12L 5s: Difference between revisions

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{{Infobox MOS
{{Infobox MOS}}
| Periods = 1
{{MOS intro|Other Names=p-enharmonic}}
| nLargeSteps = 12
Temperaments supported by this scale include those under the [[Pythagorean tuning|Pythagorean]] and [[Schismatic family|schismic]] families, characterized by a diesis (the difference between a large step and two small steps) that is smaller than the [[chroma]].
| nSmallSteps = 5
 
| Equalized = 7
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).
| Paucitonic = 5
 
| Pattern = LLLsLLsLLLsLLsLLs
== Scale properties ==
}}
{{TAMNAMS use}}


'''12L 5s''' is the MOS pattern of the [[Pythagorean tuning|Pythagorean]]/[[Schismatic family|schismic]] enharmonic or mega-chromatic scale. In contrast to the [[5L 12s|superpyth enharmonic scale]], in which the enharmonic diesis (negative diminished second) is larger than the chromatic semitone, here the reverse is true: the enharmonic diesis is smaller than the chromatic semitone, so the [[5L 7s|diatonic scale]] subset is actually [[Rothenberg propriety|proper]].
=== Intervals ===
{{MOS intervals}}


This MOS separates its small steps by intervals of 3L-2L-3L-2L-2L. Its major third of -4 generators approximates an interval between [[24/19]] and [[32/25]], thus its generator is a perfect fourth between 7\17 (494.118 cents) and 5\12 (500 cents).
=== Generator chain ===
{{MOS genchain}}


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).
=== Modes ===
{{MOS mode degrees}}


== Modes ==
=== Proposed tuning-specific names ===
* 16|0 LLLsLLsLLLsLLsLLs
[[Declan Paul Boushy]] has proposed names for these modes corresponding to step ratios [[Subaru scale|3:1]] and [[Tanegashima scale|4:1]].
* 15|1 LLLsLLsLLsLLLsLLs
{{todo|add etymology|add Template:MOS modes and annotate each row using Boushy’s names}}
* 14|2 LLsLLLsLLsLLLsLLs
* 13|3 LLsLLLsLLsLLsLLLs
* 12|4 LLsLLsLLLsLLsLLLs
* 11|5 LLsLLsLLLsLLsLLsL
* 10|6 LLsLLsLLsLLLsLLsL
* 9|7 LsLLLsLLsLLLsLLsL
* 8|8 LsLLLsLLsLLsLLLsL
* 7|9 LsLLsLLLsLLsLLLsL
* 6|10 LsLLsLLLsLLsLLsLL
* 5|11 LsLLsLLsLLLsLLsLL
* 4|12 sLLLsLLsLLLsLLsLL
* 3|13 sLLLsLLsLLsLLLsLL
* 2|14 sLLsLLLsLLsLLLsLL
* 1|15 sLLsLLLsLLsLLsLLL
* 0|16 sLLsLLsLLLsLLsLLL


== Scales ==
== Scales ==
* [[Edson17]] – 29edo tuning
* [[Subaru scale]] – 41edo tuning
* [[Cotoneum17]] – 217edo tuning
* [[Garibaldi17]] – 94edo tuning
* [[Pythagorean17]] – Pythagorean tuning
* [[Pythagorean17]] – Pythagorean tuning
* [[Tanegashima scale]] – 53edo tuning
* [[Nestoria17]] – 171edo tuning
* [[Nestoria17]] – 171edo tuning
* [[Cotoneum17]] – 217edo tuning
* [[Garibaldi17]] – 94edo tuning


== Scale tree ==
== Scale tree ==
Generator ranges:
{{MOS tuning spectrum
* Chroma-positive generator: 494.1176 cents (7\17) to 500 cents (5\12)
| 4/3 = [[Leapfrog]]  
* Chroma-negative generator: 700 cents (7\12) to 705.8824 cents (10\17)
| 7/5 = [[Leapweek]]  
 
| 3/2 = [[Leapday]]  
{| class="wikitable center-all"
| 11/7 = [[Polypyth]]  
! colspan="6" | Generator
| 13/8 = Golden neogothic (495.904{{c}})  
! Cents
| 7/3 = [[Undecental]]  
! L
| 12/5 = Argent tuning (497.056{{c}})  
! s
| 13/5 = Unnamed golden tuning (497.254{{c}})  
! L/s
| 11/4 = [[Kwai]]  
! Comments
| 3/1 = [[Garibaldi]] / [[andromeda]]  
|-
| 7/2 = Garibaldi / [[cassandra]]  
| 7\17 || || || || || || 494.118 || 1 || 1 || 1.000 ||
| 4/1 = Garibaldi / [[helenus]], Pythagorean tuning (498.045{{c}})  
|-
| 9/2 = [[Pontiac]]  
| || || || || || 40\97 || 494.845 || 6 || 5 || 1.200 ||
| 5/1 = [[Photia]]  
|-
| 6/1 = ↓ [[Grackle]], ↓↓ [[gracecordial]]
| || || || || 33\80 || || 495.000 || 5 || 4 || 1.250 ||
}}
|-
| || || || || || 59\143 || 495.105 || 9 || 7 || 1.286 ||
|-
| || || || 26\63 || || || 495.238 || 4 || 3 || 1.333 || [[Leapfrog]]
|-
| || || || || || 71\172 || 495.349 || 11 || 8 || 1.375 ||
|-
| || || || || 45\109 || || 495.413 || 7 || 5 || 1.400 || [[Leapweek]]
|-
| || || || || || 64\155 || 495.484 || 10 || 7 || 1.428 ||
|-
| || || 19\46 || || || || 495.652 || 3 || 2 || 1.500 ||
|-
| || || || || || 69\167 || 495.808 || 11 || 7 || 1.571 || [[Leapday]] / [[polypyth]]
|-
| || || || || 50\121 || || 495.868 || 8 || 5 || 1.600 ||
|-
| || || || || || 81\196 || 495.918 || 13 || 8 || 1.625 || Golden neogothic (495.9044¢)
|-
| || || || 31\75 || || || 496.000 || 5 || 3 || 1.667 ||
|-
| || || || || || 74\179 || 496.089 || 12 || 7 || 1.714 ||
|-
| || || || || 43\104 || || 496.154 || 7 || 4 || 1.750 ||
|-
| || || || || || 55\133 || 496.241 || 9 || 5 || 1.800 ||
|-
| || 12\29 || || || || || 496.552 || 2 || 1 || 2.000 || Basic 12L 5s <br>(Generators smaller than this are proper)
|-
| || || || || || 53\128 || 496.875 || 9 || 4 || 2.250 ||
|-
| || || || || 41\99 || || 496.970 || 7 || 3 || 2.333 || [[Undecental]]
|-
| || || || || || 70\169 || 497.041 || 12 || 5 || 2.400 || Argent tuning (497.0563¢)
|-
| || || || 29\70 || || || 497.143 || 5 || 2 || 2.500 ||
|-
| || || || || || 75\181 || 497.238 || 13 || 5 || 2.600 || Unnamed golden tuning (497.2540¢)
|-
| || || || || 46\111 || || 497.297 || 8 || 3 || 2.667 ||
|-
| || || || || || 63\152 || 497.368 || 11 || 4 || 2.750 || [[Kwai]]
|-
| || || 17\41 || || || || 497.561 || 3 || 1 || 3.000 || [[Garibaldi]] / [[andromeda]]
|-
| || || || || || 56\135 || 497.778 || 10 || 3 || 3.333 ||
|-
| || || || || 39\94 || || 497.872 || 7 || 2 || 3.500 || Garibaldi / [[cassandra]]
|-
| || || || || || 61\147 || 497.959 || 11 || 3 || 3.667 ||
|-
| || || || 22\53 || || || 498.113 || 4 || 1 || 4.000 || Garibaldi / [[helenus]], [[Pythagorean tuning]] (498.0450¢)
|-
| || || || || || 49\118 || 498.305 || 9 || 2 || 4.500 || [[Pontiac]]
|-
| || || || || 27\65 || || 498.462 || 5 || 1 || 5.000 || [[Photia]]
|-
| || || || || || 32\77 || 498.701 || 6 || 1 || 6.000 || [[Grackle]]↓
|-
| 5\12 || || || || || || 500.000 || 1 || 0 || → inf ||
|}


[[Category:Abstract MOS patterns]]
[[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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