms 5L + 3s 500.0 ¢ to 564.7 ¢
Major 8-mosstep M8ms 6L + 2s 564.7 ¢ to 600.0 ¢
9-mosstep Minor 9-mosstep m9ms 6L + 3s 600.0 ¢ to 635.3 ¢
Major 9-mosstep M9ms 7L + 2s 635.3 ¢ to 700.0 ¢
10-mosstep Perfect 10-mosstep P10ms 7L + 3s 700.0 ¢ to 705.9 ¢
Augmented 10-mosstep A10ms 8L + 2s 705.9 ¢ to 800.0 ¢
11-mosstep Minor 11-mosstep m11ms 7L + 4s 700.0 ¢ to 776.5 ¢
Major 11-mosstep M11ms 8L + 3s 776.5 ¢ to 800.0 ¢
12-mosstep Minor 12-mosstep m12ms 8L + 4s 800.0 ¢ to 847.1 ¢
Major 12-mosstep M12ms 9L + 3s 847.1 ¢ to 900.0 ¢
13-mosstep Minor 13-mosstep m13ms 9L + 4s 900.0 ¢ to 917.6 ¢
Major 13-mosstep M13ms 10L + 3s 917.6 ¢ to 1000.0 ¢
14-mosstep Minor 14-mosstep m14ms 9L + 5s 900.0 ¢ to 988.2 ¢
Major 14-mosstep M14ms 10L + 4s 988.2 ¢ to 1000.0 ¢
15-mosstep Minor 15-mosstep m15ms 10L + 5s 1000.0 ¢ to 1058.8 ¢
Major 15-mosstep M15ms 11L + 4s 1058.8 ¢ to 1100.0 ¢
16-mosstep Minor 16-mosstep m16ms 11L + 5s 1100.0 ¢ to 1129.4 ¢
Major 16-mosstep M16ms 12L + 4s 1129.4 ¢ to 1200.0 ¢
17-mosstep Perfect 17-mosstep P17ms 12L + 5s 1200.0 ¢

Generator chain

Generator chain of 12L 5s
Bright gens Scale degree Abbrev.
28 Augmented 9-mosdegree A9md
27 Augmented 2-mosdegree A2md
26 Augmented 12-mosdegree A12md
25 Augmented 5-mosdegree A5md
24 Augmented 15-mosdegree A15md
23 Augmented 8-mosdegree A8md
22 Augmented 1-mosdegree A1md
21 Augmented 11-mosdegree A11md
20 Augmented 4-mosdegree A4md
19 Augmented 14-mosdegree A14md
18 Augmented 7-mosdegree A7md
17 Augmented 0-mosdegree A0md
16 Augmented 10-mosdegree A10md
15 Major 3-mosdegree M3md
14 Major 13-mosdegree M13md
13 Major 6-mosdegree M6md
12 Major 16-mosdegree M16md
11 Major 9-mosdegree M9md
10 Major 2-mosdegree M2md
9 Major 12-mosdegree M12md
8 Major 5-mosdegree M5md
7 Major 15-mosdegree M15md
6 Major 8-mosdegree M8md
5 Major 1-mosdegree M1md
4 Major 11-mosdegree M11md
3 Major 4-mosdegree M4md
2 Major 14-mosdegree M14md
1 Perfect 7-mosdegree P7md
0 Perfect 0-mosdegree
Perfect 17-mosdegree		 P0md
P17md
−1 Perfect 10-mosdegree P10md
−2 Minor 3-mosdegree m3md
−3 Minor 13-mosdegree m13md
−4 Minor 6-mosdegree m6md
−5 Minor 16-mosdegree m16md
−6 Minor 9-mosdegree m9md
−7 Minor 2-mosdegree m2md
−8 Minor 12-mosdegree m12md
−9 Minor 5-mosdegree m5md
−10 Minor 15-mosdegree m15md
−11 Minor 8-mosdegree m8md
−12 Minor 1-mosdegree m1md
−13 Minor 11-mosdegree m11md
−14 Minor 4-mosdegree m4md
−15 Minor 14-mosdegree m14md
−16 Diminished 7-mosdegree d7md
−17 Diminished 17-mosdegree d17md
−18 Diminished 10-mosdegree d10md
−19 Diminished 3-mosdegree d3md
−20 Diminished 13-mosdegree d13md
−21 Diminished 6-mosdegree d6md
−22 Diminished 16-mosdegree d16md
−23 Diminished 9-mosdegree d9md
−24 Diminished 2-mosdegree d2md
−25 Diminished 12-mosdegree d12md
−26 Diminished 5-mosdegree d5md
−27 Diminished 15-mosdegree d15md
−28 Diminished 8-mosdegree d8md

Modes

Scale degrees of the modes of 12L 5s
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
16|0 1 LLLsLLsLLLsLLsLLs Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Aug. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
15|1 8 LLLsLLsLLsLLLsLLs Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
14|2 15 LLsLLLsLLsLLLsLLs Perf. Maj. Maj. Min. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
13|3 5 LLsLLLsLLsLLsLLLs Perf. Maj. Maj. Min. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Maj. Maj. Min. Maj. Maj. Maj. Perf.
12|4 12 LLsLLsLLLsLLsLLLs Perf. Maj. Maj. Min. Maj. Maj. Min. Perf. Maj. Maj. Perf. Maj. Maj. Min. Maj. Maj. Maj. Perf.
11|5 2 LLsLLsLLLsLLsLLsL Perf. Maj. Maj. Min. Maj. Maj. Min. Perf. Maj. Maj. Perf. Maj. Maj. Min. Maj. Maj. Min. Perf.
10|6 9 LLsLLsLLsLLLsLLsL Perf. Maj. Maj. Min. Maj. Maj. Min. Perf. Maj. Min. Perf. Maj. Maj. Min. Maj. Maj. Min. Perf.
9|7 16 LsLLLsLLsLLLsLLsL Perf. Maj. Min. Min. Maj. Maj. Min. Perf. Maj. Min. Perf. Maj. Maj. Min. Maj. Maj. Min. Perf.
8|8 6 LsLLLsLLsLLsLLLsL Perf. Maj. Min. Min. Maj. Maj. Min. Perf. Maj. Min. Perf. Maj. Min. Min. Maj. Maj. Min. Perf.
7|9 13 LsLLsLLLsLLsLLLsL Perf. Maj. Min. Min. Maj. Min. Min. Perf. Maj. Min. Perf. Maj. Min. Min. Maj. Maj. Min. Perf.
6|10 3 LsLLsLLLsLLsLLsLL Perf. Maj. Min. Min. Maj. Min. Min. Perf. Maj. Min. Perf. Maj. Min. Min. Maj. Min. Min. Perf.
5|11 10 LsLLsLLsLLLsLLsLL Perf. Maj. Min. Min. Maj. Min. Min. Perf. Min. Min. Perf. Maj. Min. Min. Maj. Min. Min. Perf.
4|12 17 sLLLsLLsLLLsLLsLL Perf. Min. Min. Min. Maj. Min. Min. Perf. Min. Min. Perf. Maj. Min. Min. Maj. Min. Min. Perf.
3|13 7 sLLLsLLsLLsLLLsLL Perf. Min. Min. Min. Maj. Min. Min. Perf. Min. Min. Perf. Min. Min. Min. Maj. Min. Min. Perf.
2|14 14 sLLsLLLsLLsLLLsLL Perf. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Perf. Min. Min. Min. Maj. Min. Min. Perf.
1|15 4 sLLsLLLsLLsLLsLLL Perf. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Perf. Min. Min. Min. Min. Min. Min. Perf.
0|16 11 sLLsLLsLLLsLLsLLL Perf. Min. Min. Min. Min. Min. Min. Dim. Min. Min. Perf. Min. Min. Min. Min. Min. Min. Perf.

Proposed tuning-specific names

Declan Paul Boushy has proposed names for these modes corresponding to step ratios 3:1 and 4:1.

Scales

Scale tree

Scale tree and tuning spectrum of 12L 5s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
7\17 494.118 705.882 1:1 1.000 Equalized 12L 5s
40\97 494.845 705.155 6:5 1.200
33\80 495.000 705.000 5:4 1.250
59\143 495.105 704.895 9:7 1.286
26\63 495.238 704.762 4:3 1.333 Supersoft 12L 5s
Leapfrog
71\172 495.349 704.651 11:8 1.375
45\109 495.413 704.587 7:5 1.400 Leapweek
64\155 495.484 704.516 10:7 1.429
19\46 495.652 704.348 3:2 1.500 Soft 12L 5s
Leapday
69\167 495.808 704.192 11:7 1.571 Polypyth
50\121 495.868 704.132 8:5 1.600
81\196 495.918 704.082 13:8 1.625 Golden neogothic (495.904 ¢)
31\75 496.000 704.000 5:3 1.667 Semisoft 12L 5s
74\179 496.089 703.911 12:7 1.714
43\104 496.154 703.846 7:4 1.750
55\133 496.241 703.759 9:5 1.800
12\29 496.552 703.448 2:1 2.000 Basic 12L 5s
Scales with tunings softer than this are proper
53\128 496.875 703.125 9:4 2.250
41\99 496.970 703.030 7:3 2.333 Undecental
70\169 497.041 702.959 12:5 2.400 Argent tuning (497.056 ¢)
29\70 497.143 702.857 5:2 2.500 Semihard 12L 5s
75\181 497.238 702.762 13:5 2.600 Unnamed golden tuning (497.254 ¢)
46\111 497.297 702.703 8:3 2.667
63\152 497.368 702.632 11:4 2.750 Kwai
17\41 497.561 702.439 3:1 3.000 Hard 12L 5s
Garibaldi / andromeda
56\135 497.778 702.222 10:3 3.333
39\94 497.872 702.128 7:2 3.500 Garibaldi / cassandra
61\147 497.959 702.041 11:3 3.667
22\53 498.113 701.887 4:1 4.000 Superhard 12L 5s
Garibaldi / helenus, Pythagorean tuning (498.045 ¢)
49\118 498.305 701.695 9:2 4.500 Pontiac
27\65 498.462 701.538 5:1 5.000 Photia
32\77 498.701 701.299 6:1 6.000 ↓ Grackle, ↓↓ gracecordial
5\12 500.000 700.000 1:0 → ∞ Collapsed 12L 5s
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