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5L 7s: Difference between revisions

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This is the MOS pattern of the Pythagorean/[[Schismatic_family|schismatic/Helmholtz/Garibaldi]] chromatic scale, and also the [[Archytas_clan|superpyth]] chromatic scale. In contrast to the [[7L_5s|meantone chromatic scale]], in which "diatonic" semitones are larger than "chromatic" semitones, here the reverse is true: diatonic semitones are smaller than chromatic semitones, so the [[5L_2s|diatonic scale]] subset is actually [[Rothenberg_propriety|improper]].
{{Infobox MOS}}
{{MOS intro|Other Names=p-chromatic}}
5L 7s represents the chromatic scales of [[Pythagorean tuning|Pythagorean]]/[[schismic]] and [[superpyth]], the former being [[Rothenberg propriety|proper]] but the latter improper until expanded by 5 more notes, producing Superpyth[17]. Such scales are characterized by having a small step ([[diatonic semitone]]) that is smaller than the [[chroma]] ([[chromatic semitone]]), the reverse of [[7L 5s]].


The two distinct harmonic entropy minima with this MOS pattern are, on the one hand, scales very close to Pythagorean such that 64/63 is not tempered out, such as the schismatic temperaments known as "Helmholtz" and "Garibaldi", and on the other hand, the much simpler and less accurate scale known as "superpyth" in which 64/63 is tempered out.
The two distinct harmonic entropy minima are, on the one hand, scales very close to Pythagorean tuning or the schismatic temperament, and on the other hand, the simpler and less accurate temperament known as superpyth in which 64/63 is tempered out.


The Pythagorean/schismatic version is proper, but the superpyth version is improper (it doesn't become proper until you add 5 more notes to form the superpyth "enharmonic" scale, superpyth[17]).
== Scale properties ==
{{TAMNAMS use}}


{| class="wikitable"
=== Intervals ===
|-
{{MOS intervals}}
! colspan="8" | Generator
! | in cents
! | Comments
|-
| style="text-align:center;" | 5\12
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 500
| |
|-
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 37\89
| style="text-align:center;" | 498.876
| |
|-
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 32\77
| |
| style="text-align:center;" | 498.702
| |
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 27\65
| |
| |
| style="text-align:center;" | 498.462
| style="text-align:center;" | Photia
|-
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 49\118
| |
| style="text-align:center;" | 498.305
| style="text-align:center;" | Helmholtz/Pontiac/Nestoria
|-
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 71\171
| style="text-align:center;" | 498.246
| style="text-align:center;" | Helmholtz/Pontiac/Nestoria
|-
| |
| |
| |
| |
| style="text-align:center;" | 22\53
| |
| |
| |
| style="text-align:center;" | 498.113
| style="text-align:center;" | Helenus
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 39\94
| |
| |
| style="text-align:center;" | 497.872
| style="text-align:center;" | Garibaldi
|-
| |
| |
| |
| style="text-align:center;" | 17\41
| |
| |
| |
| |
| style="text-align:center;" | 497.591
| style="text-align:center;" | Cassandra
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 46\111
| |
| |
| style="text-align:center;" | 497.297
| |
|-
| |
| |
| |
| |
| style="text-align:center;" | 29\70
| |
| |
| |
| style="text-align:center;" | 497.143
| style="text-align:center;" | Undecental
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 41\99
| |
| |
| style="text-align:center;" | 496.97
| style="text-align:center;" | Undecental
|-
| |
| |
| style="text-align:center;" | 12\29
| |
| |
| |
| |
| |
| style="text-align:center;" | 496.552
| style="text-align:center;" | Optimum rank range (L/s=3/2)


Edson
=== Generator chain ===
|-
{{MOS genchain}}
| |
| |
| |
| |
| |
| style="text-align:center;" | 43\104
| |
| |
| style="text-align:center;" | 496.154
| |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 496.157
| style="text-align:center;" | L/s = pi/2
|-
| |
| |
| |
| |
| style="text-align:center;" | 31\75
| |
| |
| |
| style="text-align:center;" | 496
| |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 495.904
| style="text-align:center;" | L/s = phi
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 50\121
| |
| |
| style="text-align:center;" | 495.868
| style="text-align:center;" | Leapday/[[Peppermint-24|Peppermint]]/Pepperoni
|-
| |
| |
| |
| style="text-align:center;" | 19\46
| |
| |
| |
| |
| style="text-align:center;" | 495.652
| style="text-align:center;" | Leapday
|-
| |
| |
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| |
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| style="text-align:center;" | 45\109
| |
| |
| style="text-align:center;" | 495.413
| |
|-
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| |
| |
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| |
| |
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| style="text-align:center;" | 495.325
| style="text-align:center;" | L/s = sqrt(3)
|-
| |
| |
| |
| |
| style="text-align:center;" | 26\63
| |
| |
| |
| style="text-align:center;" | 495.238
| |
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 33\80
| |
| |
| style="text-align:center;" | 495
| |
|-
| |
| style="text-align:center;" | 7\17
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 494.118
| style="text-align:center;" | Boundary of propriety (generators larger than this are proper)


Supraphon
=== Modes ===
|-
{{MOS mode degrees}}
| |
 
| |
=== Proposed names ===
| |
The modes are named by [[Eliora]] after Chinese zodiac animals. 5L 7s is the opposite mos to [[7L 5s]], named after a Western concept, Gregorian months, therefore this mos scale has Eastern nomenclature. Furthermore, 12edo (equalized tuning of this MOS) was independently discovered in China.
| |
 
| |
{{MOS modes
| style="text-align:center;" | 30\73
| Mode Names=
| |
Rat $
| |
Ox $
| style="text-align:center;" | 493.151
Tiger $
| |
Rabbit $
|-
Dragon $
| |
Snake $
| |
Horse $
| |
Goat $
| |
Monkey $
| style="text-align:center;" | 23\56
Rooster $
| |
Dog $
| |
Pig $
| |
}}
| style="text-align:center;" | 492.857
 
| |
== Scales ==
|-
* [[Pythagorean12]] – Pythagorean tuning
| |
* [[Garibaldi12]] – 94edo tuning
| |
* [[Cotoneum12]] – 217edo tuning
| |
* [[Edson12]] – 29edo tuning
| |
* [[Pepperoni12]] – 271edo tuning
| |
* [[Supra12]] – 56edo tuning
| style="text-align:center;" | 39\95
* [[Archy12]] – 472edo tuning
| |
* [[12-22a]] – 22edo tuning
| |
 
| style="text-align:center;" | 492.632
== Scale tree ==
| |
{{MOS tuning spectrum
|-
| 6/5 = [[Photia]], ↑ [[grackle]]
| |
| 5/4 = [[Helmholtz (temperament)|Helmholtz]], [[Pythagorean tuning]] (701.955{{c}})
| |
| 9/7 = [[Garibaldi]] / [[cassandra]]
| |
| 4/3 = Garibaldi / [[andromeda]]
| style="text-align:center;" | 16\39
| 11/8 = [[Kwai]]
| |
| 10/7 = [[Undecental]], argent tuning (702.944{{c}})
| |
| 3/2 = [[Edson]]
| |
| 13/8 = [[Polypyth]], golden neogothic (704.096{{c}})
| |
| 5/3 = [[Leapday]]
| style="text-align:center;" | 492.308
| 12/7 = [[Leapweek]]
| |
| 7/3 = [[Supra]]
|-
| 13/5 = Golden supra (708.054{{c}})
| |
| 8/3 = [[Quasisuper]] / [[quasisupra]]
| |
| 3/1 = [[Suprapyth]]
| |
| 7/2 = [[Superpyth]]
| |
| 6/1 = ↓ [[Oceanfront]] / [[ultrapyth]]
| |
}}
| style="text-align:center;" | 41\100
 
| |
[[Category:12-tone scales]]
| |
[[Category:P-chromatic| ]]<!-- main article -->
| style="text-align:center;" | 492
[[Category:Chromatic scales]]
| |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 491.946
| style="text-align:center;" | L/s = phi+1
|-
| |
| |
| |
| |
| style="text-align:center;" | 25\61
| |
| |
| |
| style="text-align:center;" | 491.803
| |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 491.655
| style="text-align:center;" | L/s = e
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 34\83
| |
| |
| style="text-align:center;" | 491.566
| |
|-
| |
| |
| style="text-align:center;" | 9\22
| |
| |
| |
| |
| |
| style="text-align:center;" | 490.909
| style="text-align:center;" | Suprapyth/Supra
|-
| |
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 490.569
| style="text-align:center;" | L/s = pi
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 29\71
| |
| |
| style="text-align:center;" | 490.141
| |
|-
| |
| |
| |
| |
| style="text-align:center;" | 20\49
| |
| |
| |
| style="text-align:center;" | 489.796
| style="text-align:center;" | Superpyth
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 31\76
| |
| |
| style="text-align:center;" | 489.474
| |
|-
| |
| |
| |
| style="text-align:center;" | 11\27
| |
| |
| |
| |
| style="text-align:center;" | 488.889
| style="text-align:center;" | Archy
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 24\59
| |
| |
| style="text-align:center;" | 488.136
| |
|-
| |
| |
| |
| |
| style="text-align:center;" | 13\32
| |
| |
| |
| style="text-align:center;" | 487.500
| |
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 15\37
| |
| |
| style="text-align:center;" | 486.486
| |
|-
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 17\42
| |
| style="text-align:center;" | 485.714
| |
|-
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 19\47
| style="text-align:center;" | 485.106
| |
|-
| style="text-align:center;" | 2\5
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 480.000
| |
|}
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