3L 2s (8/5-equivalent): Difference between revisions

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Created page with "{{Infobox MOS|Equalized=2|Equave=8/5|Name=Diatonic|Paucitonic=1|Pattern=LLsLs|nLargeSteps=3|nSmallSteps=2}} '''3L 2s<8/5>''' (sometimes called '''diatonic'''), is a minor six..."
 
CompactStar (talk | contribs)
5,020 edits
Why is there a 14/9-repeating temperament??
 
(29 intermediate revisions by 7 users not shown)
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{{Infobox MOS|Equalized=2|Equave=8/5|Name=Diatonic|Paucitonic=1|Pattern=LLsLs|nLargeSteps=3|nSmallSteps=2}}
{{Infobox MOS}}


'''3L 2s<8/5>''' (sometimes called '''diatonic'''), is a minor sixth-repeating MOS scale. The notation "<8/5>" means the period of the MOS is 8/5, disambiguating it from octave-repeating [[3L 2s]]. The name of the period interval is called the '''sextave''' (by analogy to the [[tritave]]).
'''3L 2s<8/5>''' is a minor sixth-repeating MOS scale. The notation "<8/5>" means the period of the MOS is 8/5, disambiguating it from octave-repeating [[3L 2s]].  


The generator range is 240 to 342.9 cents, placing it on the [[6/5|diatonic minor third]], usually representing a minor third of some type (like [[6/5]]). The bright (chroma-positive) generator is, however, its minor sixth complement (480 to 514.3 cents).  
The generator range is 240 to 342.9 cents, placing it on the [[6/5|diatonic minor third]], usually representing a minor third of some type (like [[6/5]]). The bright (chroma-positive) generator is, however, its minor sixth complement (480 to 514.3 cents).  


Because this diatonic is a minor sixth-repeating scale, each tone has an 8/5 minor sixth above it. The scale has one major chord, one minor chord and three diminished chords. This diatonic also has two diminished 7th chords, making it a warped melodic minor scale.
Because this is a minor sixth-repeating scale, each tone has an 8/5 minor sixth above it. The scale has one major chord, one minor chord and three diminished chords. This scale also has two diminished 7th chords, making it a warped melodic minor scale.


[[Basic]] diatonic is in [[8ed8/5]], which is a very good minor sixth-based equal tuning similar to [[12edo]].
[[Basic]] 3L 2s<8/5> is in [[8ed8/5]], which is a very good minor sixth-based equal tuning similar to [[12edo]].
==Notation==
==Notation==
There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (minor sixth) repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi). Given that 1-7/6-3/2 is minor sixth-equivalent to a tone cluster of 1-16/15-7/6, it may be more convenient to notate these diatonic scales as repeating at the double sextave (diminished eleventh~tenth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 16\15. Notating this way produces a tenth which is the Dorian mode of Annapolis[6L 4s] or Oriole[6L 4s]. Since there are exactly 10 naturals in double sesquitave notation, Greek numerals 1-10 may be used.
There are 2 main ways to notate this scale. One method uses a simple sixth repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi). Given that 1-7/6-3/2 is minor sixth-equivalent to a tone cluster of 1-16/15-7/6, it may be more convenient to notate these diatonic scales as repeating at the double sixth (diminished eleventh~tenth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 16\15. Notating this way produces a tenth which is the Dorian mode of Annapolis[6L 4s] or Oriole[6L 4s]. Since there are exactly 10 naturals in double sixth notation, Greek numerals 1-10 may be used.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 45: Line 45:
124.1379
124.1379
|2\11
|2\11
[[Tel:127.6282|127.6282]]
141.1765
|3\14
|3\14
[[Tel:150.4189|150.4189]]
163.{{Overline|63}}
|-
|-
|Sib
|Sib
Line 60: Line 60:
82.7586
82.7586
|1\11
|1\11
63.814
70.5882
|1\14
|1\14
50.1396
54.{{Overline|54}}
|-
|-
|Si
|Si
Line 77: Line 77:
206.89655
206.89655
|3\11
|3\11
[[Tel:191.4423|191.4423]]
211.7647
|4\14
|4\14
[[Tel:200.5586|200.5586]]
218.{{Overline|18}}
|-
|-
|Si#
|Si#
Line 94: Line 94:
331.0345
331.0345
|5\11
|5\11
319.07045
352.9412
|7\14
|7\14
[[Tel:350.9775|350.9775]]
381.{{Overline|81}}
|-
|-
|Dob
|Dob
Line 109: Line 109:
165.5172
165.5172
|2\11
|2\11
[[Tel:127.6282|127.6282]]
141.1765
|2\14
|2\14
[[Tel:100.2793|100.2793]]
109.{{Overline|09}}
|-
|-
|'''Do'''
|'''Do'''
Line 126: Line 126:
'''289.6552'''
'''289.6552'''
|'''4\11'''
|'''4\11'''
'''[[Tel:255.2564|255.2564]]'''
'''282.3529'''
|'''5\14'''
|'''5\14'''
'''[[Tel:250.6982|250.6982]]'''
'''272.{{Overline|72}}'''
|-
|-
|Do#
|Do#
Line 143: Line 143:
413.7931
413.7931
|6\11
|6\11
382.88455
423.5294
|8\14
|8\14
[[Tel:401.1171|401.1171]]
436.{{Overline|36}}
|-
|-
|Reb
|Reb
Line 158: Line 158:
372.4138
372.4138
|5\11
|5\11
319.07045
352.9412
|6\14
|6\14
[[Tel:300.8379|300.8379]]
327.{{Overline|27}}
|-
|-
|'''Re'''
|'''Re'''
Line 175: Line 175:
'''496.5517'''
'''496.5517'''
|'''7\11'''
|'''7\11'''
'''[[Tel:446.6986|446.6986]]'''
'''494.11765'''
|'''9\14'''
|'''9\14'''
'''[[Tel:451.2568|451.2568]]'''
'''490.{{Overline|90}}'''
|-
|-
|Re#
|Re#
Line 192: Line 192:
620.6897
620.6897
|9\11
|9\11
[[Tel:574.3268|574.3268]]
635.2941
|12\14
|12\14
[[Tel:601.6757|601.6757]]
654.{{Overline|54}}
|-
|-
|Mib
|Mib
Line 207: Line 207:
579.3103
579.3103
|8\11
|8\11
[[Tel:510.5128|510.5128]]
564.7059
|10\14
|10\14
[[Tel:501.3964|501.3964]]
545.{{Overline|45}}
|-
|-
|Mi
|Mi
Line 224: Line 224:
703.4483
703.4483
|10\11
|10\11
[[Tel:638.1409|638.1409]]
705.88235
|13\14
|13\14
[[Tel:651.8154|651.8154]]
709.{{Overline|09}}
|-
|-
|Mi#
|Mi#
Line 241: Line 241:
827.5862
827.5862
|12\11
|12\11
765.769
847.0588
|16\14
|16\14
[[Tel:802.2343|802.2343]]
872.{{Overline|72}}
|-
|-
|Lab
|Lab
Line 256: Line 256:
662.069
662.069
|9\11
|9\11
[[Tel:574.3268|574.3268]]
635.2941
|11\14
|11\14
551.636
600
|-
|-
!La
!La
Line 273: Line 273:
!19\19
!19\19
786.2069
786.2069
!
!11\11
!
776.4706
!14\14
 
763.{{Overline|63}}
|-
|-
|La#
|La#
Line 289: Line 292:
910.3448
910.3448
|13\11
|13\11
[[Tel:829.5832|829.5832]]
917.6471
|17\14
|17\14
[[Tel:852.3739|852.3739]]
927.{{Overline|27}}
|-
|-
|Sib
|Sib
Line 304: Line 307:
868.9655
868.9655
|12\11
|12\11
765.769
847.0588
|15\14
|15\14
[[Tel:752.0946|752.0946]]
818.{{Overline|18}}
|-
|-
|Si
|Si
Line 321: Line 324:
993.10345
993.10345
|14\11
|14\11
[[Tel:893.3973|893.3973]]
988.2353
|18\14
|18\14
[[Tel:902.5136|902.5136]]
981.{{Overline|81}}
|-
|-
|Si#
|Si#
Line 338: Line 341:
1117.2414
1117.2414
|16\11
|16\11
1021.02545
1129.4118
|21\14
|21\14
1052.9235
1145.{{Overline|45}}
|-
|-
|Dob
|Dob
Line 353: Line 356:
951.7241
951.7241
|13\11
|13\11
[[Tel:829.5832|829.5832]]
917.6471
|16\14
|16\14
[[Tel:802.2343|802.2343]]
872.{{Overline|72}}
|-
|-
|'''Do'''
|'''Do'''
Line 370: Line 373:
'''1075.8621'''
'''1075.8621'''
|'''15\11'''
|'''15\11'''
'''[[Tel:957.2114|957.2114]]'''
'''1052.8235'''
|'''19\14'''
|'''19\14'''
'''[[Tel:952.6532|952.6532]]'''
'''1036.{{Overline|36}}'''
|-
|-
|Do#
|Do#
Line 402: Line 405:
1158.6207
1158.6207
|16\11
|16\11
1021.02545
1129.4118
|20\14
|20\14
1002.7929
1090.{{Overline|90}}
|-
|-
|'''Re'''
|'''Re'''
Line 419: Line 422:
'''1282.7586'''
'''1282.7586'''
|'''18\11'''
|'''18\11'''
'''1148.6536'''
'''1270.5882'''
|'''23\14'''
|'''23\14'''
'''1153.2118'''
'''1254.{{Overline|54}}'''
|-
|-
|Re#
|Re#
Line 434: Line 437:
1400
1400
|34\19
|34\19
1406.8965f
1406.89655
|20\11
|20\11
1276.2818
1411.7647
|26\14
|26\14
1303.6307
1418.{{Overline|18}}
|-
|-
|Mib
|Mib
Line 451: Line 454:
1365.5172
1365.5172
|19\11
|19\11
1212.5678
1341.1765
|24\14
|24\14
1203.3514
1309.{{Overline|09}}
|-
|-
|Mi
|Mi
Line 468: Line 471:
1489.6551
1489.6551
|21\11
|21\11
1340.0959
1482.3529
|27\14
|27\14
1353.8704
1472.{{Overline|72}}
|-
|-
|Mi#
|Mi#
Line 485: Line 488:
1613.7931
1613.7931
|23\11
|23\11
1468.724
1623.5294
|30\14
|30\14
1504.1892
1636.{{Overline|36}}
|-
|-
|Lab
|Lab
Line 500: Line 503:
1448.2859
1448.2859
|20\11
|20\11
1276.2818
1411.7647
|25\14
|25\14
1253.591
1363.{{Overline|63}}
|-
|-
!La
!La
Line 517: Line 520:
!38\19
!38\19
1572.4138
1572.4138
!
!22\11
!
1552.9412
!28\14
1527.{{Overline|27}}
|}
|}
{| class="wikitable"
{| class="wikitable"
|+''ed2\3''
|+''ed8\12 (→ed2\3)''
! colspan="2" |Notation
! colspan="2" |Notation
!Supersoft
!Supersoft
Line 544: Line 549:
|Α#
|Α#
|''1\18''
|''1\18''
''38.9975''
''44.{{Overline|4}}''
|''1\13''
|''1\13''
''53.9965''
''61.5385''
|''2\21''
|''2\21''
''66.8529''
''76.1905''
| rowspan="2" |''1\8''
| rowspan="2" |''1\8''
''87.7444''
''100''
|''3\19''
|''3\19''
''110.835''
''126.3158''
|''2\11''
|''2\11''
[[Tel:127.6282|''127.6282'']]
''145.{{Overline|45}}''
|''3\14''
|''3\14''
[[Tel:150.4189|''150.4189'']]
''171.4286''
|-
|-
|Sib
|Sib
|Βb
|Βb
|''3\18''
|''3\18''
[[Tel:116.9925|''116.9925'']]
''133.{{Overline|3}}''
|''2\13''
|''2\13''
[[Tel:107.9931|''107.9931'']]
''123.0769''
|''3\21''
|''3\21''
[[Tel:100.2793|''100.2793'']]
''114.2857''
|''2\19''
|''2\19''
''73.89''
''84.2105''
|''1\11''
|''1\11''
''63.814''
''72.{{Overline|72}}''
|''1\14''
|''1\14''
''50.1396''
''57.1429''
|-
|-
|Si
|Si
|Β
|Β
|''4\18''
|''4\18''
''155.99''
''177.{{Overline|7}}''
|''3\13''
|''3\13''
[[Tel:161.9896|''161.9896'']]
''184.6154''
|''5\21''
|''5\21''
[[Tel:167.1321|''167.1321'']]
''190.4762''
|''2\8''
|''2\8''
''175.48875''
''200''
|''5\19''
|''5\19''
''184.725''
''210.5263''
|''3\11''
|''3\11''
[[Tel:191.4423|''191.4423'']]
''218.{{Overline|18}}''
|''4\14''
|''4\14''
[[Tel:200.5586|''200.5586'']]
''228.5714''
|-
|-
|Si#
|Si#
|Β#
|Β#
|''5\18''
|''5\18''
[[Tel:194.9875|''194.9875'']]
''222.''{{Overline|2}}
| rowspan="2" |''4\13''
| rowspan="2" |''4\13''
[[Tel:215.9862|''215.9862'']]
''246.15385''
|''7\21''
|''7\21''
''233.985''
''266.{{Overline|6}}''
|''3\8''
|''3\8''
[[Tel:263.2331|''263.2331'']]
''300''
|''8\19''
|''8\19''
''295.56''
''336.8421''
|''5\11''
|''5\11''
''319.07045''
''363.{{Overline|63}}''
|''7\14''
|''7\14''
[[Tel:350.9775|''350.9775'']]
''400''
|-
|-
|Dob
|Dob
|Γb
|Γb
|''6\18''
|''6\18''
''233.985''
''266.{{Overline|6}}''
|''6\21''
|''6\21''
[[Tel:200.5586|''200.5586'']]
''228.5714''
|''2\8''
|''2\8''
''175.48875''
''200''
|''4\19''
|''4\19''
''147.78''
''168.42105''
|''2\11''
|''2\11''
[[Tel:127.6282|''127.6282'']]
''145.{{Overline|45}}''
|''2\14''
|''2\14''
[[Tel:100.2793|''100.2793'']]
''114.2857''
|-
|-
|'''Do'''
|'''Do'''
|'''Γ'''
|'''Γ'''
|'''''7\18'''''
|'''''7\18'''''
'''[[Tel:272.9825|''272.9825'']]'''
'''''311.{{Overline|1}}'''''
|'''''5\13'''''
|'''''5\13'''''
'''[[Tel:269.9829|''269.9829'']]'''
'''''307.6923'''''
|'''''8\21'''''
|'''''8\21'''''
'''[[Tel:267.4114|''267.4114'']]'''
'''''304.7619'''''
|'''''3\8'''''
|'''''3\8'''''
'''[[Tel:263.2331|''263.2331'']]'''
'''''300'''''
|'''''7\19'''''
|'''''7\19'''''
'''''258.615'''''
'''''294.7368'''''
|'''''4\11'''''
|'''''4\11'''''
'''[[Tel:255.2564|''255.2564'']]'''
'''''290.{{Overline|90}}'''''
|'''''5\14'''''
|'''''5\14'''''
'''[[Tel:250.6982|''250.6982'']]'''
'''''285.7143'''''
|-
|-
|Do#
|Do#
|Γ#
|Γ#
|''8\18''
|''8\18''
''311.98''
''355.{{Overline|5}}''
|''6\13''
|''6\13''
[[Tel:323.9792|''323.9792'']]
''369.2308''
|''10\21''
|''10\21''
[[Tel:334.2643|''334.2643'']]
''380.9524''
| rowspan="2" |''4\8''
| rowspan="2" |''4\8''
[[Tel:350.9775|''350.9775'']]
''400''
|''9\19''
|''10\19''
''332.505''
''421.0526''
|''6\11''
|''6\11''
''382.88455''
''436.{{Overline|36}}''
|''8\14''
|''8\14''
[[Tel:401.1171|''401.1171'']]
''457.1429''
|-
|-
|Reb
|Reb
|Δb
|Δb
|''10\18''
|''10\18''
''389.975''
''444.{{Overline|4}}''
|''7\13''
|''7\13''
[[Tel:377.9758|''377.9758'']]
''430.7692''
|''11\21''
|''11\21''
[[Tel:367.9607|''367.9607'']]
''419.0476''
|''10\19''
|''9\19''
''369.45''
''378.9474''
|''5\11''
|''5\11''
''319.07045''
''363.{{Overline|63}}''
|''6\14''
|''6\14''
[[Tel:300.8379|''300.8379'']]
''342.8571''
|-
|-
|'''Re'''
|'''Re'''
|'''Δ'''
|'''Δ'''
|'''''11\18'''''
|'''''11\18'''''
'''[[Tel:428.9725|''428.9725'']]'''
'''''488.{{Overline|8}}'''''
|'''''8\13'''''
|'''''8\13'''''
'''[[Tel:431.9723|''431.9723'']]'''
'''''492.3077'''''
|'''''13\21'''''
|'''''13\21'''''
'''[[Tel:434.5436|''434.5436'']]'''
'''''495.2381'''''
|'''''5\8'''''
|'''''5\8'''''
'''[[Tel:438.7219|''438.7219'']]'''
'''''500'''''
|'''''12\19'''''
|'''''12\19'''''
'''''470.285'''''
'''''505.2632'''''
|'''''7\11'''''
|'''''7\11'''''
'''[[Tel:446.6986|''446.6986'']]'''
'''''509.{{Overline|09}}'''''
|'''''9\14'''''
|'''''9\14'''''
'''[[Tel:451.2568|''451.2568'']]'''
'''''514.2857'''''
|-
|-
|Re#
|Re#
|Δ#
|Δ#
|''12\18''
|''12\18''
''467.97''
''533.{{Overline|3}}''
|''9\13''
|''9\13''
[[Tel:485.9688|''485.9688'']]
''553.84615''
|''15\21''
|''15\21''
[[Tel:501.3964|''501.3964'']]
''571.4286''
| rowspan="2" |''6\8''
| rowspan="2" |''6\8''
''526.46625''
''600''
|''15\19''
|''15\19''
''554.175''
''631.42105''
|''9\11''
|''9\11''
[[Tel:574.3268|''574.3268'']]
''654.{{Overline|54}}''
|''12\14''
|''12\14''
[[Tel:601.6757|''601.6757'']]
''685.7143''
|-
|-
|Mib
|Mib
|Εb
|Εb
|''14\18''
|''14\18''
''545.965''
''622.{{Overline|2}}''
|''10\13''
|''10\13''
[[Tel:539.9653|''539.9653'']]
''615.3846''
|''16\21''
|''16\21''
[[Tel:534.8229|''534.8229'']]
''609.5238''
|''14\19''
|''14\19''
''516.23''
''589.4737''
|''8\11''
|''8\11''
[[Tel:510.5128|''510.5128'']]
''581.{{Overline|81}}''
|''10\14''
|''10\14''
[[Tel:501.3964|''501.3964'']]
''571.4286''
|-
|-
|Mi
|Mi
|Ε
|Ε
|''15\18''
|''15\18''
[[Tel:584.9625|''584.9625'']]
''666.{{Overline|6}}''
|''11\13''
|''11\13''
[[Tel:593.9619|''593.9619'']]
''676.9231''
|''18\21''
|''18\21''
[[Tel:601.6757|''601.6757'']]
''685.7143''
|''7\8''
|''7\8''
[[Tel:614.2106|''614.2106'']]
''700''
|''17\19''
|''17\19''
''628.065''
''715.7895''
|''10\11''
|''10\11''
[[Tel:638.1409|''638.1409'']]
''727.{{Overline|27}}''
|''13\14''
|''13\14''
[[Tel:651.8154|''651.8154'']]
''742.8571''
|-
|-
|Mi#
|Mi#
|Ε#
|Ε#
|''16\18''
|''16\18''
''622.96''
''711.{{Overline|1}}''
| rowspan="2" |''12\13''
| rowspan="2" |''12\13''
[[Tel:646.9585|''646.9585'']]
''738.4615''
|''20\21''
|''20\21''
[[Tel:668.5286|''668.5286'']]
''761.9048''
|''8\8''
|''8\8''
''701.955''
''800''
|''20\19''
|''20\19''
''738.9''
''842.1053''
|''12\11''
|''12\11''
''765.769''
''872.{{Overline|72}}''
|''16\14''
|''16\14''
[[Tel:802.2343|''802.2343'']]
''914.2857''
|-
|-
|Lab
|Lab
|Ϛb/Ϝb
|Ϛb/Ϝb
|''17\18''
|''17\18''
[[Tel:662.9575|''662.9575'']]
''755.{{Overline|5}}''
|''19\21''
|''19\21''
[[Tel:635.1021|''635.1021'']]
''723.8095''
|''7\8''
|''7\8''
[[Tel:614.2106|''614.2106'']]
''700''
|''16\19''
|''16\19''
''591.12''
''673.6842''
|''9\11''
|''8\11''
[[Tel:574.3268|''574.3268'']]
''581.{{Overline|81}}''
|''11\14''
|''11\14''
''551.636''
''628.5714''
|-
|-
!La
!La
!Ϛ/Ϝ
!Ϛ/Ϝ
! colspan="7" |''701.955''
! colspan="7" |''800''
|-
|-
|La#
|La#
|Ϛ#/Ϝ#
|Ϛ#/Ϝ#
|''19\18''
|''19\18''
[[Tel:740.9525|''740.9525'']]
''844.{{Overline|4}}''
|''14\13''
|''14\13''
[[Tel:754.9515|''754.9515'']]
''861.5385''
|''23\21''
|''23\21''
[[Tel:768.8021|''768.8021'']]
''876.1905''
| rowspan="2" |''9\8''
| rowspan="2" |''9\8''
[[Tel:789.6994|''789.6994'']]
''900''
|''22\19''
|''22\19''
''812.79''
''926.3158''
|''13\11''
|''13\11''
[[Tel:829.5832|''829.5832'']]
''945.{{Overline|45}}''
|''17\14''
|''17\14''
[[Tel:852.3739|''852.3739'']]
''971.4286''
|-
|-
|Sib
|Sib
|Ζb
|Ζb
|''21\18''
|''21\18''
[[Tel:818.9475|''818.9475'']]
''933.{{Overline|3}}''
|''15\13''
|''15\13''
[[Tel:809.9481|''809.9481'']]
''923.0769''
|''24\21''
|''24\21''
[[Tel:802.2343|''802.2343'']]
''914.2857''
|''21\19''
|''21\19''
''775.845''
''884.2105''
|''12\11''
|''12\11''
''765.769''
''872.{{Overline|72}}''
|''15\14''
|''15\14''
[[Tel:752.0946|''752.0946'']]
''857.1429''
|-
|-
|Si
|Si
|Ζ
|Ζ
|''22\18''
|''22\18''
''857.945''
''977.{{Overline|7}}''
|''16\13''
|''16\13''
[[Tel:862.9446|''862.9446'']]
''984.6154''
|''26\21''
|''26\21''
[[Tel:868.0871|''868.0871'']]
''990.4762''
|''10\8''
|''10\8''
''877.44375''
''1000''
|''24\19''
|''24\19''
''886.68''
''1010.5263''
|''14\11''
|''14\11''
[[Tel:893.3973|''893.3973'']]
''1018.{{Overline|18}}''
|''18\14''
|''18\14''
[[Tel:902.5136|''902.5136'']]
''1028.5714''
|-
|-
|Si#
|Si#
|Ζ#
|Ζ#
|''23\18''
|''23\18''
[[Tel:896.9425|''896.9425'']]
''1022.{{Overline|2}}''
| rowspan="2" |''17\13''
| rowspan="2" |''17\13''
[[Tel:917.9412|''917.9412'']]
''1046.15385''
|''28\21''
|''28\21''
[[Tel:935.9406|''935.9406'']]
''1066.{{Overline|6}}''
|''11\8''
|''11\8''
[[Tel:965.1881|''965.1881'']]
''1100''
|''27\19''
|''27\19''
''997.515''
''1136.8421''
|''16\11''
|''16\11''
''1021.02545''
''1163.{{Overline|63}}''
|''21\14''
|''21\14''
''1052.9235''
''1200''
|-
|-
|Dob
|Dob
|Ηb
|Ηb
|''24\18''
|''24\18''
''935.94''
''1066.{{Overline|6}}''
|''27\21''
|''27\21''
[[Tel:902.5136|''902.5136'']]
''1028.5714''
|''10\8''
|''10\8''
''877.44375''
''1000''
|''23\19''
|''23\19''
''849.753''
''968.42105''
|''13\11''
|''13\11''
[[Tel:829.5832|''829.5832'']]
''945.{{Overline|45}}''
|''16\14''
|''16\14''
[[Tel:802.2343|''802.2343'']]
''914.2857''
|-
|-
|'''Do'''
|'''Do'''
|'''Η'''
|'''Η'''
|'''''25\18'''''
|'''''25\18'''''
'''[[Tel:974.9375|''974.9375'']]'''
'''''1111.{{Overline|1}}'''''
|'''''18\13'''''
|'''''18\13'''''
'''[[Tel:971.9379|''971.9379'']]'''
'''''1107.6923'''''
|'''''29\21'''''
|'''''29\21'''''
'''[[Tel:969.3664|''969.3664'']]'''
'''''1104.7619'''''
|'''''11\8'''''
|'''''11\8'''''
'''[[Tel:965.1881|''965.1881'']]'''
'''''1100'''''
|'''''26\19'''''
|'''''26\19'''''
'''''960.57'''''
'''''1094.7368'''''
|'''''15\11'''''
|'''''15\11'''''
'''[[Tel:957.2114|''957.2114'']]'''
'''''1090.{{Overline|90}}'''''
|'''''19\14'''''
|'''''19\14'''''
'''[[Tel:952.6532|''952.6532'']]'''
'''''1085.7143'''''
|-
|-
|Do#
|Do#
|Η#
|Η#
|''26\18''
|''26\18''
''1012.935''
''1155.{{Overline|5}}''
|''19\13''
|''19\13''
''1025.9342''
''1169.2308''
|''31\21''
|''31\21''
''1036.2193''
''1180.9524''
| rowspan="2" |''12\8''
| rowspan="2" |''12\8''
''1052.9235''
''1200''
|''29\19''
|''29\19''
''1071.405''
''1221.0526''
|''17\11''
|''17\11''
''1084.83955''
''1236.{{Overline|36}}''
|''22\14''
|''22\14''
''1103.0721''
''1257.1429''
|-
|-
|Reb
|Reb
|Θb
|Θb
|''28\18''
|''28\18''
''1091.93''
''1244.{{Overline|4}}''
|''20\13''
|''20\13''
''1079.9308''
''1230.7692''
|''32\21''
|''32\21''
''1069.9157''
''1219.0476''
|''28\19''
|''28\19''
''1034.46''
''1178.9474''
|''16\11''
|''16\11''
''1021.02545''
''1163.{{Overline|63}}''
|''20\14''
|''20\14''
''1002.7929''
''1142.8571''
|-
|-
|'''Re'''
|'''Re'''
|'''Θ'''
|'''Θ'''
|'''''29\18'''''
|'''''29\18'''''
'''''1130.9275'''''
'''''1288.{{Overline|8}}'''''
|'''''21\13'''''
|'''''21\13'''''
'''''1133.9273'''''
'''''1292.3077'''''
|'''''34\21'''''
|'''''34\21'''''
'''''1136.4986'''''
'''''1295.2381'''''
|'''''13\8'''''
|'''''13\8'''''
'''''1140.7769'''''
'''''1300'''''  
|'''''31\19'''''
|'''''31\19'''''
'''''1145.295'''''
'''''1305.2632'''''
|'''''18\11'''''
|'''''18\11'''''
'''''1148.6536'''''
'''''1309.{{Overline|09}}'''''
|'''''23\14'''''
|'''''23\14'''''
'''''1153.2118'''''
'''''1314.2857'''''
|-
|-
|Re#
|Re#
|Θ#
|Θ#
|''30\18''
|''30\18''
''1169.925''
''1333.{{Overline|3}}''
|''22\13''
|''22\13''
''1187.9238''
''1187.9238''
|''36\21''
|''36\21''
''1203.3514''
''1371.4286''
| rowspan="2" |''14\8''
| rowspan="2" |''14\8''
''1228.42125''
''1400''
|''34\19''
|''34\19''
''1256.13''
''1431.42105''
|''20\11''
|''20\11''
''1276.2818''
''1454.{{Overline|54}}''
|''26\14''
|''26\14''
''1303.6307''
''1485.7143''
|-
|-
|Mib
|Mib
|Ιb
|Ιb
|''32\18''
|''32\18''
''1247.92''
''1422.{{Overline|2}}''
|''23\13''
|''23\13''
''1241.9203''
''1415.3846''
|''37\21''
|''37\21''
''1236.7779''
''1409.5238''
|''33\19''
|''33\19''
''1218.285''
''1389.4737''
|''19\11''
|''19\11''
''1212.5678''
''1381.{{Overline|81}}''
|''24\14''
|''24\14''
''1203.3514''
''1371.4286''
|-
|-
|Mi
|Mi
|Ι
|Ι
|''33\18''
|''33\18''
''1286.9175''
''1466.{{Overline|6}}''
|''24\13''
|''24\13''
''1295.9169''
''1476.9231''
|''39\21''
|''39\21''
''1303.6307''
''1485.7143''
|''15\8''
|''15\8''
''1316.1656''
''1500''
|''36\19''
|''36\19''
''1330.02''
''1515.7895''
|''21\11''
|''21\11''
''1340.0959''
''1527.{{Overline|27}}''
|''27\14''
|''27\14''
''1353.8704''
''1542.8571''
|-
|-
|Mi#
|Mi#
|Ι#
|Ι#
|''34\18''
|''34\18''
''1323.915''
''1511.{{Overline|1}}''
| rowspan="2" |''25\13''
| rowspan="2" |''25\13''
''1348.9135''
''1538.4615''
|''41\21''
|''41\21''
''1370.4836''
''1561.9048''
|''16\8''
|''16\8''
''1403.91''
''1600''
|''39\19''
|''39\19''
''1440.855''
''1642.1053''
|''23\11''
|''23\11''
''1468.724''
''1672.{{Overline|72}}''
|''30\14''
|''30\14''
''1504.1892''
''1714.2857''
|-
|-
|Lab
|Lab
|Αb
|Αb
|''35\18''
|''35\18''
''1364.9125''
''1555.{{Overline|5}}''
|''40\21''
|''40\21''
''1337.0571''
''1523.8095''
|''15\8''
|''15\8''
''1316.1656''
''1500''
|''35\19''
|''35\19''
''1293.075''
''1473.6842''
|''20\11''
|''20\11''
''1276.2818''
''1454.{{Overline|54}}''
|''25\14''
|''25\14''
''1253.591''
''1428.5714''
|-
|-
!La
!La
!Α
!Α
! colspan="7" |''1403.91''
! colspan="7" |''1600''
|}
|}


Line 1,009: Line 1,014:
{| class="wikitable"
{| class="wikitable"
!Generators
!Generators
!Sesquitave notation
!Sixth notation
!Interval category name
!Interval category name
!Generators
!Generators
!Notation of 3/2 inverse
!Notation of sixth inverse
!Interval category name
!Interval category name
|-
|-
Line 1,018: Line 1,023:
|-
|-
|0
|0
|A
|La
|perfect sixth (minor sixth)
|0
|La
|perfect unison
|perfect unison
|0
|A
|sesquitave (just fifth)
|-
|-
|1
|1
|C
|Re
|perfect mosthird (min third)
|perfect fourth
| -1
| -1
|D
|Do
|perfect mosfourth (maj third)
|minor third
|-
|-
|2
|2
|Eb
|Si
|minor mosfifth
|major second
| -2
| -2
|B
|Mib
|major mossecond
|diminished fifth
|-
|-
|3
|3
|Bb
|Mi
|minor mossecond
|perfect fifth
| -3
| -3
|E
|Sib
|major mosfifth
|minor second
|-
|-
|4
|4
|Db
|Do#
|diminished mosfourth
|major third
| -4
| -4
|C#
|Reb
|augmented mosthird
|diminished fourth
|-
|-
| colspan="6" |The chromatic 8-note MOS also has the following intervals (from some root):
| colspan="6" |The chromatic 8-note MOS also has the following intervals (from some root):
|-
|-
|5
|5
|Ab
|La#
|diminished sesquitave
|augmented unison (chroma)
| -5
| -5
|A#
|Lab
|augmented unison (chroma)
|diminished sixth
|-
|-
|6
|6
|Cb
|Re#
|diminished mosthird
|augmented fourth
| -6
| -6
|D#
|Dob
|augmented mosfourth
|diminished third
|-
|-
|7
|7
|Ebb
|Si#
|diminished mosfifth
|augmented second
| -7
| -7
|B#
|Mibb
|augmented mossecond
|doubly diminished fifth
|}
|}
==Genchain==
==Genchain==
The generator chain for this scale is as follows:
The generator chain for this scale is as follows:
{| class="wikitable"
{| class="wikitable"
|Bbb
|Sibb
|Ebb
|Mibb
|Cb
|Dob
|Ab
|Lab
|Db
|Reb
|Bb
|Sib
|Eb
|Mib
|C
|Do
|A
|La
|D
|Re
|B
|Si
|E
|Mi
|C#
|Do#
|A#
|La#
|D#
|Re#
|B#
|Si#
|E#
|Mi#
|-
|-
|d2
|d2
|d5
|dd5
|d3
|d3
|d6
|d6
|d4
|d4
|m2
|m2
|m5
|d5
|P3
|m3
|P1
|P1
|P4
|P4
|M2
|M2
|M5
|P5
|A3
|M3
|A1
|A1
|A4
|A4
Line 1,115: Line 1,120:
|}
|}
==Modes==
==Modes==
The mode names are based on the major satellites of Uranus, in order of size:
The mode names are based on the modes of the diatonic scale , in order of size:
{| class="wikitable"
{| class="wikitable"
!Mode
!Mode
!Scale
!Scale
![[Modal UDP Notation|UDP]]
![[Modal UDP Notation|UDP]]
! colspan="4" |Interval type (mos-)
! colspan="4" |Interval type
|-
|-
!name
!name
Line 1,130: Line 1,135:
!5th
!5th
|-
|-
|Titanian
|Hindu
|LLsLs
|LLsLs
|<nowiki>4|0</nowiki>
|<nowiki>4|0</nowiki>
|M
|M
|A
|M
|P
|P
|P
|M
|-
|-
|Oberonan
|Minor
|LsLLs
|LsLLs
|<nowiki>3|1</nowiki>
|<nowiki>3|1</nowiki>
|M
|M
|m
|P
|P
|P
|P
|M
|-
|-
|Umbrielan
|Half diminished
|LsLsL
|LsLsL
|<nowiki>2|2</nowiki>
|<nowiki>2|2</nowiki>
|M
|M
|m
|P
|P
|P
|d
|m
|-
|-
|Arielan
|Diminished
|sLLsL
|sLLsL
|<nowiki>1|3</nowiki>
|<nowiki>1|3</nowiki>
|m
|m
|m
|P
|P
|P
|d
|m
|-
|-
|Mirandan
|Altered
|sLsLL
|sLsLL
|<nowiki>0|4</nowiki>
|<nowiki>0|4</nowiki>
|m
|m
|P
|m
|d
|d
|d
|m
|}
|}
==Temperaments==
==Temperaments==
The most basic rank-2 temperament interpretation of uranian is '''semiwolf''', which has 4:7:10 chords spelled <code>root-(p+1g)-(3p-2g)</code> (p = 3/2, g = the approximate 7/6). The name "semiwolf" comes from two [[7/6]] generators approximating a [[27/20]]<nowiki/>wolf fourth. This is further extended to the 11-limit in two interpretations: '''semilupine''' where 2 major mos2nds (LL) equal 11/9, and '''hemilycan''' where 1 major and 2 minor mos2nds (sLs) equal 11/9. Basic 8edf fits both extensions.
The most basic rank-2 temperament interpretation of this diatonic is '''Aeolianic''', which has septimal 6:7:9 or pental 10:12:15 chords spelled <code>root-(p-1g)-(3g)</code> (p = the minor sixth, g = the approximate 4/3). The name "Aeolianic" comes from the Aeolian minor mode having the minor sixth as its characteristic interval.
===Semiwolf===
==='''Aeolianic-Meantone'''===
[[Subgroup]]: 3/2.7/4.5/2
[[Subgroup]]: 8/5.4/3.3/2


[[Comma]] list: [[245/243]]
[[Comma]] list: [[81/80]]


[[POL2]] generator: ~7/6 = [[Tel:262.1728|262.1728]]
[[POL2]] generator: ~6/5 = 308.3057


[[Mapping]]: [{{val|1 1 3}}, {{val|0 1 -2}}]
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -1 -3}}]


[[Vals]]: {{val list|8edf, 11edf, 13edf}}
[[Optimal ET sequence]]: 5ed8/5, 8ed8/5, 13ed8/5
====Semilupine====
[[Subgroup]]: 3/2.7/4.5/2.11/4


[[Comma]] list: [[245/243]], [[100/99]]
[[POL2]] generator: ~7/6 = [[Tel:264.3771|264.3771]]
[[Mapping]]: [{{val|1 1 3 4}}, {{val|0 1 -2 -4}}]
[[Vals]]: {{val list|8edf, 13edf}}
====Hemilycan====
[[Subgroup]]: 3/2.7/4.5/2.11/4
[[Comma]] list: [[245/243]], [[441/440]]
[[POL2]] generator: ~7/6 = [[Tel:261.5939|261.5939]]
[[Mapping]]: [{{val|1 1 3 1}}, {{val|0 1 -2 4}}]
[[Vals]]: {{val list|8edf, 11edf}}
==Scale tree==
==Scale tree==
The spectrum looks like this:
The spectrum looks like this:
{| class="wikitable"
{| class="wikitable"
! colspan="6" rowspan="2" |Generator
! colspan="3" rowspan="2" |Generator
(bright)
(bright)
! colspan="2" |Cents
! colspan="2" |Normalised
! colspan="2" |''ed8\12 (→ed2\3)''
! rowspan="2" |L
! rowspan="2" |L
! rowspan="2" |s
! rowspan="2" |s
Line 1,213: Line 1,201:
! rowspan="2" |Comments
! rowspan="2" |Comments
|-
|-
!Chroma-positive
!Chroma-negative
!Chroma-positive
!Chroma-positive
!Chroma-negative
!Chroma-negative
Line 1,219: Line 1,209:
|
|
|
|
|
|514.286
|
|342.857
|
|''480''
|421.173
|''320''
|280.782
|1
|1
|1
|1
Line 1,229: Line 1,218:
|Equalised
|Equalised
|-
|-
|11\18
|17\28
|
|
|510
|330
|''485.714''
|''314.286''
|6
|5
|1.200
|
|-
|
|48\79
|
|509.7345
|329.2035
|''486.076''
|''313.924''
|17
|14
|1.214
|
|-
|
|31\51
|
|509.589
|328.767
|''486.2745''
|''313.7255''
|11
|9
|1.222
|
|-
|14\23
|
|
|509.{{Overline|09}}
|327.{{Overline|27}}
|''486.9565''
|''313.0435''
|5
|4
|1.250
|
|-
|
|39\64
|
|508.966
|326.087
|''487.5''
|''312.5''
|14
|11
|1.273
|
|-
|
|25\41
|
|
|508.475
|325.424
|''487.805''
|''312.195''
|9
|7
|1.286
|
|
|-
|
|
|36\59
|
|
|508.235
|324.706
|''488.136''
|''311.864''
|13
|10
|1.300
|
|
|428.973
|-
|272.983
|11\18
|
|
|507.692
|323.077
|''488.{{Overline|8}}''
|''311.{{Overline|1}}''
|4
|4
|3
|3
Line 1,243: Line 1,315:
|-
|-
|
|
|30\49
|63\103
|
|507.383
|322.148
|''489.32''
|''310.68''
|23
|17
|1.353
|
|-
|
|52\85
|
|
|507.317
|321.951
|''489.412''
|''310.588''
|19
|14
|1.357
|
|
|-
|
|
|41\67
|
|
|429.768
|507.2165
|272.187
|321.6495
|''489.552''
|''310.448''
|15
|11
|1.364
|
|-
|
|30\49
|
|507.062
|321.127
|''489.796''
|''310.204''
|11
|11
|8
|8
Line 1,258: Line 1,365:
|19\31
|19\31
|
|
|506.{{Overline|6}}
|320
|''490.323''
|''309.678''
|7
|5
|1.400
|
|-
|
|
|46\75
|506.422
|319.266
|''490.{{Overline|6}}''
|''309.{{Overline|3}}''
|17
|12
|1.417
|
|
|-
|
|
|27\44
|
|
|[[Tel:430.2305|430.2305]]
|506.25
|[[Tel:271.7255|271.7255]]
|318.75
|''490.{{Overline|90}}''
|''309.{{Overline|09}}''
|10
|7
|7
|5
|1.429
|1.400
|
|-
|
|35\57
|
|506.024
|318.072
|''491.228''
|''308.712''
|13
|9
|1.444
|
|-
|
|43\70
|
|505.882
|317.647
|''491.429''
|''308.571''
|16
|11
|1.4545
|
|
|-
|-
|8\13
|
|
|51\83
|
|
|505.785
|317.355
|''491.566''
|''308.434''
|19
|13
|1.4615
|
|
|-
|8\13
|
|
|
|
|431.972
|505.263
|269.983
|315.79
|''492.308''
|''307.692''
|3
|3
|2
|2
|1.500
|1.500
|Semiwolf and Semilupine start here
|Aeolianic-Meantone starts here
|-
|-
|
|
|45\73
|
|
|37\60
|504.673
|314.019
|''493.151''
|''306.849''
|17
|11
|1.5455
|
|
|-
|
|
|37\60
|
|
|432.872
|504.{{Overline|54}}
|269.083
|313.{{Overline|63}}
|''493.{{Overline|3}}''
|''306.{{Overline|6}}''
|14
|14
|9
|9
Line 1,297: Line 1,473:
|29\47
|29\47
|
|
|
|504.348
|
|313.043
|
|''493.617''
|433.121
|''306.383''
|268.834
|11
|11
|7
|7
Line 1,310: Line 1,485:
|21\34
|21\34
|
|
|
|504
|
|312
|
|''494.118''
|433.56
|''305.882''
|268.395
|8
|8
|5
|5
Line 1,323: Line 1,497:
|
|
|34\55
|34\55
|
|503.{{Overline|703}}
|
|311.{{Overline|1}}
|
|''494.{{Overline|54}}''
|433.935
|''305.{{Overline|45}}''
|268.02
|13
|13
|8
|8
Line 1,334: Line 1,507:
|-
|-
|
|
|13\21
|
|
|47\76
|503.571
|310.714
|''494.737''
|''305.263''
|18
|11
|1.636
|
|
|-
|
|
|13\21
|
|
|435.084
|503.226
|266.871
|309.678
|''495.238''
|''304.762''
|5
|5
|3
|3
Line 1,347: Line 1,531:
|-
|-
|
|
|18\29
|
|
|31\50
|502.{{Overline|702}}
|308.{{Overline|108}}
|''496''
|''304''
|12
|7
|1.714
|
|-
|
|
|49\79
|502.564
|307.692
|''496.2025''
|''303.7975''
|19
|11
|1.727
|
|
|-
|
|
|18\29
|
|
|435.696
|502.326
|266.259
|306.977
|''496.552''
|''303.448''
|7
|7
|4
|4
Line 1,362: Line 1,569:
|23\37
|23\37
|
|
|
|501.{{Overline|81}}
|
|305.{{Overline|45}}
|
|''497.{{Overline|297}}''
|436.35
|''302.{{Overline|702}}''
|265.605
|9
|9
|5
|5
Line 1,375: Line 1,581:
|28\45
|28\45
|
|
|
|501.492
|
|304.478
|
|''497.{{Overline|7}}''
|436.772
|''302.{{Overline|2}}''
|265.183
|11
|11
|6
|6
Line 1,385: Line 1,590:
|
|
|-
|-
|
|
|
|33\53
|33\53
|
|
|
|501.265
|
|303.797
|437.066
|''498.113''
|264.889
|''301.887''
|13
|13
|7
|7
Line 1,398: Line 1,602:
|
|
|-
|-
|5\8
|
|
|38\61
|
|
|501.09
|303.297
|''498.361''
|''301.639''
|15
|8
|1.875
|
|
|
|
|438.722
|263.233
|2
|1
|2.000
|Semilupine ends, Hemilycan begins
|-
|-
|
|
|43\69
|
|
|
|500.971
|
|302.913
|
|''498.551''
|47\75
|''301.449''
|439.892
|17
|262.063
|19
|9
|9
|2.111
|1.889
|
|
|-
|-
|5\8
|
|
|
|
|
|500
|300
|''500''
|''300''
|2
|1
|2.000
|Aeolianic-Meantone ends, Aeolianic-Pythagorean begins
|-
|
|
|42\67
|42\67
|
|
|440.031
|499.01
|261.924
|297.03
|''501.4925''
|''298.5075''
|17
|17
|8
|8
Line 1,437: Line 1,650:
|
|
|-
|-
|
|
|
|
|37\59
|37\59
|
|
|
|498.876
|440.209
|296.629
|261.746
|''501.695''
|''298.305''
|15
|15
|7
|7
Line 1,450: Line 1,662:
|
|
|-
|-
|
|
|
|32\51
|32\51
|
|
|
|498.701
|
|296.104
|440.442
|''501.961''
|261.513
|''298.039''
|13
|13
|6
|6
Line 1,466: Line 1,677:
|27\43
|27\43
|
|
|
|498.461
|
|295.385
|
|''502.326''
|440.762
|''297.674''
|261.193
|11
|11
|5
|5
Line 1,479: Line 1,689:
|22\35
|22\35
|
|
|
|498.113
|
|294.34
|
|''502.857''
|441.229
|''297.143''
|260.726
|9
|9
|4
|4
Line 1,490: Line 1,699:
|-
|-
|
|
|17\27
|
|
|39\62
|497.872
|293.617
|''503.226''
|''296.774''
|16
|7
|2.286
|
|
|-
|
|
|17\27
|
|
|441.972
|497.561
|259.973
|292.683
|''503.{{Overline|703}}''
|''296.{{Overline|296}}''
|7
|7
|3
|3
Line 1,505: Line 1,725:
|
|
|29\46
|29\46
|
|497.143
|
|291.429
|
|''504.348''
|442.537
|''295.652''
|259.418
|12
|12
|5
|5
Line 1,516: Line 1,735:
|-
|-
|
|
|12\19
|
|
|41\65
|496.{{Overline|96}}
|290.{{Overline|90}}
|''504.615''
|''295.385''
|17
|7
|2.429
|
|
|-
|
|
|12\19
|
|
|443.34
|496.552
|258.615
|289.655
|''505.263''
|''294.737''
|5
|5
|2
|2
Line 1,529: Line 1,759:
|-
|-
|
|
|19\30
|
|
|31\49
|496
|288
|''506.122''
|''293.878''
|13
|5
|2.600
|
|
|-
|
|
|
|
|[[Tel:444.5715|444.5715]]
|50\79
|[[Tel:257.3835|257.3835]]
|495.868
|287.633
|''506.329''
|''293.671''
|21
|8
|2.625
|
|-
|
|19\30
|
|495.652
|286.957
|''506.{{Overline|6}}''
|''293.{{Overline|3}}''
|8
|8
|3
|3
Line 1,544: Line 1,797:
|26\41
|26\41
|
|
|495.238
|285.714
|''507.317''
|''292.683''
|11
|4
|2.750
|
|-
|
|33\52
|
|495
|285
|''507.692''
|''292.308''
|14
|5
|2.800
|
|-
|
|40\63
|
|494.536
|284.536
|''507.9365''
|''292.0635''
|17
|6
|2.833
|
|-
|
|47\74
|
|494.737
|284.211
|''508.{{Overline|108}}''
|''291.{{Overline|891}}''
|20
|7
|2.857
|
|
|-
|
|
|54\85
|
|
|445.142
|494.6565
|256.813
|283.9695
|11
|''508.235''
|4
|''291.765''
|2.750
|23
|8
|2.875
|
|
|-
|-
|7\11
|
|
|61\96
|
|
|494.{{Overline|594}}
|283.{{Overline|783}}
|''508.{{Overline|3}}''
|''291.{{Overline|6}}''
|26
|9
|2.889
|
|
|-
|7\11
|
|
|
|
|446.699
|494.118
|255.256
|282.353
|''509.{{Overline|09}}''
|''290.{{Overline|90}}''
|3
|3
|1
|1
|3.000
|3.000
|Semiwolf and Hemilycan end here
|Aeolianic-Pythagorean ends, Aeolianic-Superpyth begins
|-
|
|65\102
|
|493.671
|281.013
|''509.804''
|''290.196''
|28
|9
|3.111
|
|-
|
|58\91
|
|493.617
|280.851
|''509.89''
|''290.11''
|25
|8
|3.125
|
|-
|
|51\80
|
|493.548
|280.645
|''510''
|''290''
|22
|7
|3.143
|
|-
|-
|
|
|44\69
|
|
|37\58
|493.458
|280.374
|''510.145''
|''289.855''
|19
|6
|3.167
|
|
|-
|
|
|37\58
|
|
|447.799
|493.{{Overline|3}}
|254.156
|280
|''510.345''
|''289.655''
|16
|16
|5
|5
Line 1,583: Line 1,941:
|30\47
|30\47
|
|
|
|493.151
|
|279.452
|
|''510.638''
|448,056
|''289.362''
|253.899
|13
|13
|4
|4
Line 1,596: Line 1,953:
|23\36
|23\36
|
|
|
|492.857
|
|278.571
|
|''511.{{Overline|1}}''
|448.471
|''288.{{Overline|8}}''
|253.484
|10
|10
|3
|3
Line 1,609: Line 1,965:
|16\25
|16\25
|
|
|
|492.308
|
|276.923
|
|''512''
|449.251
|''288''
|252.704
|7
|7
|2
|2
Line 1,622: Line 1,977:
|25\39
|25\39
|
|
|
|491.803
|
|275.41
|
|''512.8205''
|449.971
|''287.1795''
|251.984
|11
|11
|3
|3
Line 1,635: Line 1,989:
|34\53
|34\53
|
|
|491.566
|274.699
|''513.2075''
|''286.7925''
|15
|4
|3.750
|
|-
|
|43\67
|
|491.429
|274.286
|''513.433''
|''286.567''
|19
|5
|3.800
|
|
|-
|
|
|52\81
|
|
|450.311
|491.339
|251.644
|274.016
|15
|''513.58''
|''286.42''
|23
|6
|3.833
|
|-
|
|61\95
|
|491.275
|273.825
|''513.684''
|''286.316''
|27
|7
|3.857
|
|-
|9\14
|
|
|490.{{Overline|90}}
|272.{{Overline|72}}
|''514.286''
|''285.714''
|4
|1
|4.000
|
|-
|
|47\73
|
|490.435
|271.304
|''515.0685''
|''284.3315''
|21
|5
|4.200
|
|-
|
|38\59
|
|490.323
|270.968
|''515.254''
|''284.746''
|17
|4
|4
|3.750
|4.250
|
|-
|
|29\45
|
|490.141
|270.422
|''515.{{Overline|5}}''
|''284.{{Overline|4}}''
|13
|3
|4.333
|
|-
|
|20\31
|
|489.795
|269.388
|''516.129''
|''283.871''
|9
|2
|4.500
|
|
|-
|-
|9\14
|
|
|31\48
|
|
|489.474
|268.421
|''516.{{Overline|6}}''
|''283.{{Overline|3}}''
|14
|3
|4.667
|
|
|-
|
|
|42\65
|
|
|451.257
|489.32
|250.698
|267.961
|''516.923''
|''283.077''
|19
|4
|4
|4.750
|
|-
|11\17
|
|
|488.{{Overline|8}}
|266.{{Overline|6}}
|''517.647''
|''282.353''
|5
|1
|1
|4.000
|5.000
|Near [[24edo]]
|Aeolianic-Superpyth ends
|-
|
|35\54
|
|488.372
|265.116
|''518.{{Overline|518}}''
|''281.{{Overline|481}}''
|16
|3
|5.333
|
|-
|
|24\37
|
|488.136
|264.407
|''518.{{Overline|918}}''
|''281.{{Overline|081}}''
|11
|2
|5.500
|
|-
|
|37\57
|
|487.912
|263.736
|''519.298''
|''280.702''
|17
|3
|5.667
|
|-
|-
|2\3
|13\20
|
|
|
|
|487.5
|262.2
|''520''
|''280''
|6
|1
|6.000
|
|
|-
|2\3
|
|
|
|
|467.97
|480
|233.985
|240
|''533.{{Overline|3}}''
|''266.{{Overline|6}}''
|1
|1
|0
|0

Latest revision as of 02:16, 23 May 2023

↖ 2L 1s⟨8/5⟩ ↑ 3L 1s⟨8/5⟩ 4L 1s⟨8/5⟩ ↗
← 2L 2s⟨8/5⟩ 3L 2s (8/5-equivalent) 4L 2s⟨8/5⟩ →
↙ 2L 3s⟨8/5⟩ ↓ 3L 3s⟨8/5⟩ 4L 3s⟨8/5⟩ ↘ 
┌╥╥┬╥┬┐
│║║│║││
│││││││
└┴┴┴┴┴┘
Scale structure
Step pattern LLsLs
sLsLL
Equave 8/5 (813.7 ¢)
Period 8/5 (813.7 ¢)
Generator size(ed8/5)
Bright 3\5 to 2\3 (488.2 ¢ to 542.5 ¢)
Dark 1\3 to 2\5 (271.2 ¢ to 325.5 ¢)
Related MOS scales
Parent 2L 1s⟨8/5⟩
Sister 2L 3s⟨8/5⟩
Daughters 5L 3s⟨8/5⟩, 3L 5s⟨8/5⟩
Neutralized 1L 4s⟨8/5⟩
2-Flought 8L 2s⟨8/5⟩, 3L 7s⟨8/5⟩
Equal tunings(ed8/5)
Equalized (L:s = 1:1) 3\5 (488.2 ¢)
Supersoft (L:s = 4:3) 11\18 (497.3 ¢)
Soft (L:s = 3:2) 8\13 (500.7 ¢)
Semisoft (L:s = 5:3) 13\21 (503.7 ¢)
Basic (L:s = 2:1) 5\8 (508.6 ¢)
Semihard (L:s = 5:2) 12\19 (513.9 ¢)
Hard (L:s = 3:1) 7\11 (517.8 ¢)
Superhard (L:s = 4:1) 9\14 (523.1 ¢)
Collapsed (L:s = 1:0) 2\3 (542.5 ¢)

3L 2s<8/5> is a minor sixth-repeating MOS scale. The notation "<8/5>" means the period of the MOS is 8/5, disambiguating it from octave-repeating 3L 2s.

The generator range is 240 to 342.9 cents, placing it on the diatonic minor third, usually representing a minor third of some type (like 6/5). The bright (chroma-positive) generator is, however, its minor sixth complement (480 to 514.3 cents).

Because this is a minor sixth-repeating scale, each tone has an 8/5 minor sixth above it. The scale has one major chord, one minor chord and three diminished chords. This scale also has two diminished 7th chords, making it a warped melodic minor scale.

Basic 3L 2s<8/5> is in 8ed8/5, which is a very good minor sixth-based equal tuning similar to 12edo.

Notation

There are 2 main ways to notate this scale. One method uses a simple sixth repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi). Given that 1-7/6-3/2 is minor sixth-equivalent to a tone cluster of 1-16/15-7/6, it may be more convenient to notate these diatonic scales as repeating at the double sixth (diminished eleventh~tenth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 16\15. Notating this way produces a tenth which is the Dorian mode of Annapolis[6L 4s] or Oriole[6L 4s]. Since there are exactly 10 naturals in double sixth notation, Greek numerals 1-10 may be used.

Normalized
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Oriole, Annapolis 18eds 13eds 21eds 8eds 19eds 11eds 14eds
La# Α# 1\18

46.15385

1\13

63.1579

2\21

77.41935

1\8

100

3\19

124.1379

2\11

141.1765

3\14

163.63

Sib Βb 3\18

138.4615

2\13

126.3158

3\21

116.129

2\19

82.7586

1\11

70.5882

1\14

54.54

Si Β 4\18

184.6154

3\13

189.4736

5\21

193.5484

2\8

200

5\19

206.89655

3\11

211.7647

4\14

218.18

Si# Β# 5\18

230.7692

4\13

252.6316

7\21

270.9677

3\8

300

8\19

331.0345

5\11

352.9412

7\14

381.81

Dob Γb 6\18

276.9231

6\21

232.2581

2\8

200

4\19

165.5172

2\11

141.1765

2\14

109.09

Do Γ 7\18

323.0769

5\13

315.7895

8\21

309.6774

3\8

300

7\19

289.6552

4\11

282.3529

5\14

272.72

Do# Γ# 8\18

369.2308

6\13

378.9474

10\21

387.0968

4\8

400

10\19

413.7931

6\11

423.5294

8\14

436.36

Reb Δb 10\18

461.5385

7\13

442.1053

11\21

425.80645

9\19

372.4138

5\11

352.9412

6\14

327.27

Re Δ 11\18

507.6923

8\13

505.2632

13\21

503.2259

5\8

500

12\19

496.5517

7\11

494.11765

9\14

490.90

Re# Δ# 12\18

553.84615

9\13

568.42105

15\21

580.6452

6\8

600

15\19

620.6897

9\11

635.2941

12\14

654.54

Mib Εb 14\18

646.15385

10\13

631.57895

16\21

619.3548

14\19

579.3103

8\11

564.7059

10\14

545.45

Mi Ε 15\18

692.3077

11\13

694.7368

18\21

696.7742

7\8

700

17\19

703.4483

10\11

705.88235

13\14

709.09

Mi# Ε# 16\18

738.4615

12\13

757.8947

20\21

774.19355

8\8

800

20\19

827.5862

12\11

847.0588

16\14

872.72

Lab Ϛb/Ϝb 17\18

784.6154

19\21

735.4839

7\8

700

16\19

662.069

9\11

635.2941

11\14

600

La Ϛ/Ϝ 18\18

830.7692

13\13

821.0526

21\21

812.9032

8\8

800

19\19

786.2069

11\11

776.4706

14\14

763.63

La# Ϛ#/Ϝ# 19\18

876.9231

14\13

884.2105

23\21

890.3226

9\8

900

22\19

910.3448

13\11

917.6471

17\14

927.27

Sib Ζb 21\18

969.2308

15\13

947.3684

24\21

929.0323

21\19

868.9655

12\11

847.0588

15\14

818.18

Si Ζ 22\18

1015.3846

16\13

1010.5263

26\21

1006.4516

10\8

1000

24\19

993.10345

14\11

988.2353

18\14

981.81

Si# Ζ# 23\18

1061.5385

17\13

1071.6842

28\21

1083.871

11\8

1100

27\19

1117.2414

16\11

1129.4118

21\14

1145.45

Dob Ηb 24\18

1107.6923

27\21

1045.1613

10\8

1000

23\19

951.7241

13\11

917.6471

16\14

872.72

Do Η 25\18

1153.84615

18\13

1136.8421

29\21

1122.58065

11\8

1100

26\19

1075.8621

15\11

1052.8235

19\14

1036.36

Do# Η# 26\18

1200

19\13

1200

31\21

1200

12\8

1200

29\19

1200

17\11

1200

22\14

1200

Reb Θb 28\18

1292.3077

20\13

1263.1579

32\21

1238.7097

28\19

1158.6207

16\11

1129.4118

20\14

1090.90

Re Θ 29\18

1338.4615

21\13

1326.3158

34\21

1316.129

13\8

1300

31\19

1282.7586

18\11

1270.5882

23\14

1254.54

Re# Θ# 30\18

1384.6154

22\13

1389.4737

36\21

1393.5484

14\8

1400

34\19

1406.89655

20\11

1411.7647

26\14

1418.18

Mib Ιb 32\18

1476.9231

23\13

1452.6316

37\21

1432.2581

33\19

1365.5172

19\11

1341.1765

24\14

1309.09

Mi Ι 33\18

1523.0769

24\13

1515.7895

39\21

1509.6774

15\8

1500

36\19

1489.6551

21\11

1482.3529

27\14

1472.72

Mi# Ι# 34\18

1569.2308

25\13

1578.9474

41\21

1587.0968

16\8

1600

39\19

1613.7931

23\11

1623.5294

30\14

1636.36

Lab Αb 35\18

1615.3846

40\21

1548.3871

15\8

1500

35\19

1448.2859

20\11

1411.7647

25\14

1363.63

La Α 36\18

1661.5385

26\13

1642.1053

42\21

1625.80645

16\8

1600

38\19

1572.4138

22\11

1552.9412

28\14

1527.27

ed8\12 (→ed2\3)
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Oriole, Annapolis 18eds 13eds 21eds 8eds 19eds 11eds 14eds
La# Α# 1\18

44.4

1\13

61.5385

2\21

76.1905

1\8

100

3\19

126.3158

2\11

145.45

3\14

171.4286

Sib Βb 3\18

133.3

2\13

123.0769

3\21

114.2857

2\19

84.2105

1\11

72.72

1\14

57.1429

Si Β 4\18

177.7

3\13

184.6154

5\21

190.4762

2\8

200

5\19

210.5263

3\11

218.18

4\14

228.5714

Si# Β# 5\18

222.2

4\13

246.15385

7\21

266.6

3\8

300

8\19

336.8421

5\11

363.63

7\14

400

Dob Γb 6\18

266.6

6\21

228.5714

2\8

200

4\19

168.42105

2\11

145.45

2\14

114.2857

Do Γ 7\18

311.1

5\13

307.6923

8\21

304.7619

3\8

300

7\19

294.7368

4\11

290.90

5\14

285.7143

Do# Γ# 8\18

355.5

6\13

369.2308

10\21

380.9524

4\8

400

10\19

421.0526

6\11

436.36

8\14

457.1429

Reb Δb 10\18

444.4

7\13

430.7692

11\21

419.0476

9\19

378.9474

5\11

363.63

6\14

342.8571

Re Δ 11\18

488.8

8\13

492.3077

13\21

495.2381

5\8

500

12\19

505.2632

7\11

509.09

9\14

514.2857

Re# Δ# 12\18

533.3

9\13

553.84615

15\21

571.4286

6\8

600

15\19

631.42105

9\11

654.54

12\14

685.7143

Mib Εb 14\18

622.2

10\13

615.3846

16\21

609.5238

14\19

589.4737

8\11

581.81

10\14

571.4286

Mi Ε 15\18

666.6

11\13

676.9231

18\21

685.7143

7\8

700

17\19

715.7895

10\11

727.27

13\14

742.8571

Mi# Ε# 16\18

711.1

12\13

738.4615

20\21

761.9048

8\8

800

20\19

842.1053

12\11

872.72

16\14

914.2857

Lab Ϛb/Ϝb 17\18

755.5

19\21

723.8095

7\8

700

16\19

673.6842

8\11

581.81

11\14

628.5714

La Ϛ/Ϝ 800
La# Ϛ#/Ϝ# 19\18

844.4

14\13

861.5385

23\21

876.1905

9\8

900

22\19

926.3158

13\11

945.45

17\14

971.4286

Sib Ζb 21\18

933.3

15\13

923.0769

24\21

914.2857

21\19

884.2105

12\11

872.72

15\14

857.1429

Si Ζ 22\18

977.7

16\13

984.6154

26\21

990.4762

10\8

1000

24\19

1010.5263

14\11

1018.18

18\14

1028.5714

Si# Ζ# 23\18

1022.2

17\13

1046.15385

28\21

1066.6

11\8

1100

27\19

1136.8421

16\11

1163.63

21\14

1200

Dob Ηb 24\18

1066.6

27\21

1028.5714

10\8

1000

23\19

968.42105

13\11

945.45

16\14

914.2857

Do Η 25\18

1111.1

18\13

1107.6923

29\21

1104.7619

11\8

1100

26\19

1094.7368

15\11

1090.90

19\14

1085.7143

Do# Η# 26\18

1155.5

19\13

1169.2308

31\21

1180.9524

12\8

1200

29\19

1221.0526

17\11

1236.36

22\14

1257.1429

Reb Θb 28\18

1244.4

20\13

1230.7692

32\21

1219.0476

28\19

1178.9474

16\11

1163.63

20\14

1142.8571

Re Θ 29\18

1288.8

21\13

1292.3077

34\21

1295.2381

13\8

1300

31\19

1305.2632

18\11

1309.09

23\14

1314.2857

Re# Θ# 30\18

1333.3

22\13

1187.9238

36\21

1371.4286

14\8

1400

34\19

1431.42105

20\11

1454.54

26\14

1485.7143

Mib Ιb 32\18

1422.2

23\13

1415.3846

37\21

1409.5238

33\19

1389.4737

19\11

1381.81

24\14

1371.4286

Mi Ι 33\18

1466.6

24\13

1476.9231

39\21

1485.7143

15\8

1500

36\19

1515.7895

21\11

1527.27

27\14

1542.8571

Mi# Ι# 34\18

1511.1

25\13

1538.4615

41\21

1561.9048

16\8

1600

39\19

1642.1053

23\11

1672.72

30\14

1714.2857

Lab Αb 35\18

1555.5

40\21

1523.8095

15\8

1500

35\19

1473.6842

20\11

1454.54

25\14

1428.5714

La Α 1600

Intervals

Generators Sixth notation Interval category name Generators Notation of sixth inverse Interval category name
The 5-note MOS has the following intervals (from some root):
0 La perfect sixth (minor sixth) 0 La perfect unison
1 Re perfect fourth -1 Do minor third
2 Si major second -2 Mib diminished fifth
3 Mi perfect fifth -3 Sib minor second
4 Do# major third -4 Reb diminished fourth
The chromatic 8-note MOS also has the following intervals (from some root):
5 La# augmented unison (chroma) -5 Lab diminished sixth
6 Re# augmented fourth -6 Dob diminished third
7 Si# augmented second -7 Mibb doubly diminished fifth

Genchain

The generator chain for this scale is as follows:

Sibb Mibb Dob Lab Reb Sib Mib Do La Re Si Mi Do# La# Re# Si# Mi#
d2 dd5 d3 d6 d4 m2 d5 m3 P1 P4 M2 P5 M3 A1 A4 A2 A5

Modes

The mode names are based on the modes of the diatonic scale , in order of size:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th
Hindu LLsLs 4|0 M M P P
Minor LsLLs 3|1 M m P P
Half diminished LsLsL 2|2 M m P d
Diminished sLLsL 1|3 m m P d
Altered sLsLL 0|4 m m d d

Temperaments

The most basic rank-2 temperament interpretation of this diatonic is Aeolianic, which has septimal 6:7:9 or pental 10:12:15 chords spelled root-(p-1g)-(3g) (p = the minor sixth, g = the approximate 4/3). The name "Aeolianic" comes from the Aeolian minor mode having the minor sixth as its characteristic interval.

Aeolianic-Meantone

Subgroup: 8/5.4/3.3/2

Comma list: 81/80

POL2 generator: ~6/5 = 308.3057

Mapping: [⟨1 1 2], ⟨0 -1 -3]]

Optimal ET sequence: 5ed8/5, 8ed8/5, 13ed8/5

Scale tree

The spectrum looks like this:

Generator

(bright)

Normalised ed8\12 (→ed2\3) L s L/s Comments
Chroma-positive Chroma-negative Chroma-positive Chroma-negative
3\5 514.286 342.857 480 320 1 1 1.000 Equalised
17\28 510 330 485.714 314.286 6 5 1.200
48\79 509.7345 329.2035 486.076 313.924 17 14 1.214
31\51 509.589 328.767 486.2745 313.7255 11 9 1.222
14\23 509.09 327.27 486.9565 313.0435 5 4 1.250
39\64 508.966 326.087 487.5 312.5 14 11 1.273
25\41 508.475 325.424 487.805 312.195 9 7 1.286
36\59 508.235 324.706 488.136 311.864 13 10 1.300
11\18 507.692 323.077 488.8 311.1 4 3 1.333
63\103 507.383 322.148 489.32 310.68 23 17 1.353
52\85 507.317 321.951 489.412 310.588 19 14 1.357
41\67 507.2165 321.6495 489.552 310.448 15 11 1.364
30\49 507.062 321.127 489.796 310.204 11 8 1.375
19\31 506.6 320 490.323 309.678 7 5 1.400
46\75 506.422 319.266 490.6 309.3 17 12 1.417
27\44 506.25 318.75 490.90 309.09 10 7 1.429
35\57 506.024 318.072 491.228 308.712 13 9 1.444
43\70 505.882 317.647 491.429 308.571 16 11 1.4545
51\83 505.785 317.355 491.566 308.434 19 13 1.4615
8\13 505.263 315.79 492.308 307.692 3 2 1.500 Aeolianic-Meantone starts here
45\73 504.673 314.019 493.151 306.849 17 11 1.5455
37\60 504.54 313.63 493.3 306.6 14 9 1.556
29\47 504.348 313.043 493.617 306.383 11 7 1.571
21\34 504 312 494.118 305.882 8 5 1.600
34\55 503.703 311.1 494.54 305.45 13 8 1.625
47\76 503.571 310.714 494.737 305.263 18 11 1.636
13\21 503.226 309.678 495.238 304.762 5 3 1.667
31\50 502.702 308.108 496 304 12 7 1.714
49\79 502.564 307.692 496.2025 303.7975 19 11 1.727
18\29 502.326 306.977 496.552 303.448 7 4 1.750
23\37 501.81 305.45 497.297 302.702 9 5 1.800
28\45 501.492 304.478 497.7 302.2 11 6 1.833
33\53 501.265 303.797 498.113 301.887 13 7 1.857
38\61 501.09 303.297 498.361 301.639 15 8 1.875
43\69 500.971 302.913 498.551 301.449 17 9 1.889
5\8 500 300 500 300 2 1 2.000 Aeolianic-Meantone ends, Aeolianic-Pythagorean begins
42\67 499.01 297.03 501.4925 298.5075 17 8 2.125
37\59 498.876 296.629 501.695 298.305 15 7 2.143
32\51 498.701 296.104 501.961 298.039 13 6 2.167
27\43 498.461 295.385 502.326 297.674 11 5 2.200
22\35 498.113 294.34 502.857 297.143 9 4 2.250
39\62 497.872 293.617 503.226 296.774 16 7 2.286
17\27 497.561 292.683 503.703 296.296 7 3 2.333
29\46 497.143 291.429 504.348 295.652 12 5 2.400
41\65 496.96 290.90 504.615 295.385 17 7 2.429
12\19 496.552 289.655 505.263 294.737 5 2 2.500
31\49 496 288 506.122 293.878 13 5 2.600
50\79 495.868 287.633 506.329 293.671 21 8 2.625
19\30 495.652 286.957 506.6 293.3 8 3 2.667
26\41 495.238 285.714 507.317 292.683 11 4 2.750
33\52 495 285 507.692 292.308 14 5 2.800
40\63 494.536 284.536 507.9365 292.0635 17 6 2.833
47\74 494.737 284.211 508.108 291.891 20 7 2.857
54\85 494.6565 283.9695 508.235 291.765 23 8 2.875
61\96 494.594 283.783 508.3 291.6 26 9 2.889
7\11 494.118 282.353 509.09 290.90 3 1 3.000 Aeolianic-Pythagorean ends, Aeolianic-Superpyth begins
65\102 493.671 281.013 509.804 290.196 28 9 3.111
58\91 493.617 280.851 509.89 290.11 25 8 3.125
51\80 493.548 280.645 510 290 22 7 3.143
44\69 493.458 280.374 510.145 289.855 19 6 3.167
37\58 493.3 280 510.345 289.655 16 5 3.200
30\47 493.151 279.452 510.638 289.362 13 4 3.250
23\36 492.857 278.571 511.1 288.8 10 3 3.333
16\25 492.308 276.923 512 288 7 2 3.500
25\39 491.803 275.41 512.8205 287.1795 11 3 3.667
34\53 491.566 274.699 513.2075 286.7925 15 4 3.750
43\67 491.429 274.286 513.433 286.567 19 5 3.800
52\81 491.339 274.016 513.58 286.42 23 6 3.833
61\95 491.275 273.825 513.684 286.316 27 7 3.857
9\14 490.90 272.72 514.286 285.714 4 1 4.000
47\73 490.435 271.304 515.0685 284.3315 21 5 4.200
38\59 490.323 270.968 515.254 284.746 17 4 4.250
29\45 490.141 270.422 515.5 284.4 13 3 4.333
20\31 489.795 269.388 516.129 283.871 9 2 4.500
31\48 489.474 268.421 516.6 283.3 14 3 4.667
42\65 489.32 267.961 516.923 283.077 19 4 4.750
11\17 488.8 266.6 517.647 282.353 5 1 5.000 Aeolianic-Superpyth ends
35\54 488.372 265.116 518.518 281.481 16 3 5.333
24\37 488.136 264.407 518.918 281.081 11 2 5.500
37\57 487.912 263.736 519.298 280.702 17 3 5.667
13\20 487.5 262.2 520 280 6 1 6.000
2\3 480 240 533.3 266.6 1 0 → inf Paucitonic
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