12L 5s: Difference between revisions

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This MOS separates its small steps by intervals of 3L-2L-2L-3L-2L. Its major third of -4 generators approximates an interval between [[~~24/19|~~24/19]] and [[~~32/25|~~32/25]], thus its generator is a perfect fourth between 7~~/17edo~~ (494.412) and 5~~/12edo~~ (500 cents).

{{Infobox MOS

| Periods = 1

| nLargeSteps = 12

| nSmallSteps = 5

| Equalized = 7

| Paucitonic = 5

| Pattern = sLLsLLsLLLsLLsLLL

| Name = Schismic mega-chromatic

}}

'''12L 5s''' is the MOS pattern of the [[Pythagorean tuning|Pythagorean]]/[[Garibaldi temperament|Helmholtz/Garibaldi]] mega-chromatic scale. In contrast to the [[5L 12s|superpyth mega-chromatic scale]], in which mega-chromatic semitones (negative diminished seconds) are larger than chromatic semitones, here the reverse is true: mega-chromatic semitones are smaller than chromatic semitones, so the [[5L 7s|diatonic scale]] subset is actually [[Rothenberg propriety|proper]].

This MOS separates its small steps by intervals of 3L-2L-2L-3L-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 (494.412 cents) and 5\12 (500 cents).

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

== Modes ==

* 16|0 LLLsLLsLLLsLLsLLs

* 15|1 LLLsLLsLLsLLLsLLs

* 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 ==

* [[Pythagorean17]] – Pythagorean tuning

* [[Nestoria17]] – 171edo tuning

* [[Cotoneum17]] – 217edo tuning

* [[Garibaldi17]] – 94edo tuning

== Scale tree ==

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| || || || || || 59\143 || 495.105 || 9 || 7 || 1.286 ||

|-

| || || || 26\63 || || || 495.238 || 4 || 3 || 1.333 || Leapfrog

| || || || 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

| || || || || 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 || ~~L/s = 3/2~~

| || || 19\46 || || || || 495.652 || 3 || 2 || 1.500 ||

|-

| || || || || || 69\167 || 495.808 || 11 || 7 || 1.571 || Leapday

| || || || || || 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

| || || || || || 81\196 || 495.918 || 13 || 8 || 1.625 || Golden neogothic (Generator = 495.9044 cents)

|-

| || || || 31\75 || || || 496.000 || 5 || 3 || 1.667 ||

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| || || || || || 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)

| || 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

| || || || || 41\99 || || 496.970 || 7 || 3 || 2.333 || [[Undecental]]

|-

| || || || || || 70\169 || 497.041 || 12 || 5 || 2.400 || Argent tuning

| || || || || || 70\169 || 497.041 || 12 || 5 || 2.400 || Argent tuning (Generator: 497.0563 cents)

|-

| || || || 29\70 || || || 497.143 || 5 || 2 || 2.500 ||

|-

| || || || || || 75\181 || 497.238 || 13 || 5 || 2.600 || Unnamed golden tuning

| || || || || || 75\181 || 497.238 || 13 || 5 || 2.600 || Unnamed golden tuning (Generator: 497.2540 cents)

|-

| || || || || 46\111 || || 497.297 || 8 || 3 || 2.667 ||

|-

| || || || || || 63\152 || 497.368 || 11 || 4 || 2.750 || Kwai

| || || || || || 63\152 || 497.368 || 11 || 4 || 2.750 || [[Kwai]]

|-

| || || 17\41 || || || || 497.561 || 3 || 1 || 3.000 || L/~~s = 3/1, garibaldi/andromeda~~

| || || 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~~

| || || || || 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~~

| || || || 22\53 || || || 498.113 || 4 || 1 || 4.000 || Garibaldi / [[Helenus]] / [[Pythagorean]]

|-

| || || || || || 49\118 || 498.305 || 9 || 2 || 4.500 || Pontiac

| || || || || || 49\118 || 498.305 || 9 || 2 || 4.500 || [[Pontiac]]

|-

| || || || || 27\65 || || 498.462 || 5 || 1 || 5.000 || Photia

| || || || || 27\65 || || 498.462 || 5 || 1 || 5.000 || [[Photia]]

|-

| || || || || || 32\77 || 498.701 || 6 || 1 || 6.000 || ~~Grackle↓~~

| || || || || || 32\77 || 498.701 || 6 || 1 || 6.000 || [[Grackle]]↓

|-

| 5\12 || || || || || || 500.000 || 1 || 0 || → inf ||

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[[Category:Abstract MOS patterns]]

[[Category:17-tone scales]]

[[Category:Mega chromatic scales]]

@@ Line 1: / Line 1: @@
-This MOS separates its small steps by intervals of 3L-2L-2L-3L-2L. Its major third of -4 generators approximates an interval between [[24/19|24/19]] and [[32/25|32/25]], thus its generator is a perfect fourth between 7/17edo (494.412) and 5/12edo (500 cents).
+{{Infobox MOS
+| Periods = 1
+| nLargeSteps = 12
+| nSmallSteps = 5
+| Equalized = 7
+| Paucitonic = 5
+| Pattern = sLLsLLsLLLsLLsLLL
+| Name = Schismic mega-chromatic
+}}
+'''12L 5s''' is the MOS pattern of the [[Pythagorean tuning|Pythagorean]]/[[Garibaldi temperament|Helmholtz/Garibaldi]] mega-chromatic scale. In contrast to the [[5L 12s|superpyth mega-chromatic scale]], in which mega-chromatic semitones (negative diminished seconds) are larger than chromatic semitones, here the reverse is true: mega-chromatic semitones are smaller than chromatic semitones, so the [[5L 7s|diatonic scale]] subset is actually [[Rothenberg propriety|proper]].
+This MOS separates its small steps by intervals of 3L-2L-2L-3L-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 (494.412 cents) and 5\12 (500 cents).
+The leapday/leapweek version is proper, but the Pythagorean/schismic version is improper (it doesn't become proper until you add 12 more notes to form the schismic 29-note scale).
+== Modes ==
+* 16|0 LLLsLLsLLLsLLsLLs
+* 15|1 LLLsLLsLLsLLLsLLs
+* 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 ==
+* [[Pythagorean17]] – Pythagorean tuning
+* [[Nestoria17]] – 171edo tuning
+* [[Cotoneum17]] – 217edo tuning
+* [[Garibaldi17]] – 94edo tuning
 == Scale tree ==
@@ Line 18: / Line 57: @@
 | || || || || || 59\143 || 495.105 || 9 || 7 || 1.286 ||
 |-
-| || || || 26\63 || || || 495.238 || 4 || 3 || 1.333 || Leapfrog
+| || || || 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
+| || || || || 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 || L/s = 3/2
+| || || 19\46 || || || || 495.652 || 3 || 2 || 1.500 ||
 |-
-| || || || || || 69\167 || 495.808 || 11 || 7 || 1.571 || Leapday
+| || || || || || 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
+| || || || || || 81\196 || 495.918 || 13 || 8 || 1.625 || Golden neogothic (Generator = 495.9044 cents)
 |-
 | || || || 31\75 || || || 496.000 || 5 || 3 || 1.667 ||
@@ Line 42: / Line 81: @@
 | || || || || || 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)
+| || 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
+| || || || || 41\99 || || 496.970 || 7 || 3 || 2.333 || [[Undecental]]
 |-
-| || || || || || 70\169 || 497.041 || 12 || 5 || 2.400 || Argent tuning
+| || || || || || 70\169 || 497.041 || 12 || 5 || 2.400 || Argent tuning (Generator: 497.0563 cents)
 |-
 | || || || 29\70 || || || 497.143 || 5 || 2 || 2.500 ||
 |-
-| || || || || || 75\181 || 497.238 || 13 || 5 || 2.600 || Unnamed golden tuning
+| || || || || || 75\181 || 497.238 || 13 || 5 || 2.600 || Unnamed golden tuning (Generator: 497.2540 cents)
 |-
 | || || || || 46\111 || || 497.297 || 8 || 3 || 2.667 ||
 |-
-| || || || || || 63\152 || 497.368 || 11 || 4 || 2.750 || Kwai
+| || || || || || 63\152 || 497.368 || 11 || 4 || 2.750 || [[Kwai]]
 |-
-| || || 17\41 || || || || 497.561 || 3 || 1 || 3.000 || L/s = 3/1, garibaldi/andromeda
+| || || 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
+| || || || || 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
+| || || || 22\53 || || || 498.113 || 4 || 1 || 4.000 || Garibaldi / [[Helenus]] / [[Pythagorean]]
 |-
-| || || || || || 49\118 || 498.305 || 9 || 2 || 4.500 || Pontiac
+| || || || || || 49\118 || 498.305 || 9 || 2 || 4.500 || [[Pontiac]]
 |-
-| || || || || 27\65 || || 498.462 || 5 || 1 || 5.000 || Photia
+| || || || || 27\65 || || 498.462 || 5 || 1 || 5.000 || [[Photia]]
 |-
-| || || || || || 32\77 || 498.701 || 6 || 1 || 6.000 || Grackle↓
+| || || || || || 32\77 || 498.701 || 6 || 1 || 6.000 || [[Grackle]]↓
 |-
 | 5\12 || || || || || || 500.000 || 1 || 0 || → inf ||
@@ Line 80: / Line 119: @@
 [[Category:Abstract MOS patterns]]
 [[Category:17-tone scales]]
+[[Category:Mega chromatic scales]]