Xenharmonic Wiki

User:Moremajorthanmajor/8L 3s (perfect twelfth-equivalent): Difference between revisions

← Older edit
VisualWikitext
No edit summary
 
(14 intermediate revisions by the same user not shown)
Line 9: Line 9:
Pyotr Ilyich Tchaikovsky drew from the Obikhod style for his ''1812 Overture'', as did Nikolai Rimsky-Korsakov in his ''Russian Easter Festival Overture.'' Anatoly Lyadov also drew from them in his ''Ten Arrangements from Obikhod'' Op.61, as did Alexander Raskatov in his  ''Obikhod'' (2002).
Pyotr Ilyich Tchaikovsky drew from the Obikhod style for his ''1812 Overture'', as did Nikolai Rimsky-Korsakov in his ''Russian Easter Festival Overture.'' Anatoly Lyadov also drew from them in his ''Ten Arrangements from Obikhod'' Op.61, as did Alexander Raskatov in his  ''Obikhod'' (2002).


The pitch set used in these chants traditionally consists of four three-note groups. Each note within a group is separated by a whole tone, and each group is separated by a semitone. If starting from G, the result is: G, A, B / C, D, E / F, G, A / B♭, C, D. Theoretically, more groups can be added either above or below, which has been done by some 20th-century Russian composers. This pitch set also influenced Russian folk music: for example, the Livenka accordion contains the pitch set on its melody side. On a common Livenka accordion, the pitch set will not span a pure tritave.<ref>[https://en.wikipedia.org/wiki/Obikhod Obikhod - Wikipedia]. ''en.wikipedia.org''. Retrieved July 28, 2021.</ref>
The pitch set used in these chants traditionally consists of four three-note groups. Each note within a group is separated by a whole tone, and each group is separated by a semitone. If starting from G, the result is: G, A, B / C, D, E / F, G, A / B♭, C, D. Theoretically, more groups can be added either above or below, which has been done by some 20th-century Russian composers. This pitch set also influenced Russian folk music: for example, the Livenka accordion contains the pitch set on its melody side. On a common Livenka accordion, the pitch set will not span a pure tritave.<ref>[https://en.wikipedia.org/wiki/Obikhod Obikhod - Wikipedia]. ''en.wikipedia.org''. Retrieved July 28, 2021.</ref> A pathological trait the pitch set exhibits is that normalization to [[edo]] collapses the range for the [[dark]] [[generator]] to the octave.
==Standing assumptions==
==Standing assumptions==
The tempered generalized Livenka accordion is used in this article to refer to tunings of the pitch set.
The tempered generalized Livenka accordion is used in this article to refer to tunings of the pitch set.
Line 15: Line 15:
The [[TAMNAMS]] system is used in this article to refer to 8L 3s (perfect twelfth equivalent) step size ratios and step ratio ranges.
The [[TAMNAMS]] system is used in this article to refer to 8L 3s (perfect twelfth equivalent) step size ratios and step ratio ranges.


The notation used in this article is GHJKLABCDEFG = LLsLLLsLLLs (Ionian #11), #/f = up/down by chroma (mnemonic f = F molle in Latin).
The notation used in this article is GHJKLABCDEFG = LLsLLLsLLLs (Ionian #11) or LLLsLLsLLLs (Lydian), #/f = up/down by chroma (mnemonic f = F molle in Latin).


Thus the [[19edt]] gamut is as follows:
Thus the [[19edt]] gamut is as follows:


'''G/F#''' G#/Hf '''H''' H#/Jf '''J K''' K#/Lf '''L''' L#/Af '''A''' ''A#/Bf'' '''B C''' C#/Df '''D''' D#/Ef '''E''' ''E#/Ff''  '''F/Gf'''
'''G/F#''' G#/Hf '''H''' H#/Jf '''J K''' K#/Lf '''L''' L#/Af '''A''' ''A#/Bf'' '''B C''' C#/Df '''D''' D#/Ef '''E''' ''E#/Ff''  '''F/Gf'''
'''G/F#''' G#/Hf '''H''' H#/Jf '''J''' J#/Kf '''K''' '''L''' L#/Af '''A''' ''A#/Bf'' '''B C''' C#/Df '''D''' D#/Ef '''E''' ''E#/Ff''  '''F/Gf'''


The [[27edt]] gamut is notated as follows:
The [[27edt]] gamut is notated as follows:


'''G''' F#/Hf G# '''H''' Jf H#/Kf '''J K''' J#/Lf K# '''L''' Af L# '''A''' ''Bf'' A#/Cf '''B''' '''C''' B#/Df C# '''D''' Ef D# '''E''' ''Ff'' E#/Gf '''F'''
'''G''' F#/Hf G# '''H''' Jf H#/Kf '''J K''' J#/Lf K# '''L''' Af L# '''A''' ''Bf'' A#/Cf '''B''' '''C''' B#/Df C# '''D''' Ef D# '''E''' ''Ff'' E#/Gf '''F'''
'''G''' F#/Hf G# '''H''' Jf H#/Kf '''J''' Kf J#/Lf '''K''' '''L''' Af L# '''A''' ''Bf'' A#/Cf '''B''' '''C''' B#/Df C# '''D''' Ef D# '''E''' ''Ff'' E#/Gf '''F'''


The [[30edt]] gamut:
The [[30edt]] gamut:


'''G''' Hf G# '''H''' Jf H# '''J''' J#/Kf '''K''' K# Lf '''L''' L# Af '''A''' A# ''Bf'' '''B''' B#/Cf '''C''' C# Df '''D''' D# Ef '''E''' E# ''Ff'' '''F''' F#/Gf
'''G''' G# Hf '''H''' H# Jf '''J''' J#/Kf '''K''' K# Lf '''L''' L# Af '''A''' A# ''Bf'' '''B''' B#/Cf '''C''' C# Df '''D''' D# Ef '''E''' E# ''Ff'' '''F''' F#/Gf
 
'''G''' G# Hf '''H''' H# Jf '''J''' J# Kf '''K''' K#/Lf '''L''' L# Af '''A''' A# ''Bf'' '''B''' B#/Cf '''C''' C# Df '''D''' D# Ef '''E''' E# ''Ff'' '''F''' F#/Gf


==Intervals==
==Intervals==
The table of Obikhodic intervals below takes the fifth as the generator. Given the size of the fifth generator ''g'', any Obikhodic interval can easily be found by noting what multiple of ''g'' it is, and multiplying the size by the number ''k'' of generators it takes to reach the interval and reducing mod 1900<ref>For relative cents</ref> if necessary (so you can use "''k''*''g'' % 1900" for search engines, for plugged-in values of ''k'' and ''g''). For example, since the major 3rd is reached by six fifth generators, [[27edt]]'s major 3rd is 6*703.7 mod 1900 = 4222.22 mod 1900 = 422.22r¢.
The table of Obikhodic intervals below takes the fifth as the generator.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 67: Line 73:
|6L+2s
|6L+2s
| -2
| -2
|K
|K, Kf
|natural 4th
|natural 4th
|2L+1s
|2L+1s
Line 126: Line 132:
|-
|-
|9
|9
|K#
|K, K#
|augmented 4th
|augmented 4th
|3L
|3L
Line 245: Line 251:
|major 2nd
|major 2nd
|2\19, 200.00
|2\19, 200.00
|3\27, 211.765
|3\27, 211.76
|3\30, 189.47
|3\30, 189.47
|H
|H
Line 268: Line 274:
|7\27, 494.12
|7\27, 494.12
|8\30, 505,26
|8\30, 505,26
|K
|K, Kf
| -2
| -2
|-
|-
Line 275: Line 281:
|9\27, 635.29
|9\27, 635.29
|9\30, 568.42
|9\30, 568.42
|K#
|K, K#
|9
|9
|-
|-
Line 341: Line 347:
|major 9th
|major 9th
|14\19, 1400.00
|14\19, 1400.00
|20\27, 1411.765
|20\27, 1411.76
|22\30, 1389.47
|22\30, 1389.47
|D
|D
Line 432: Line 438:
|major 2nd
|major 2nd
|2\19, 200.00
|2\19, 200.00
|3\27, 211.765
|3\27, 211.76
|5\46, 206.90
|5\46, 206.90
|H
|H
Line 455: Line 461:
|7\27, 494.12
|7\27, 494.12
|12\46, 496.55
|12\46, 496.55
|K
|K, Kf
| -2
| -2
|-
|-
Line 462: Line 468:
|9\27, 635.29
|9\27, 635.29
|15\46, 620.69
|15\46, 620.69
|K#
|K, K#
|9
|9
|-
|-
Line 488: Line 494:
|9\19, 900.00
|9\19, 900.00
|13\27, 917.65
|13\27, 917.65
|22\46, 910.345
|22\46, 910.34
|A
|A
|4
|4
Line 528: Line 534:
|major 9th
|major 9th
|14\19, 1400.00
|14\19, 1400.00
|20\27, 1411.765
|20\27, 1411.76
|34\46, 1406.90
|34\46, 1406.90
|D
|D
Line 536: Line 542:
|15\19, 1500.00
|15\19, 1500.00
|21\27, 1482.35
|21\27, 1482.35
|36\46, 1489.655
|36\46, 1489.66
|Ef
|Ef
| -6
| -6
Line 565: Line 571:
*The large step is between near the meantone and near the Pythagorean 9/8 whole tone, somewhere between as in [[19edo]] and as in [[17edo|12edo]].
*The large step is between near the meantone and near the Pythagorean 9/8 whole tone, somewhere between as in [[19edo]] and as in [[17edo|12edo]].
*The major 3rd (made of two large steps) is a near-[[Just intonation|just]] to near-[[Pythagorean]] major third.
*The major 3rd (made of two large steps) is a near-[[Just intonation|just]] to near-[[Pythagorean]] major third.
The sizes of the generator, large step and small step of oneirotonic are as follows in various hyposoft Obikhod tunings (~19edt not shown).
The sizes of the generator, large step and small step of Obikhodic are as follows in various hyposoft Obikhod tunings (~19edt not shown).
{| class="wikitable right-2 right-3 right-4 right-5"
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
Line 633: Line 639:
|8\30, 505,26
|8\30, 505,26
|13\49, 503.23
|13\49, 503.23
|K
|K, Kf
|4/3
|4/3
| -2
| -2
Line 639: Line 645:
|augmented 4th
|augmented 4th
|9\30, 568.42
|9\30, 568.42
|15\49, 580.645
|15\49, 580.65
|K#
|K, K#
|7/5
|7/5
|9
|9
Line 646: Line 652:
|diminished 5th
|diminished 5th
|10\30, 631.58
|10\30, 631.58
|16\49, 619.355
|16\49, 619.35
|Lf
|Lf
|10/7
|10/7
Line 737: Line 743:
|augmented 11th
|augmented 11th
|28\30, 1768.42
|28\30, 1768.42
|46\49, 1780.645
|46\49, 1780.65
|F
|F
|14/5
|14/5
Line 757: Line 763:
|-
|-
|L (3g - ~tritave)
|L (3g - ~tritave)
|4\41, 184.615
|4\41, 184.62
|5\52, 181.81
|5\52, 181.81
|-
|-
Line 799: Line 805:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 2nd
|major 2nd
|4\41, 184.615
|4\41, 184.62
|H
|H
|[[9/8]], [[10/9]], [[11/10]]
|[[9/8]], [[10/9]], [[11/10]]
Line 829: Line 835:
|-
|-
|augmented 3rd
|augmented 3rd
|9\41, 415.385
|9\41, 415.38
|J#
|J#
|[[9/7]], [[14/11]], [[33/26]]
|[[9/7]], [[14/11]], [[33/26]]
Line 836: Line 842:
|diminished 4th
|diminished 4th
|10\41, 461.54
|10\41, 461.54
|Kff
|Kf, Kff
|[[21/16]], [[13/10]]
|[[21/16]], [[13/10]]
| -13
| -13
Line 842: Line 848:
|natural 4th
|natural 4th
|11\41, 507.69
|11\41, 507.69
|Kf
|K, Kf
|[[75/56]], [[4/3]]
|[[75/56]], [[4/3]]
| -2
| -2
Line 848: Line 854:
|augmented 4th
|augmented 4th
|12\41, 553.85
|12\41, 553.85
|K
|K, K#
|[[11/8]], [[18/13]]
|[[11/8]], [[18/13]]
|9
|9
Line 854: Line 860:
|doubly augmented 4th, doubly diminished 5th
|doubly augmented 4th, doubly diminished 5th
|13\41, 600.00
|13\41, 600.00
|K#, Lff
|K#, Kx, Lff
|[[7/5]], [[10/7]]
|[[7/5]], [[10/7]]
|20,-21
|20,-21
Line 877: Line 883:
|-
|-
|diminished 6th
|diminished 6th
|17\41, 784.615
|17\41, 784.62
|Aff
|Aff
|[[11/7]], [[14/9]]
|[[11/7]], [[14/9]]
Line 907: Line 913:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|22\41, 1015.385
|22\41, 1015.38
|Bf
|Bf
|[[9/5]], [[16/9]], [[20/11]]
|[[9/5]], [[16/9]], [[20/11]]
Line 955: Line 961:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 9th
|major 9th
|30\41, 1384.615
|30\41, 1384.62
|D
|D
|9/4, 20/9, 11/5
|9/4, 20/9, 11/5
Line 985: Line 991:
|-
|-
|augmented 10th
|augmented 10th
|35\41, 1615.385
|35\41, 1615.38
|E#
|E#
|18/7, 28/11, 33/13
|18/7, 28/11, 33/13
Line 1,045: Line 1,051:
|-
|-
|minor 2nd
|minor 2nd
|1\35, 54.545
|1\35, 54.55
|Hf
|Hf
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
Line 1,088: Line 1,094:
|diminished 4th
|diminished 4th
|6\35, 327.27
|6\35, 327.27
|Kff
|Kf, Kff
|[[6/5]], [[17/14]], [[11/9]]
|[[6/5]], [[17/14]], [[11/9]]
| -13
| -13
Line 1,094: Line 1,100:
|natural 4th
|natural 4th
|9\35, 490.91
|9\35, 490.91
|Kf
|K, Kf
|[[4/3]]
|[[4/3]]
| -2
| -2
|-
|-
|augmented 4th
|augmented 4th
|12\35, 654.545
|12\35, 654.55
|K
|K, K#
|[[16/11]], [[22/15]]
|[[16/11]], [[22/15]]
|9
|9
|-
|-
|diminished 5th
|diminished 5th
|10\35, 545.455
|10\35, 545.45
|Lf
|Lf
|[[15/11]], [[11/8]]
|[[15/11]], [[11/8]]
Line 1,159: Line 1,165:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 7th
|major 7th
|21\35, 1145.455
|21\35, 1145.45
|B
|B
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
Line 1,189: Line 1,195:
|-
|-
|minor 9th
|minor 9th
|23\35, 1254.545
|23\35, 1254.55
|Df
|Df
|72/35, 68/33, 33/16, 64/31
|72/35, 68/33, 33/16, 64/31
Line 1,243: Line 1,249:
|-
|-
|augmented 11th
|augmented 11th
|34\35, 1854.545
|34\35, 1854.55
|F#
|F#
|32/11, 44/15
|32/11, 44/15
Line 1,249: Line 1,255:
|-
|-
|diminished 12th
|diminished 12th
|32\35, 1745.455
|32\35, 1745.45
|Gf
|Gf
|30/11, 11/4
|30/11, 11/4
Line 1,274: Line 1,280:
|generator (g)
|generator (g)
|22\59, 713.51
|22\59, 713.51
|31\83, 715.385
|31\83, 715.38
|34\91, 715.79
|34\91, 715.79
|37\99, 716.13
|37\99, 716.13
Line 1,307: Line 1,313:
!Size in PHTE tuning
!Size in PHTE tuning
!Note name on D
!Note name on D
!Note name on H
! class="unsortable" |Approximate ratios
! class="unsortable" |Approximate ratios
!#Gens up
!#Gens up
Line 1,317: Line 1,324:
|0.00
|0.00
|D
|D
|H
|1/1
|1/1
|0
|0
Line 1,327: Line 1,335:
|230.55
|230.55
|E
|E
|J
|8/7
|8/7
| +3
| +3
Line 1,337: Line 1,346:
|461.10
|461.10
|F
|F
|K
|13/10, 9/7
|13/10, 9/7
| +6
| +6
Line 1,342: Line 1,352:
|4
|4
|15\59, 486.49
|15\59, 486.49
|21\83, 484.615
|21\83, 484.62
|23\91, 484.21
|23\91, 484.21
|25\99, 483.87
|25\99, 483.87
|482.06
|482.06
|G
|G
|L
|4/3
|4/3
| -2
| -2
Line 1,352: Line 1,363:
|5
|5
|22\59, 713.51
|22\59, 713.51
|31\83, 715.385
|31\83, 715.38
|34\91, 715.79
|34\91, 715.79
|37\99, 716.13
|37\99, 716.13
|712.61
|712.61
|H
|H
|A
|3/2
|3/2
| +1
| +1
Line 1,367: Line 1,379:
|943.16
|943.16
|J
|J
|B
|12/7, 26/15
|12/7, 26/15
| +4
| +4
Line 1,377: Line 1,390:
|964.12
|964.12
|K
|K
|C
|7/4
|7/4
| -4
| -4
Line 1,387: Line 1,401:
|1194.67
|1194.67
|L
|L
|D
|2/1
|2/1
| -1
| -1
Line 1,393: Line 1,408:
|44\59, 1427.03
|44\59, 1427.03
|62\83, 1430.77
|62\83, 1430.77
|68\91, 1430.58
|68\91, 1431.58
|74\99, 1432.26
|74\99, 1432.26
|1425.22
|1425.22
|A
|A
|E
|16/7
|16/7
| +2
| +2
Line 1,407: Line 1,423:
|1655.77
|1655.77
|B
|B
|F
|13/5, 18/7
|13/5, 18/7
| +5
| +5
Line 1,413: Line 1,430:
|52\59, 1686.49
|52\59, 1686.49
|73\83,
|73\83,
1684.615
1684.62
|80\91,
|80\91,
1684.21
1684.21
Line 1,419: Line 1,436:
|1676.32
|1676.32
|C
|C
|G
|4/3
|4/3
| -3
| -3
Line 1,426: Line 1,444:
{| class="wikitable"
{| class="wikitable"
|-
|-
| style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
| style="text-align:center;" |'''Mode'''
| style="text-align:center;" |'''Mode'''
| style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
| style="text-align:center;" |'''Name'''
| style="text-align:center;" |'''Name'''
|-
|-
| style="text-align:center;" |<nowiki>10|0</nowiki>
|LLLsLLLsLLs
|LLLsLLLsLLs
| style="text-align:center;" |<nowiki>10|0</nowiki>
|(Great) Lydian #8 (Tanagran)
|(Great) Lydian #8 (Tanagran)
|-
|-
| style="text-align:center;" |<nowiki>9|1</nowiki>
|LLLsLLsLLLs
|LLLsLLsLLLs
| style="text-align:center;" |<nowiki>9|1</nowiki>
|(Great) Lydian
|(Great) Lydian
|-
|-
| style="text-align:center;" |<nowiki>8|2</nowiki>
|LLsLLLsLLLs
|LLsLLLsLLLs
| style="text-align:center;" |<nowiki>8|2</nowiki>
|(Great) Lydian f4, Ionian #11 (Distomian)
|(Great) Lydian b4, Ionian #11 (Distomian)
|-
|-
| style="text-align:center;" |<nowiki>7|3</nowiki>
| |LLsLLLsLLsL
| |LLsLLLsLLsL
| style="text-align:center;" |<nowiki>7|3</nowiki>
| |(Great) Ionian
| |(Great) Ionian
|-
|-
| style="text-align:center;" |<nowiki>6|4</nowiki>
| |LLsLLsLLLsL
| |LLsLLsLLLsL
| style="text-align:center;" |<nowiki>6|4</nowiki>
| |(Great) Mixolydian
| |(Great) Mixolydian
|-
|-
| style="text-align:center;" |<nowiki>5|5</nowiki>
| |LsLLLsLLLsL
| |LsLLLsLLLsL
| style="text-align:center;" |<nowiki>5|5</nowiki>
| |(Great) Mixolydian f3, Dorian #10 (Livadeian)
| |(Great) Mixolydian b3, Dorian #10 (Livadeian)
|-
|-
| style="text-align:center;" |<nowiki>4|6</nowiki>
| |LsLLLsLLsLL
| |LsLLLsLLsLL
| style="text-align:center;" |<nowiki>4|6</nowiki>
| |(Great) Dorian
| |(Great) Dorian
|-
|-
| style="text-align:center;" |<nowiki>3|7</nowiki>
| |LsLLsLLLsLL
| |LsLLsLLLsLL
| style="text-align:center;" |<nowiki>3|7</nowiki>
| |(Great) Aeolian
| |(Great) Aeolian
|-
|-
| style="text-align:center;" |<nowiki>2|8</nowiki>
| |sLLLsLLLsLL
| |sLLLsLLLsLL
| style="text-align:center;" |<nowiki>2|8</nowiki>
| |(Great) Aeolian f2, Phrygian #9 (Theban)
| |(Great) Aeolian b2, Phrygian #9 (Theban)
|-
|-
| style="text-align:center;" |<nowiki>1|9</nowiki>
| |sLLLsLLsLLL
| |sLLLsLLsLLL
| style="text-align:center;" |<nowiki>1|9</nowiki>
| |(Great) Phrygian
| |(Great) Phrygian
|-
|-
| style="text-align:center;" |<nowiki>0|10</nowiki>
| |sLLsLLLsLLL
| |sLLsLLLsLLL
| style="text-align:center;" |<nowiki>0|10</nowiki>
| |(Great) Locrian
| |(Great) Locrian
|}
|}
===Cyclic Permutation order===
This temperament is named Obikhodic because the Obikhod pitch set is the Mixolydian mode with the tenth flattened or the Dorian mode with the third sharpened.
{| class="wikitable"
{| class="wikitable"
|+
|+
!Spelling
|'''Mode'''
!'''Mode'''
|[[Modal UDP Notation|'''UDP''']] 1
![[Modal UDP Notation|'''UDP''']]
|[[Modal UDP Notation|'''UDP''']] 2
!'''Name'''
|'''Name 1'''
|'''Name 2'''
|-
|LLsLLsLLsLL
|<nowiki>6|4 b10</nowiki>
|<nowiki>4|6 #3</nowiki>
|(Great) Mixolydian f10
|(Great) Dorian #3
|-
|-
|GHJKLABCDEFG
|LLsLLsLLLLs
|<nowiki>8|2 b7</nowiki>
|<nowiki>6|4 #11</nowiki>
|(Great) Lydian f4 f7, Ionian f7 #11 (Distomian Dominant)
|(Great) Mixolydian #11
|-
|LLsLLLLsLLs
|<nowiki>10|0 b4</nowiki>
|<nowiki>8|2 #8</nowiki>
|(Great) Lydian f4 #8 (Tanagran f4)
|(Great) Lydian f4 #8, Ionian #8 #11 (Distomian #8)
|-
|LLLLsLLsLLs
|<nowiki>1|9 *b1</nowiki>
|<nowiki>10|0 #5</nowiki>
|(Great) Phrygian *f1
|(Great) Lydian #5 #8 (Tanagran #5)
|-
|LsLLsLLsLLL
|<nowiki>3|7 b9</nowiki>
|<nowiki>1|9 #2</nowiki>
|(Great) Aeolian f9
|(Great) Phrygian #2
|-
|LsLLsLLLLsL
|<nowiki>5|5 b6</nowiki>
|<nowiki>3|7 #10</nowiki>
|(Great) Mixolydian f3 f6, Dorian f6 #10 (Livadeian f6)
|(Great) Aeolian #10
|-
|sLLLLsLLsLL
|<nowiki>7|3 b3</nowiki>
|<nowiki>5|5 #7</nowiki>
|(Great) Ionian f3
|(Great) Mixolydian f3 #7, Dorian #7 #10 (Livadeian #7)
|-
|LLLsLLsLLsL
|<nowiki>9|1 b11</nowiki>
|<nowiki>7|3 #4</nowiki>
|(Great) Lydian f11
|(Great) Ionian #4
|-
|sLLsLLsLLLL
|<nowiki>0|10 b8</nowiki>
|<nowiki>9|1 *#1</nowiki>
|(Great) Locrian f8
|(Great) Lydian *#1
|-
|sLLsLLLLsLL
|<nowiki>2|8 b5</nowiki>
|<nowiki>0|10 #9</nowiki>
|(Great) Aeolian f2 f5, Phrygian f5 #9 (Theban f5)
|(Great) Locrian #9
|-
|LsLLLLsLLsL
|<nowiki>4|6 b2</nowiki>
|<nowiki>2|8 #6</nowiki>
|(Great) Dorian f2
|(Great) Aeolian f2 #6, Phrygian #6 #9 (Theban #6)
|}
 
===Cyclic Permutation order===
{| class="wikitable"
|+
!Spelling 1
!Spelling 2
!'''Mode'''
![[Modal UDP Notation|'''UDP''']]
!'''Name'''
|-
|GHJKLABCDEFG
|LABCDEFGHJKL
|LLsLLLsLLLs
|LLsLLLsLLLs
| style="text-align:center;" |<nowiki>8|2</nowiki>
|<nowiki>8|2</nowiki>
|(Great) Distomian
|(Great) Distomian
|-
|-
|HJKLABCDEFGH
|HJKLABCDEFGH
|ABCDEFGHJKLA
|LsLLLsLLLsL
|LsLLLsLLLsL
| style="text-align:center;" |<nowiki>5|5</nowiki>
|<nowiki>5|5</nowiki>
|(Great) Livadeian
|(Great) Livadeian
|-
|-
|JKLABCDEFGHJ
|JKLABCDEFGHJ
|BCDEFGHJKLAB
|sLLLsLLLsLL
|sLLLsLLLsLL
| style="text-align:center;" |<nowiki>2|8</nowiki>
|<nowiki>2|8</nowiki>
|(Great) Theban
|(Great) Theban
|-
|-
|KLABCDEFGHJK
|KLABCDEFGHJK
|CDEFGHJKLABC
|LLLsLLLsLLs
|LLLsLLLsLLs
| style="text-align:center;" |<nowiki>10|0</nowiki>
|<nowiki>10|0</nowiki>
|(Great) Tanagran
|(Great) Tanagran
|-
|-
|LABCDEFGHJKL
|LABCDEFGHJKL
|DEFGHJKLABCD
|LLsLLLsLLsL
|LLsLLLsLLsL
| style="text-align:center;" |<nowiki>7|3</nowiki>
|<nowiki>7|3</nowiki>
|(Great) Ionian
|(Great) Ionian
|-
|-
|ABCDEFGHJKLA
|ABCDEFGHJKLA
|EFGHJKLABCDE
|LsLLLsLLsLL
|LsLLLsLLsLL
| style="text-align:center;" |<nowiki>4|6</nowiki>
|<nowiki>4|6</nowiki>
|(Great) Dorian
|(Great) Dorian
|-
|-
|BCDEFGHJKLAB
|BCDEFGHJKLAB
|FGHJKLABCDEF
|sLLLsLLsLLL
|sLLLsLLsLLL
| style="text-align:center;" |<nowiki>1|9</nowiki>
|<nowiki>1|9</nowiki>
|(Great) Phrygian
|(Great) Phrygian
|-
|-
|CDEFGHJKLABC
|CDEFGHJKLABC
|GHJKLABCDEFG
|LLLsLLsLLLs
|LLLsLLsLLLs
| style="text-align:center;" |<nowiki>9|1</nowiki>
|<nowiki>9|1</nowiki>
|(Great) Lydian
|(Great) Lydian
|-
|-
|DEFGHJKLABCD
|DEFGHJKLABCD
|HJKLABCDEFGH
|LLsLLsLLLsL
|LLsLLsLLLsL
| style="text-align:center;" |<nowiki>6|4</nowiki>
|<nowiki>6|4</nowiki>
|(Great) Mixolydian
|(Great) Mixolydian
|-
|-
|EFGHJKLABCDE
|EFGHJKLABCDE
|JKLABCDEFGHJ
|LsLLsLLLsLL
|LsLLsLLLsLL
| style="text-align:center;" |<nowiki>3|7</nowiki>
|<nowiki>3|7</nowiki>
|(Great) Aeolian
|(Great) Aeolian
|-
|-
|FGHJKLABCDEF
|FGHJKLABCDEF
|KLABCDEFGHJK
|sLLsLLLsLLL
|sLLsLLLsLLL
| style="text-align:center;" |<nowiki>0|10</nowiki>
|<nowiki>0|10</nowiki>
|(Great) Locrian
|(Great) Locrian
|}
|}
===Notes on Naming===
The modes of the Obikhodic scale are named after the existing modes, but contain the "Great" prefix (e.g. Great Ionian, Great Aeolian, etc.). The "Great" prefixes can be left in to explicitly distinguish which MOS's modes you're talking about, or can be omitted for convention.
Each Obikhodic mode contains its corresponding mode in the diatonic scale. This leads to a pattern: LLsLLLsLLLs and LLsLLLsLLsL both contain the meantone LLsLLLs Ionian mode. Additionally, sLLsLLLsLLL contains the diatonic sLLsLLL Locrian mode.
Since there are only seven diatonic modes, four of the superdiatonic modes need additional names and cannot reference any mode of the diatonic scale. These four modes present themselves as "altered" modes, which have an accidental the mode below them lacks, or vice versa. These are the only four modes to exhibit this behavior. They're interspersed on the ranking above and below Lydian, between Dorian and Mixolydian and between Aeolian and Phrygian and on the rotational continuum between Locrian and Ionian.
As were the original modes named after regions of ancient Greece, so are these new Obikhodic extensions. They are called after regions of Boeotia, set up so that the Locrian -> Distomian -> Livadeian -&gt; Theban -&gt; Tanagran -&gt; Ionian cyclic sequence will resemble the geography of ancient Greece.
==Scale tree==
{| class="wikitable"
{| class="wikitable"
|+
|+
! colspan="2" |Generator
!Spelling 1
!Normalized
!Spelling 2
!'''Mode'''
![[Modal UDP Notation|'''UDP''']]
!'''Name'''
|-
|-
|3\8
|GHJKLABfCDEFG
|
|LABCDEFfGHJKL
|<u>720</u>
|LLsLLsLLLLs
| style="text-align:center;" |<nowiki>8|2 b7</nowiki>
|(Great) Distomian Dominant
|-
|-
|19\51
|HJKLABfCDEFGH
|
|ABCDEFfGHJKLA
|<u>712.5</u>
|LsLLLLsLLsL
| style="text-align:center;" |<nowiki>5|5 b6</nowiki>
|(Great) Livadeian f6
|-
|-
|
|JKLABfCDEFGHJ
|35\94
|BCDEFfGHJKLAB
|<u>711.864</u>
|sLLsLLLLsLL
| style="text-align:center;" |<nowiki>2|8 b5</nowiki>
|(Great) Theban f5
|-
|-
|16\43
|KLABfCDEFGHJK
|
|CDEFfGHJKLABC
|<u>711.111</u>
|LLsLLLLsLLs
| style="text-align:center;" |<nowiki>10|0 b4</nowiki>
|(Great) Tanagran f4
|-
|-
|
|LABfCDEFGHJKL
|29\78
|DEFfGHJKLABCD
|<u>710.204</u>
|LsLLLLsLLsL
| style="text-align:center;" |<nowiki>7|3 b3</nowiki>
|(Great) Ionian f3
|-
|-
|13\35
|ABfCDEFGHJKLA
|
|EFfGHJKLABCDE
|<u>709.091</u>
|sLLLLsLLsLL
| style="text-align:center;" |<nowiki>4|6 b2</nowiki>
|(Great) Dorian f2
|-
|-
|
|BfCDEFGHJKLABf
|36\97
|FfGHJKLABCDEFf
|<u>708.197</u>
|LsLLsLLsLLL
| style="text-align:center;" |<nowiki>1|9 *b1</nowiki>
|(Great) Phrygian *f1
|-
|-
|
|CDEFGHJKLABfC
|23\62
|GHJKLABCDEFfG
|<u>707.692</u>
|LLLsLLsLLsL
| style="text-align:center;" |<nowiki>9|1 b11</nowiki>
|(Great) Lydian f11
|-
|-
|
|DEFGHJKLABfCD
|33\89
|HJKLABCDEFfGH
|<u>707.143</u>
|LLsLLsLLsLL
| style="text-align:center;" |<nowiki>6|4 b10</nowiki>
|(Great) Mixolydian f10
|-
|-
|
|EFGHJKLABfCDE
|43\116
|JKLABCDEFfGHJ
|<u>706.849</u>
|LsLLsLLsLLL
| style="text-align:center;" |<nowiki>3|7 b9</nowiki>
|(Great) Aeolian f9
|-
|-
|10\27
|FGHJKLABfCDEF
|
|KLABCDEFfGHJK
|<u>705.882</u>
|sLLsLLsLLLL
| style="text-align:center;" |<nowiki>0|10 b8</nowiki>
|(Great) Locrian f8
|}
{| class="wikitable"
!Spelling 1
!Spelling 2
!'''Mode'''
![[Modal UDP Notation|'''UDP''']]
!'''Name'''
|-
|-
|
|GHJKLABC#DEFG
|27\73
|LABCDEFG#HJKL
|<u>704.348</u>
|LLsLLsLLLLs
|<nowiki>8|2 #8</nowiki>
|(Great) Distomian #8
|-
|-
|
|HJKLABC#DEFGH
|17\46
|ABCDEFG#HJKLA
|<u>703.448</u>
|LsLLLLsLLsL
|<nowiki>5|5 #7</nowiki>
|(Great) Livadeian #7
|-
|-
|
|JKLABC#DEFGHJ
|24\65
|BCDEFG#HJKLAB
|<u>702.439</u>
|sLLLLsLLsLL
|<nowiki>2|8 #6</nowiki>
|(Great) Theban #6
|-
|-
|
|KLABC#DEFGHJK
|31\84
|CDEFG#HJKLABC
|<u>701.887</u>
|LLLLsLLsLLs
|<nowiki>10|0 #5</nowiki>
|(Great) Tanagran #5
|-
|-
|7\19
|LABC#DEFGHJKL
|
|DEFG#HJKLABCD
|<u>700</u>
|LLLsLLsLLsL
|<nowiki>7|3 #4</nowiki>
|(Great) Ionian #4
|-
|-
|
|ABC#DEFGHJKLA
|60\163
|EFG#HJKLABCDE
|<u>699.029</u>
|LLsLLsLLsLL
|<nowiki>4|6 #3</nowiki>
|(Great) Dorian #3
|-
|-
|
|BC#DEFGHJKLAB
|53\144
|FG#HJKLABCDEF
|<u>698.901</u>
|LsLLsLLsLLL
|<nowiki>1|9 #2</nowiki>
|(Great) Phrygian #2
|-
|-
|
|C#DEFGHJKLABC#
|46\125
|G#HJKLABCDEFG#
|<u>698.734</u>
|sLLsLLsLLLL
|<nowiki>9|1 *#1</nowiki>
|(Great) Lydian *#1
|-
|-
|
|DEFGHJKLABC#D
|39\106
|HJKLABCDEFG#H
|<u>698.5075</u>
|LLsLLsLLLLs
|<nowiki>6|4 #11</nowiki>
|(Great) Mixolydian #11
|-
|-
|
|EFGHJKLABC#DE
|32\87
|JKLABCDEFG#HJ
|<u>698.182</u>
|LsLLsLLLLsL
|<nowiki>3|7 #10</nowiki>
|(Great) Aeolian #10
|-
|-
|
|FGHJKLABC#DEF
|25\68
|KLABCDEFG#HJK
|<u>697.674</u>
|sLLsLLLLsLL
|<nowiki>0|10 #9</nowiki>
|(Great) Locrian #9
|}
===Notes on Naming===
The modes of the Obikhodic scale are named after the existing modes, but contain the "Great" prefix (e.g. Great Ionian, Great Aeolian, etc.). The "Great" prefixes can be left in to explicitly distinguish which MOS's modes you're talking about, or can be omitted for convention.
 
Each Obikhodic mode contains its corresponding mode in the diatonic scale. This leads to a pattern: LLsLLLsLLLs and LLsLLLsLLsL both contain the meantone LLsLLLs Ionian mode. Additionally, sLLsLLLsLLL contains the diatonic sLLsLLL Locrian mode.
 
Since there are only seven diatonic modes, four of the superdiatonic modes need additional names and cannot reference any mode of the diatonic scale. These four modes present themselves as "altered" modes, which have an accidental the mode below them lacks, or vice versa. These are the only four modes to exhibit this behavior. They're interspersed on the ranking above and below Lydian, between Dorian and Mixolydian and between Aeolian and Phrygian and on the rotational continuum between Locrian and Ionian.
 
As were the original modes named after regions of ancient Greece, so are these new Obikhodic extensions. They are called after regions of Boeotia, set up so that the Locrian -> Distomian -> Livadeian -&gt; Theban -&gt; Tanagran -&gt; Ionian cyclic sequence will resemble the geography of ancient Greece.
==Scale tree==
{| class="wikitable"
|+
!Generator
!Normalized
!Large step
!Small step
|-
|-
|
|3\8
|18\49
|<u>720.000</u>
|<u>696.77</u>
|1\5, <u>240.000</u>
|0
|-
|-
|
|19\51
|29\79
|<u>712.500</u>
|<u>696</u>
|6\51, <u>225.000</u>
|1\51, <u>37.500</u>
|-
|-
|
|54\145
|40\109
|<u>712.088</u>
|<u>695.652</u>
|17\145, <u>224.175</u>
|3\145, <u>39.560</u>
|-
|-
|11\30
|35\94
|
|<u>711.864</u>
|<u>694.737</u>
|11\94, <u>223.729</u>
|2\94, <u>40.678</u>
|-
|-
|
|16\43
|26\71
|<u>711.111</u>
|<u>693.333</u>
|5\43, <u>222.222</u>
|1\43, <u>44.444</u>
|-
|-
|15\41
|45\121
|
|<u>710.526</u>
|<u>692.308</u>
|14\121, <u>221.053</u>
|3\121, <u>47.368</u>
|-
|-
|
|29\78
|34\93
|<u>710.204</u>
|<u>691.525</u>
|9\78, <u>220.408</u>
|2\78, <u>48.980</u>
|-
|-
|19\52
|42\113
|
|<u>709.859</u>
|<u>690.909</u>
|13\113, <u>219.718</u>
|3\113, <u>50.704</u>
|-
|-
|
|13\35
|42\115
|<u>709.091</u>
|<u>690.411</u>
|4\35, <u>218.182</u>
|1\35, <u>54.545</u>
|-
|-
|23\63
|49\132
|
|<u>708.434</u>
|<u>690</u>
|15\132, <u>216.867</u>
|4\132, <u>57.831</u>
|-
|-
|4\11
|36\97
|
|<u>708.197</u>
|<u>685.714</u>
|11\97, <u>216.393</u>
|}
|3\97, <u>59.016</u>
 
|-
|23\62
|<u>707.692</u>
|7\62, <u>215.385</u>
|2\62, <u>61.538</u>
|-
|33\89
|<u>707.143</u>
|10\89, <u>214.286</u>
|3\89, <u>64.286</u>
|-
|43\116
|<u>706.849</u>
|13\116, <u>213.699</u>
|4\116, <u>65.753</u>
|-
|53\143
|<u>706.667</u>
|16\143, <u>213.333</u>
|5\143, <u>66.667</u>
|-
|63\170
|<u>706.542</u>
|19\170, <u>213.084</u>
|6\170, <u>67.290</u>
|-
|73\197
|<u>706.452</u>
|22\197, <u>212.903</u>
|7\197, <u>67.742</u>
|-
|10\27
|<u>705.882</u>
|3\27, <u>211.765</u>
|1\27, <u>70.588</u>
|-
|47\127
|<u>705.000</u>
|14\127, <u>210.000</u>
|5\127, <u>75.000</u>
|-
|37\100
|<u>704.762</u>
|11\100, <u>209.524</u>
|4\100, <u>76.190</u>
|-
|27\73
|<u>704.348</u>
|8\73, <u>208.696</u>
|3\27, <u>78.261</u>
|-
|17\46
|<u>703.448</u>
|5\46, <u>206.897</u>
|2\46, <u>82.759</u>
|-
|41\111
|<u>702.857</u>
|12\111, <u>205.714</u>
|5\111, <u>85.714</u>
|-
|24\65
|<u>702.439</u>
|7\65, <u>204.878</u>
|3\65, <u>87.805</u>
|-
|31\84
|<u>701.887</u>
|9\84, <u>203.774</u>
|4\84, <u>90.566</u>
|-
|…
|…
|…
|…
|-
|59\160
|<u>700.990</u>
|17\160, <u>201.980</u>
|8\160, <u>95.050</u>
|-
|…
|…
|…
|…
|-
|122\331
|<u>700.478</u>
|35\331, <u>200.960</u>
|18\334, <u>97.608</u>
|-
|7\19
|<u>700.000</u>
|2\19, <u>200.000</u>
|1\19, <u>100.000</u>
|-
|123\334
|<u>699.526</u>
|35\334, <u>199.052</u>
|18\334, <u>102.370</u>
|-
|…
|…
|…
|…
|-
|63\163
|<u>699.029</u>
|17\163, <u>198.058</u>
|9\163, <u>104.854</u>
|-
|…
|…
|…
|…
|-
|32\87
|<u>698.182</u>
|9\87, <u>196.364</u>
|5\87, <u>109.091</u>
|-
|25\68
|<u>697.674</u>
|7\68, <u>195.349</u>
|4\68, <u>111.628</u>
|-
|43\117
|<u>697.297</u>
|12\117, <u>194.594</u>
|7\117, <u>113.514</u>
|-
|18\49
|<u>696.774</u>
|5\49, <u>193.548</u>
|3\49, <u>116.129</u>
|-
|29\79
|<u>696.000</u>
|8\79, <u>192.000</u>
|5\79, <u>120.000</u>
|-
|40\109
|<u>695.652</u>
|11\109, <u>191.304</u>
|7\109, <u>121.739</u>
|-
|51\139
|<u>695.455</u>
|14\139, <u>190.909</u>
|9\139, <u>122.727</u>
|-
|11\30
|<u>694.737</u>
|3\30, <u>189.474</u>
|2\30, <u>126.316</u>
|-
|59\161
|<u>694.118</u>
|16\161, <u>188.235</u>
|11\161, <u>129.412</u>
|-
|48\131
|<u>693.976</u>
|13\131, <u>187.952</u>
|9\131, <u>130.120</u>
|-
|37\101
|<u>693.750</u>
|10\101, <u>187.500</u>
|7\101, <u>131.250</u>
|-
|26\71
|<u>693.333</u>
|7\71, <u>186.667</u>
|5\71, <u>133.333</u>
|-
|41\112
|<u>692.958</u>
|11\112, <u>185.915</u>
|8\112, <u>135.211</u>
|-
|15\41
|<u>692.308</u>
|4\41, <u>184.615</u>
|3\41, <u>138.462</u>
|-
|34\93
|<u>691.525</u>
|9\93, <u>183.051</u>
|7\93, <u>142.373</u>
|-
|53\145
|<u>691.304</u>
|14\145, <u>182.609</u>
|11\145, <u>143.478</u>
|-
|19\52
|<u>690.909</u>
|5\52, <u>181.818</u>
|4\52, <u>145.455</u>
|-
|23\63
|<u>690.000</u>
|6\63, <u>180.000</u>
|5\63, <u>150.000</u>
|-
|4\11
|<u>685.714</u>
|1\11, <u>171.429</u>
|1\11, <u>171.429</u>
|}
 
== See also ==
== See also ==
[[8L 3s (3/1-equivalent)]]
[[8L 3s (3/1-equivalent)]]
[[16L 6s (80/9-equivalent)]]
[[16L 6s (352/39-equivalent)]]
[[16L 6s (64/7-equivalent)]]
Retrieved from "https://en.xen.wiki/w/User:Moremajorthanmajor/8L_3s_(perfect_twelfth-equivalent)"