8L 3s (3/1-equivalent): Difference between revisions

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{{Todo|rework|inline=1|comment=Rewrite according to [[Xenharmonic Wiki: Conventions]] and [[Xenharmonic Wiki: Article guidelines]]}}
{{Infobox MOS|Scale Signature=8L 3s<3/1>}}
{{Todo|cleanup|inline=1|comment=Russian obikhod (LLsLLs(LLs)<sup>+</sup>LL) and 8L 3s <3/1> (LLLsLLLsLLs) are unrelated step patterns, so references to the former should be (re)moved.}}
{{MOS intro|Scale Signature=8L 3s<3/1>}}
{{Infobox MOS
{{mos scalesig|8L 3s<3/1>}} scale pattern includes the well-known {{mos scalesig|5L 2s|link=1}} pattern within it.
| Name =
| Equave = 3/1
| nLargeSteps = 8
| nSmallSteps = 3
| Equalized = 4
| Collapsed = 3
| Pattern = LLLsLLLsLLs
}}  
{{MOS intro}}
==Standing assumptions==
The tempered generalized Livenka accordion is used in this article to refer to tunings of the pitch set.


The [[TAMNAMS]] system is used in this article to refer to {{PAGENAME}} step size ratios and step ratio ranges.
== Scale properties ==
{{TAMNAMS use}}


The notation used in this article is GHJKLABCDEFG = LLsLLLsLLLs (Ionian #11), #/f = up/down by chroma (mnemonic f = F molle in Latin).
=== Intervals ===
{{MOS intervals|Scale Signature=8L 3s<3/1>}}


Thus the [[19edt]] gamut is as follows:
=== Generator chain ===
{{MOS genchain|Scale Signature=8L 3s<3/1>}}


'''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'''
=== Modes ===
{{MOS mode degrees|Scale Signature=8L 3s<3/1>}}


The [[27edt]] gamut is notated as follows:
== Theory ==
By dividing the {{mos scalesig|5L 2s}} of LLsLLLs into A=LLs and B=LLLs, and combining them as ABBABB..., it becomes {{mos scalesig|8L 3s<3/1>}}. This scale has octaves that are too frequent for the listener to feel a tritave equivalence. The range of possible dark generators will likely feel sufficiently pseudo-octave. Similar to [[Angel]], it would be good to utilize a finite-length chain of octaves and make use of existing diatonic music theory.


'''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'''
=== Low harmonic entropy scales ===
* Pythagorean tuning (period = 3/1, generator = 3/2): L/s = 2.260
* Tritave-equivalent meantone tunings:
** 1/6-comma 3eantone{{idiosyncratic}} tuning ([[unchanged-interval]]s: {3/1, 5/4}): L/s = 1.625


The [[30edt]] gamut:
== Tuning ranges ==
=== Simple tunings ===
{{MOS tunings|Scale Signature=8L 3s<3/1>|Step Ratios=2/1; 3/1; 3/2}}


'''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
=== Soft-of-basic tunings ===
==Intervals==
{{MOS tunings|Scale Signature=8L 3s<3/1>|Step Ratios=5/4; 4/3; 3/2; 5/3}}
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¢.
{| class="wikitable"
|+
!# generators up
!Notation (1/1 = G)
!name
!In L's and s's
!# generators up
!Notation of ~3/1 inverse
!name
!In L's and s's
|-
| colspan="8" style="text-align:center" |The 11-note MOS has the following intervals (from some root):
|-
|0
|G
| perfect unison
|0
|0
|G
|perfect 12th
|8L+3s
|-
|1
|L
|perfect 5th
|3L+1s
| -1
|C
|perfect octave
|5L+2s
|-
|2
|D
|major 9th
|6L+2s
| -2
|K
|natural 4th
|2L+1s
|-
|3
|H
| major 2nd
|1L
|  -3
|Ff
|natural 11th
|7L+3s
|-
|4
|A
|major 6th
|4L+1s
| -4
|Bf
|minor 7th
|4L+2s
|-
|5
|E
|major 10th
|7L+2s
| -5
|Jf
|minor 3rd
|1L+1s
|-
|6
|J
|major 3rd
|2L
| -6
|Ef
|minor 10th
|6L+3s
|-
|7
|B
| major 7th
|5L+1s
|  -7
|Af
|minor 6th
|3L+2s
|-
|8
|F
|augmented 11th
|8L+2s
| -8
|Hf
|minor 2nd
|1s
|-
|9
|K#
| augmented 4th
|3L
| -9
|Df
|minor 2nd
|5L+3s
|-
|10
|C#
|augmented octave
|6L+1s
| -10
|Lf
|diminished 5th
|2L+2s
|-
|11
|G#
|augmented unison
|1L-1s
| -11
|Gf
|diminished unison
|7L+4s
|-
| colspan="8" style="text-align:center" |The chromatic 19-note MOS (either [[8L 11s (tritave equivalent)|8L 11s]], [[11L 8s (tritave equivalent)|11L 8s]], or [[19edt]]) also has the following intervals (from some root):
|-
|12
|L#
|augmented 5th
|4L
| -12
|Cf
|diminished octave
|4L+3s
|-
|13
|D#
|augmented 9th
|7L+1s
| -13
|Kf
|diminished 4th
|1L+2s
|-
|14
|H#
|augmented 2nd
|2L-1s
| -14
|Fff
|diminished 11th
| 6L+4s
|-
|15
|A#
|augmented 6th
|5L
| -15
|Bff
|diminished 7th
|3L+3s
|-
|16
|E#
|augmented 10th
|8L+1s
| -16
|Jff
|diminished 3rd
|2s
|-
|17
|J#
|augmented 3rd
|3L-1s
|  -17
|Eff
|diminished 10th
|5L+4s
|-
|18
| B#
|augmented 7th
|6L
| -18
|Aff
|diminished 6th
|2L+3s
|}
==Tuning ranges==
===Simple tunings ===
Table of intervals in the simplest Obikhodic tunings:
{| class="wikitable right-2 right-3 right-4 sortable"
|-
! class="unsortable" |Degree
!Size in [[19edt|~19edt]] (basic)
!Size in [[27edt|~27edt]] (hard)
!Size in ~[[30edt]] (soft)
! class="unsortable" |Note name on G
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\19, 0.00
|0\27, 0.00
|0\30, 0.00
|G
|0
|-
|minor 2nd
|1\19, 100.00
|1\27, 70.59 (70.37)
|2\30, 126.32 (126.67)
|Hf
| -8
|- bgcolor="#eaeaff"
|major 2nd
|2\19, 200.00
|3\27, 211.765 (211.11)
| 3\30, 189.47 (190.00)
|H
|3
|-
|minor 3rd
|3\19, 300.00
|4\27, 282.35 (281.48)
|5\30, 315.79 (316.67)
|Jf
| -5
|- bgcolor="#eaeaff"
|major 3rd
|4\19, 400.00
| 6\27, 423.53 (422.22)
|6\30, 378.95 (380.00)
|J
|6
|- bgcolor="#eaeaff"
|natural 4th
|5\19, 500.00
|7\27, 494.12 (493.59)
|8\30, 505,26 (506.67)
|K
| -2
|-
|augmented 4th
| rowspan="2" |6\19, 600.00
|9\27, 635.29 (633.33)
|9\30, 568.42 (570.00)
|K#
|9
|-
|diminished 5th
|8\27, 564.71 (562.96)
| 10\30, 631.58 (633.33)
|Lf
| -10
|- bgcolor="#eaeaff"
|perfect 5th
|7\19, 700.00
|10\27, 705.88 (703.70)
|11\30, 694.74 (696.67)
| L
|1
|-
|minor 6th
|8\19, 800.00
|11\27, 776.47 (774.07)
|13\30, 821.05 (823.33)
|Af
| -7
|- bgcolor="#eaeaff"
|major 6th
|9\19, 900.00
|13\27, 917.65 (914.81)
|14\30, 884.21 (886.67)
|A
|4
|- bgcolor="#eaeaff"
|minor 7th
|10\19, 1000.00
|14\27, 988.235 (985.19)
|16\30, 1010.53 (1013.33)
|Bf
| -4
|-
|major 7th
|11\19, 1100.00
|16\27, 1129.42 (1125.93)
|17\30, 1073.68 (1076.67)
|B
|7
|- bgcolor="#eaeaff"
|perfect octave
|12\19, 1200.00
|17\27, 1200.00 (1196.30)
|19\30, 1200.00 (1203.33)
|C
| -1
|-
|augmented octave
| rowspan="2" |13\19, 1300.00
|19\27, 1341.18 (1337.04)
|20\30, 1263.16 (1266.67)
|C#
|10
|-
|minor 9th
|18\27, 1270.59 (1266.67)
|21\30, 1326.32 (1330.00)
|Df
| -9
|- bgcolor="#eaeaff"
|major 9th
|14\19, 1400.00
|20\27, 1411.765 (1406.07)
|22\30, 1389.47 (1393.33)
|D
|2
|-
| minor 10th
|15\19, 1500.00
|21\27, 1482.35 (1477.78)
|24\30, 1515.79 (1520.00)
|Ef
|  -6
|- bgcolor="#eaeaff"
|major 10th
|16\19, 1600.00
|23\27, 1623.53 (1618.52)
|25\30, 1578.95 (1583.33)
|E
|5
|-
|natural 11th
|17\19, 1700.00
|24\27, 1694.12 (1688.89)
|27\30, 1705.26 (1710.00)
|Ff
| -3
|- bgcolor="#eaeaff"
|augmented 11th
|18\19, 1800.00
|26\27, 1835.29 (1829.63)
|28\30, 1768.42 (1773.33)
|F
|8
|}
===Hypohard ===
[[Hypohard]] Obikhodic tunings (with generator between 7\19 and 10\27) have step ratios between 2/1 and 3/1.


Hypohard Obikhodic can be considered "superpythagorean Obikhodic". This is because these tunings share the following features with [[Superpyth|superpythagorean]] diatonic tunings:
=== Hard-of-basic tunings ===
*The large step is near the Pythagorean 9/8 whole tone, somewhere between as in [[12edo]] and as in [[17edo]].
{{MOS tunings|Scale Signature=8L 3s<3/1>|Step Ratios=5/2; 3/1; 4/1; 5/1}}
*The major 3rd (made of two large steps) is a near-[[Pythagorean]] to [[Neogothic]] major third.
~EDTs that are in the hypohard range include ~[[19edt]], ~[[27edt]], and ~[[46edt]].
 
The sizes of the generator, large step and small step of Obikhodic are as follows in various hypohard Obikhod tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
!~[[19edt]] (basic)
!~[[27edt]] (hard)
!~[[46edt]] (semihard)
|-
|generator (g)
|7\19, 700.00
|10\27, 705.88 (703.70)
|17\46, 703.45 (702.17)
|-
|L (3g - ~tritave)
|2\19, 200.00
|3\27, 211.765 (211.11)
| 5\46, 206.90 (206.52)
|-
|s (-8g + 3 ~tritaves)
|1\19, 100.00
|1\27, 70.59 (70.37)
|2\46, 82.76 (82.61)
|}
====Intervals====
Sortable table of major and minor intervals in hypohard Obikhod tunings:
{| class="wikitable"
|+
!Degree
!Size in [[19edt|~19edt]] (basic)
!Size in [[27edt|~27edt]] (hard)
!Size in ~[[46edt]] (semihard)
! Note name on G
!#Gens up
|- bgcolor="#eaeaff"
|unison
| 0\19, 0.00
|0\27, 0.00
|0\46, 0.00
|G
|0
|-
|minor 2nd
|1\19, 100.00
|1\27, 70.59 (70.37)
|2\46, 82.76 (82.61)
|Hf
| -8
|- bgcolor="#eaeaff"
| major 2nd
|2\19, 200.00
|3\27, 211.765 (211.11)
|5\46, 206.90 (206.52)
|H
| 3
|-
|minor 3rd
|3\19, 300.00
|4\27, 282.35 (281.48)
|7\46, 289.655 (289.13)
| Jf
| -5
|- bgcolor="#eaeaff"
|major 3rd
|4\19, 400.00
|6\27, 423.53 (422.22)
|10\46, 413.79 (413.04)
| J
|6
|- bgcolor="#eaeaff"
|natural 4th
|5\19, 500.00
|7\27, 494.12 (493.59)
| 12\46, 496.55 (495.65)
|K
| -2
|-
|augmented 4th
| rowspan="2" |6\19, 600.00
|9\27, 635.29 (633.33)
|15\46, 620.69 (619.565)
|K#
|9
|-
|diminished 5th
|8\27, 564.71 (562.96)
|14\46, 579.31 (578.26)
|Lf
| -10
|- bgcolor="#eaeaff"
|perfect 5th
|7\19, 700.00
|10\27, 705.88 (703.70)
| 17\46, 703.45 (702.17)
|L
|1
|-
|minor 6th
|8\19, 800.00
|11\27, 776.47 (774.07)
|19\46, 786.21 (784.78)
|Af
| -7
|- bgcolor="#eaeaff"
|major 6th
|9\19, 900.00
|13\27, 917.65 (914.81)
| 22\46, 910.345 (908.70)
|A
|4
|- bgcolor="#eaeaff"
|minor 7th
|10\19, 1000.00
|14\27, 988.235 (985.19)
|24\46, 993.10 (991.30)
|Bf
| -4
|-
|major 7th
|11\19, 1100.00
|16\27, 1129.42 (1125.93)
|27\46, 1117.24 (1115.22)
|B
|7
|- bgcolor="#eaeaff"
|perfect octave
| 12\19, 1200.00
|17\27, 1200.00 (1196.30)
| 29\46, 1200.00 (1197.83)
|C
| -1
|-
|augmented octave
| rowspan="2" |13\19, 1300.00
|19\27, 1341.18 (1337.04)
|32\46, 1324.14 (1321.74)
|C#
|10
|-
|minor 9th
| 18\27, 1270.59 (1266.67)
|31\46, 1282.76 (1280.435)
|Df
| -9
|- bgcolor="#eaeaff"
|major 9th
|14\19, 1400.00
|20\27, 1411.765 (1406.07)
|34\46, 1406.90 (1404.35)
|D
|2
|-
|minor 10th
|15\19, 1500.00
|21\27, 1482.35 (1477.78)
|36\46, 1489.655 (1486.96)
|Ef
| -6
|- bgcolor="#eaeaff"
|major 10th
|16\19, 1600.00
|23\27, 1623.53 (1618.52)
|39\46, 1613.79 (1610.87)
|E
|5
|- bgcolor="#eaeaff"
|natural 11th
|17\19, 1700.00
|24\27, 1694.12 (1688.89)
|41\46, 1696.55 (1693.48)
|Ff
| -3
|-
|augmented 11th
|18\19, 1800.00
|26\27, 1835.29 (1829.63)
|44\46, 1820.69 (1817.39)
|F
|8
|}
===Hyposoft===
[[Hyposoft]] Obikhodic tunings (with generator between 11\30 and 7\19) have step ratios between 3/2 and 2/1. The 11\30-to-7\19 range of Obikhodic tunings can be considered "meantone Obikhodic". This is because these tunings share the following features with meantone diatonic tunings:
*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 sizes of the generator, large step and small step of oneirotonic are as follows in various hyposoft Obikhod tunings (~19edt not shown).
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
!~[[30edt]] (soft)
!~[[49edt]] (semisoft)
|-
|generator (g)
|11\30, 694.74 (696.67)
|18\49, 696.77 (697.96)
|-
|L (3g - ~tritave)
|3\30, 189.47 (190.00)
|5\49, 193.55 (193.87)
|-
|s (-8g + 3 ~tritaves)
|2\30, 126.32 (126.67)
|3\49, 116.13 (116.33)
|}
==== Intervals ====
Sortable table of major and minor intervals in hyposoft Obikhod tunings (~19edt not shown):
{| class="wikitable"
|+
! Degree
!Size in ~[[30edt]] (soft)
!~[[49edt]] (semisoft)
!Note name on G
!Approximate ratios
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\30, 0.00
|0\49, 0.00
|G
|1/1
|0
|-
|minor 2nd
|2\30, 126.32 (126.67)
|3\49, 116.13 (116.33)
|Hf
|16/15
| -8
|- bgcolor="#eaeaff"
|major 2nd
|3\30, 189.47 (190.00)
|5\49, 193.55 (193.87)
|H
|10/9, 9/8
|3
|-
| minor 3rd
|5\30, 315.79 (316.67)
|8\49, 309.68 (310.20)
|Jf
|6/5
| -5
|- bgcolor="#eaeaff"
|major 3rd
|6\30, 378.95 (380.00)
|10\49, 387.10 (387.755)
|J
|5/4
|6
|- bgcolor="#eaeaff"
| natural 4th
|8\30, 505,26 (506.67)
|13\49, 503.23 (504.08)
|K
|4/3
| -2
|-
|augmented 4th
|9\30, 568.42 (570.00)
|15\49, 580.645 (581.63)
|K#
|7/5
|9
|-
|diminished 5th
|10\30, 631.58 (633.33)
|16\49, 619.355 (620.41)
|Lf
|10/7
| -10
|- bgcolor="#eaeaff"
|perfect 5th
|11\30, 694.74 (696.67)
|18\49, 696.77 (697.96)
|L
|3/2
|1
|-
|minor 6th
|13\30, 821.05 (823.33)
|21\49, 812.90 (814.29)
|Af
|8/5
| -7
|- bgcolor="#eaeaff"
|major 6th
|14\30, 884.21 (886.67)
|23\49, 890.32 (891.84)
|A
|5/3
|4
|- bgcolor="#eaeaff"
|minor 7th
|16\30, 1010.53 (1013.33)
|26\49, 1006.45 (1008.16)
|Bf
|16/9, 9/5
| -4
|-
|major 7th
|17\30, 1073.68 (1076.67)
|28\49, 1083.87 (1085.71)
|B
|15/8
|7
|- bgcolor="#eaeaff"
|perfect octave
|19\30, 1200.00 (1203.33)
|31\49, 1200.00 (1202.04)
|C
|2/1
| -1
|-
|augmented octave
|20\30, 1263.16 (1266.67)
|33\49, 1277.42 (1279.59)
|C#
|25/24
|10
|-
|minor 9th
| 21\30, 1326.32 (1330.00)
|34\49, 1316.13 (1318.37)
|Df
|15/7
|  -9
|- bgcolor="#eaeaff"
|major 9th
| 22\30, 1389.47 (1393.33)
|36\49, 1393.55 (1395.92)
|D
|20/9, 9/4
|2
|-
| minor 10th
|24\30, 1515.79 (1520.00)
|39\49, 1508.68 (1512.245)
|Ef
|12/5
| -6
|- bgcolor="#eaeaff"
|major 10th
|25\30, 1578.95 (1583.33)
|41\49, 1587.10 (1590.80)
| E
|5/2
|5
|- bgcolor="#eaeaff"
|natural 11th
|27\30, 1705.26 (1710.00)
|44\49, 1703.23 (1706.13j
|Ff
|8/3
| -3
|-
|augmented 11th
|28\30, 1768.42 (1773.33)
|46\49, 1780.645 (1783.67)
| F
|14/5
|8
|}
 
===Parasoft to ultrasoft tunings===
The range of Obikhodic tunings of step ratio between 6/5 and 3/2 (thus in the [[parasoft]] to [[ultrasoft]] range) may be of interest because it is closely related to [[Meantone family|flattone]] temperament.
 
The sizes of the generator, large step and small step of Obikhodic are as follows in various tunings in this range.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
!~[[41edt]] (supersoft)
!~[[52edt]]
|-
|generator (g)
|15\41, 692.31 (695.12)
|19\52, 690.91 (694.23)
|-
|L (3g - ~tritave)
|4\41, 184.615 (185.37)
|5\52, 181.81 (182.69)
|-
|s (-8g + 3 ~tritaves)
|3\41, 138.46 (139.02)
|4\52, 145.455 (146.15)
|}
====Intervals====
The intervals of the extended generator chain (-21 to +21 generators) are as follows in various softer-than-soft Obikhodic tunings.
{| class="wikitable"
|+
!Degree
!Size in ~[[41edt]] (supersoft)
!Note name on G
!Approximate ratios
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\41, 0.00
|G
|1/1
|0
|-
|chroma
|1\41, 46.15 (46.34)
|G#
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
|11
|-
|diminished 2nd
|2\41, 92.31 (92.68)
|Hff
|[[21/20]], [[22/21]], [[26/25]]
| -19
|-
|minor 2nd
|3\41, 138.46 (139.02)
|Hf
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
| -8
|- bgcolor="#eaeaff"
|major 2nd
|4\41, 184.615 (185.37)
|H
|[[9/8]], [[10/9]], [[11/10]]
|3
|-
|augmented 2nd
|5\41, 230.77 (231.71)
|H#
|[[8/7]], [[15/13]]
|14
|-
|diminished 3rd
|6\41, 276.92 (278.05)
|Jff
|[[7/6]], [[13/11]], [[33/28]]
| -16
|-
|minor 3rd
|7\41, 323.08 (324.39)
|Jf
|[[135/112]], [[6/5]]
| -5
|- bgcolor="#eaeaff"
|major 3rd
|8\41, 369.23 (370.73)
|J
|[[5/4]], [[11/9]], [[16/13]]
|6
|-
|augmented 3rd
|9\41, 415.385 (417.07)
|J#
|[[9/7]], [[14/11]], [[33/26]]
|17
|-
|diminished 4th
|10\41, 461.54 (463.415)
|Kff
|[[21/16]], [[13/10]]
| -13
|- bgcolor="#eaeaff"
|natural 4th
|11\41, 507.69 (509.76)
|Kf
|[[75/56]], [[4/3]]
| -2
|-
|augmented 4th
|12\41, 553.85 (556.10)
| K
|[[11/8]], [[18/13]]
|9
|-
|doubly augmented 4th, doubly diminished 5th
|13\41, 600.00 (602.44)
|K#, Lff
|[[7/5]], [[10/7]]
| 20,-21
|-
|diminished 5th
|14\41, 646.15 (648.78)
| Lf
|[[16/11]], [[13/9]]
| -10
|- bgcolor="#eaeaff"
|perfect 5th
|15\41, 692.31 (695.12)
|L
|[[112/75]], [[3/2]]
|1
|-
| augmented 5th
|16\41, 738.46 (741.46)
|L#
|[[32/21]], [[20/13]]
|12
|-
|diminished 6th
|17\41, 784.615 (787.805)
|Aff
|[[11/7]], [[14/9]]
| -18
|-
| minor 6th
|18\41, 830.77 (834.15)
|Af
|[[13/8]], [[8/5]]
| -7
|- bgcolor="#eaeaff"
|major 6th
|19\41, 876.92 (880.49)
|A
|[[5/3]], [[224/135]]
|4
|-
|augmented 6th
|20\41, 923.08 (926.83)
|A#
|[[12/7]], [[22/13]]
|15
|-
|diminished 7th
|21\41, 969.23 (973.17)
|Bff
|[[7/4]], [[26/15]]
| -15
|- bgcolor="#eaeaff"
|minor 7th
|22\41, 1015.385 (1019.51)
|Bf
|[[9/5]], [[16/9]], [[20/11]]
| -4
|-
|major 7th
|23\41, 1061.54 (1065.85)
|B
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
|7
|-
|augmented 7th
|24\41, 1107.69 (1112.195)
|B#
|[[21/11]], [[25/13]], [[40/21]]
|18
|-
|diminished octave
| 25\41, 1153.85 (1158.54)
|Cf
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
| -12
|- bgcolor="#eaeaff"
|perfect octave
|26\41, 1200.00 (1204.88)
|C
|2/1
| -1
|-
|augmented octave
|27\41, 1246.15 (1251.22)
|C#
|33/16, 49/24, 72/35, 25/12
| 10
|-
|doubly augmented octave, diminished 9th
|28\41, 1292.31 (1297.56)
|Cx, Dff
|21/10, 44/21, 52/25
|21,-20
|-
|minor 9th
|29\41, 1338.46 (1343.90)
|Df
|24/11, 13/6, 28/13, 32/15
| -9
|- bgcolor="#eaeaff"
|major 9th
|30\41, 1384.615 (1390.24)
|D
|9/4, 20/9, 11/5
|2
|-
|augmented 9th
|31\41, 1430.77 (1436.595)
| D#
|16/7, 30/13
|13
|-
|diminished 10th
|32\41, 1476.92 (1492.93)
|Eff
|7/3, 26/11, 33/14
| -17
|-
|minor 10th
|33\41, 1523.08 (1529.27)
|Ef
|135/56, 12/5
| -6
|- bgcolor="#eaeaff"
| major 10th
|34\41, 1569.23 (1575.61)
|E
|5/2, 22/9, 32/13
|5
|-
|augmented 10th
|35\41, 1615.385 (1621.95)
|E#
| 18/7, 28/11, 33/13
|16
|-
|diminished 11th
|36\41, 1661.54 (1668.29)
|Ff
|21/8, 13/5
| -14
|- bgcolor="#eaeaff"
|natural 11th
|37\41, 1709.69 (1714.63)
|F
|75/28, 8/3
| -3
|-
|augmented 11th
|38\41, 1753.85 (1760.98)
| F#
|11/4, 36/13
|8
|-
|doubly augmented 11th, doubly diminished 12th
|39\41, 1800.00 (1807.32)
|Fx, Gff
|14/5, 20/7
|19
|-
|diminished 12th
|40\41, 1846.15 (1853.66)
|Gf
|32/11, 26/9
| -11
|}
===Parahard===
~[[35edt]] Obikhod combines the sound of the 9/4 major ninth and the sound of the [[8/7]] whole tone. This is because ~35edt Obikhodic has a large step of ~218.2¢, close to [[22edo]]'s superpythagorean major second, and is both a warped Pythagorean [[9/8]] whole tone and a warped [[8/7]] septimal whole tone.
====Intervals====
The intervals of the extended generator chain (-18 to +18 generators) are as follows in various Obikhodic tunings close to ~[[35edt]].
{| class="wikitable"
|+
!Degree
!Size in ~[[35edt]]
!Note name on G
!Approximate Ratios*
! #Gens up
|- bgcolor="#eaeaff"
|unison
|0\35, 0.00
|G
|[[1/1]]
|0
|-
|chroma
|3\35, 163.64 (162.86)
|G#
|[[12/11]], [[11/10]], [[10/9]]
|11
|-
|minor 2nd
|1\35, 54.545 (54.29)
|Hf
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
| -8
|- bgcolor="#eaeaff"
|major 2nd
|4\35, 218.18 (217.14)
|H
|[[9/8]], [[17/15]], [[8/7]]
|3
|-
|augmented 2nd
|7\35, 381.818 (380.00)
|H#
|[[5/4]], [[96/77]]
|14
|-
|diminished 3rd
| 2\35, 109.09 (108.57)
|Jff
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
| -16
|-
|minor 3rd
|5\35, 272.73 (271.43)
|Jf
|[[20/17]], [[7/6]]
| -5
|- bgcolor="#eaeaff"
|major 3rd
|8\35, 436.36 (434.29)
|J
|[[14/11]], [[9/7]], [[22/17]]
|6
|-
|augmented 3rd
|11\35, 600.00 (542.86)
|J#
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|17
|-
| diminished 4th
|6\35, 327.27 (325.71)
|Kff
|[[6/5]], [[17/14]], [[11/9]]
| -13
|- bgcolor="#eaeaff"
|natural 4th
|9\35, 490.91 (488.57)
|Kf
|[[4/3]]
| -2
|-
|augmented 4th
|12\35, 654.545 (651.43)
|K
|[[16/11]], [[22/15]]
|9
|-
|diminished 5th
|10\35, 545.455 (542.86)
|Lf
|[[15/11]], [[11/8]]
| -10
|- bgcolor="#eaeaff"
|perfect 5th
|13\35, 709.09 (705.71)
| L
|[[3/2]]
|1
|-
| augmented 5th
|16\35, 872.73 (868.57)
|L#
|[[18/11]], [[28/17]], [[5/3]]
|12
|-
| diminished 6th
|11\35, 600.00 (597.14)
|Aff
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
| -18
|-
|minor 6th
|14\35, 763.64 (760.00)
|Af
|[[17/11]], [[14/9]], [[11/7]]
| -7
|- bgcolor="#eaeaff"
|major 6th
|17\35, 927.27 (822.86)
|A
|[[17/10]], [[12/7]]
|4
|-
|augmented 6th
|20\35, 1090.909 (1085.71)
|A#
|[[28/15]], [[15/8]], [[32/17]], [[17/9]]
|15
|-
|diminished 7th
|15\35, 818.182 (814.29)
|Bff
|[[8/5]], [[77/48]]
| -15
|-
|minor 7th
|18/35, 981.82 (977.14)
|Bf
|[[7/4]], [[30/17]], [[16/9]]
| -4
|- bgcolor="#eaeaff"
|major 7th
|21\35, 1145.455 (1140.00)
| B
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
|7
|-
|augmented 7th
|24\35, 1309.09 (1302.86)
|B#
|36/17, 17/8, 32/15, 15/7
|18
|-
| diminished octave
|19\22, 1036.36 (1031.43)
| Cf
|[[9/5]], [[11/6]], [[20/11]]
| -12
|- bgcolor="#eaeaff"
|perfect octave
|22\35, 1200.00 (1194.29)
|C
|[[2/1]]
| -1
|-
|augmented octave
|25\35, 1363.64 (1357.14)
|C#
|24/11, 11/5, 20/9
|10
|-
|minor 9th
|23\35, 1254.55 (1248.57)
|Df
|72/35, 68/33, 33/16, 64/31
| -9
|- bgcolor="#eaeaff"
|major 9th
|26\35, 1418.18 (1411.43)
| D
|9/4, 34/15, 16/7
|2
|-
|augmented 9th
|29\35, 1581.81 (1574.29)
|D#
|5/2, 192/77
|13
|-
|diminished 10th
|24\35, 1309.09 (1302.86)
|Eff
|36/17, 17/8, 32/15, 15/7
| -17
|-
|minor 10th
|27\35, 1472.72 (1465.71)
|Ef
|40/17, 7/3
|  -6
|- bgcolor="#eaeaff"
|major 10th
|30\35, 1636.36 (1628.57)
|E
| 28/11, 18/7, 44/17
|5
|-
|augmented 10th
|33\35, 1800.00 (1791.43)
|E#
|14/5, 48/17, 17/6, 20/7
|16
|-
|diminished 11th
| 28\35, 1527.27 (1520.00)
|Ff
|12/5, 17/7, 22/9
| -14
|- bgcolor="#eaeaff"
|natural 11th
|31\35, 1690.91 (1682.86)
|F
|8/3
| -3
|-
|augmented 11th
|34\35, 1854.545 (1845.71)
|F#
|32/11, 44/15
|8
|-
|diminished 12th
|32\35, 1745.455 (1737.14)
|Gf
|30/11, 11/4
| -11
|}
===Ultrahard===
[[Archytas clan#Ultrapyth|Ultrapythagorean]] Obikhodic is a rank-2 temperament in the [[Step ratio|pseudopaucitonic]] range. It represents the [[harmonic entropy]] minimum of the Obikhodic spectrum where [[7/4]] is the minor seventh.
 
In the broad sense, Ultrapyth can be viewed as any tuning that divides a 16/7 into 2 equal parts. ~[[35edt]] and ~[[43edt]] can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between ~[[27edt]] and true Ultrapyth in terms of harmonies. ~[[51edt]] & ~[[59edt]] are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but ~[[67edt]] is where it really comes into its own in terms of harmonies, providing not only an excellent [[6/5]], but also 7:8:9 melodies, as by shifting one whole tone done a comma, it shifts from [[The Archipelago|archipelago]] to septimal harmonies.
 
Beyond that, it's a question of which intervals you want to favor. ~[[75edt]] has an essentially perfect [[9/8]], either ~[[83edt]] or ~[[91edt]] has an essentially perfect [[7/4]] and multiple chains of essentially perfect [[meantone]], and while ~[[99edt]] does not have an essentially perfect [[7/4]], it has a double chain of essentially perfect quarter-comma meantone. You could in theory go up to ~[[131edt]] if you want to favor the [[3/2]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.
 
The sizes of the generator, large step and small step of Obikhodic are as follows in various ultrapyth tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
![[38edo|~]][[59edt]]
!~[[83edt]]
!~[[91edt]]
!~[[99edt]]
!Optimal ([[POTE|PHTE]]) Ultrapyth tuning
!JI intervals represented (2.3.5.7.13 subgroup)
|-
| generator (g)
|22\59, 713.51
|31\83, 715.385
|34\91, 715.79
| 37\99, 716.13
|712.61
| 3/2
|-
|L (3g - ~tritave)
| 7\39, 227.03
|10\83, 230.77
|11\91, 231.58
|12\99, 232.26
| 230.55
|8/7
|-
|s (-8g + 3 ~tritaves)
| 1\59, 32.43
|1\83, 23.08
|1\91, 21.05
|1/99, 19.355
|20.96
| 50/49 81/80 91/90
|}
====Intervals ====
Sortable table of intervals in the Great Mixolydian mode and their Ultrapyth interpretations:
{| class="wikitable right-2 right-3 right-4 right-5 sortable"
|-
!Degree
!Size in ~[[59edt]]
!Size in ~[[83edt]]
!Size in ~[[91edt]]
!Size in ~[[99edt]]
! Size in PHTE tuning
! Note name on D
! class="unsortable" |Approximate ratios
!#Gens up
|-
|1
|0\59, 0.00
|0\83, 0.00
|0\91, 0.00
|0\99, 0.00
|0.00
|D
|1/1
|0
|-
|2
|7\59, 227.03
|10\83, 230.77
|11\91, 231.58
|12\99, 232.26
|230.55
|E
|8/7
| +3
|-
|3
|14\59, 454.05
|20\83, 461.54
|22\91, 463.16
|24\99, 464.52
|461.10
|F
| 13/10, 9/7
| +6
|-
|4
|15\59, 486.49
|21\83, 484.615
|23\91, 484.21
|25\99, 483.87
|482.06
|G
|4/3
| -2
|-
|5
|22\59, 713.51
|31\83, 715.385
|34\91, 715.79
|37\99, 716.13
|712.61
|H
|3/2
| +1
|-
|6
|29\59, 940.54
|41\83, 946.15
|45\91, 947.36
|49\99, 948.39
|943.16
|J
|12/7, 26/15
| +4
|-
|7
|30\38, 972.97
|42\83, 969.23
|46\91, 968.42
|50\99, 967.74
|964.12
|K
|7/4
| -4
|-
|8
|37\59, 1200.00
|52\83, 1200.00
|57\91, 1200.00
|62\99, 1200.00
|1194.67
|L
|2/1
| -1
|-
|9
|44\59, 1427.03
|62\83, 1430.77
|68\91, 1430.58
|74\99, 1432.26
|1425.22
|A
|16/7
| +2
|-
|10
|51\59, 1654.05
|72\83, 1661.54
|79\91, 1663.16
|86\99, 1664.52
|1655.77
|B
| 13/5, 18/7
| +5
|-
|11
|52\59, 1686.49
|73\83,
1684.615
|80\91,
1684.21
|87\99, 1683.87
|1676.32
|C
|4/3
| -3
|}
==Modes==
Obikhodic modes are named after the Church modes, but with a “Great” prefix.
{| class="wikitable"
|-
| style="text-align:center;" |'''Mode'''
| style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
| style="text-align:center;" |'''Name'''
|-
|LLLsLLLsLLs
| style="text-align:center;" |<nowiki>10|0</nowiki>
|(Great) Lydian #8 (Tanagran)
|-
|LLLsLLsLLLs
| style="text-align:center;" |<nowiki>9|1</nowiki>
|(Great) Lydian
|-
|LLsLLLsLLLs
| style="text-align:center;" |<nowiki>8|2</nowiki>
|(Great) Lydian b4, Ionian #11 (Distomian)
|-
| |LLsLLLsLLsL
| style="text-align:center;" |<nowiki>7|3</nowiki>
| |(Great) Ionian
|-
| |LLsLLsLLLsL
| style="text-align:center;" |<nowiki>6|4</nowiki>
| |(Great) Mixolydian
|-
| |LsLLLsLLLsL
| style="text-align:center;" |<nowiki>5|5</nowiki>
| |(Great) Mixolydian b3, Dorian #10 (Livadeian)
|-
| |LsLLLsLLsLL
| style="text-align:center;" |<nowiki>4|6</nowiki>
| |(Great) Dorian
|-
| |LsLLsLLLsLL
| style="text-align:center;" |<nowiki>3|7</nowiki>
| |(Great) Aeolian
|-
| | sLLLsLLLsLL
| style="text-align:center;" |<nowiki>2|8</nowiki>
| |(Great) Aeolian b2, Phrygian #9 (Theban)
|-
| |sLLLsLLsLLL
| style="text-align:center;" |<nowiki>1|9</nowiki>
| |(Great) Phrygian
|-
| |sLLsLLLsLLL
| style="text-align:center;" |<nowiki>0|10</nowiki>
| |(Great) Locrian
|}
 
===Cyclic Permutation order===
{| class="wikitable"
|+
!Spelling
!'''Mode'''
![[Modal UDP Notation|'''UDP''']]
!'''Name'''
|-
|GHJKLABCDEFG
|LLsLLLsLLLs
| style="text-align:center;" |<nowiki>8|2</nowiki>
|(Great) Distomian
|-
|HJKLABCDEFGH
|LsLLLsLLLsL
| style="text-align:center;" |<nowiki>5|5</nowiki>
|(Great) Livadeian
|-
|JKLABCDEFGHJ
|sLLLsLLLsLL
| style="text-align:center;" |<nowiki>2|8</nowiki>
|(Great) Theban
|-
|KLABCDEFGHJK
|LLLsLLLsLLs
| style="text-align:center;" |<nowiki>10|0</nowiki>
|(Great) Tanagran
|-
|LABCDEFGHJKL
|LLsLLLsLLsL
| style="text-align:center;" |<nowiki>7|3</nowiki>
|(Great) Ionian
|-
|ABCDEFGHJKLA
|LsLLLsLLsLL
| style="text-align:center;" |<nowiki>4|6</nowiki>
|(Great) Dorian
|-
|BCDEFGHJKLAB
|sLLLsLLsLLL
| style="text-align:center;" |<nowiki>1|9</nowiki>
|(Great) Phrygian
|-
|CDEFGHJKLABC
|LLLsLLsLLLs
| style="text-align:center;" |<nowiki>9|1</nowiki>
|(Great) Lydian
|-
|DEFGHJKLABCD
|LLsLLsLLLsL
| style="text-align:center;" |<nowiki>6|4</nowiki>
|(Great) Mixolydian
|-
|EFGHJKLABCDE
|LsLLsLLLsLL
| style="text-align:center;" |<nowiki>3|7</nowiki>
|(Great) Aeolian
|-
|FGHJKLABCDEF
|sLLsLLLsLLL
| style="text-align:center;" |<nowiki>0|10</nowiki>
|(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 ==
== Scale tree ==
{| class="wikitable"
{{MOS tuning spectrum|Scale Signature=8L 3s<3/1>|9/4=Pythagorean tuning is around here}}
|+
! colspan="2" |Generator
!Normalized
!''ed19\12''
|-
|3\8
|
|<u>720</u>
|''712.5''
|-
|19\51
|
|<u>712.5</u>
|''707.843''
|-
|
|35\94
|<u>711.864</u>
|''707.447''
|-
|16\43
|
|<u>711.111</u>
|''706.977''
|-
|
|29\78
|<u>710.204</u>
|''706.41''
|-
|13\35
|
|<u>709.091</u>
|''705.714''
|-
|
|36\97
|<u>708.197</u>
|705.155
|-
|
|23\62
|<u>707.692</u>
|''704.839''
|-
|
|33\89
|<u>707.143</u>
|''704.494''
|-
|
|43\116
|<u>706.849</u>
|''704.31''
|-
|10\27
|
|<u>705.882</u>
|''703.704''
|-
|
|27\73
|<u>704.348</u>
|''702.738''
|-
|
|17\46
|<u>703.448</u>
|''702.174''
|-
|
|24\65
|<u>702.439</u>
|''701.5385''
|-
|
|31\84
|<u>701.887</u>
|701.1905
|-
|7\19
|
|<u>700</u>
|''700''
|-
|
|60\163
|<u>699.029</u>
|''699.3865''
|-
|
|53\144
|<u>698.901</u>
|''699.306''
|-
|
|46\125
|<u>698.734</u>
|''699.2''
|-
|
|39\106
|<u>698.5075</u>
|''699.057''
|-
|
|32\87
|<u>698.182</u>
|698.852
|-
|
|25\68
|<u>697.674</u>
|''698.529''
|-
|
|18\49
|<u>696.77</u>
|''697.959''
|-
|
|29\79
|<u>696</u>
|''697.468''
|-
|
|40\109
|<u>695.652</u>
|''697.248''
|-
|11\30
|
|<u>694.737</u>
|''696.667''
|-
|
|26\71
|<u>693.333</u>
|''695.775''
|-
|15\41
|
|<u>692.308</u>
|''695.122''
|-
|
|34\93
|<u>691.525</u>
|''694.623''
|-
|19\52
|
|<u>690.909</u>
|''694.231''
|-
|
|42\115
|<u>690.411</u>
|''693.913''
|-
|23\63
|
|<u>690</u>
|''693.651''
|-
|4\11
|
|<u>685.714</u>
|''690.909''
|}
 
== References ==
 
<references />
 
[[Category:Nonoctave]]
[[Category:11-tone scales]]

Revision as of 15:41, 19 April 2025

↖ 7L 2s⟨3/1⟩ ↑ 8L 2s⟨3/1⟩ 9L 2s⟨3/1⟩ ↗
← 7L 3s⟨3/1⟩ 8L 3s (3/1-equivalent) 9L 3s⟨3/1⟩ →
↙ 7L 4s⟨3/1⟩ ↓ 8L 4s⟨3/1⟩ 9L 4s⟨3/1⟩ ↘
Scale structure
Step pattern LLLsLLLsLLs
sLLsLLLsLLL
Equave 3/1 (1902.0 ¢)
Period 3/1 (1902.0 ¢)
Generator size(edt)
Bright 4\11 to 3\8 (691.6 ¢ to 713.2 ¢)
Dark 5\8 to 7\11 (1188.7 ¢ to 1210.3 ¢)
Related MOS scales
Parent 3L 5s⟨3/1⟩
Sister 3L 8s⟨3/1⟩
Daughters 11L 8s⟨3/1⟩, 8L 11s⟨3/1⟩
Neutralized 5L 6s⟨3/1⟩
2-Flought 19L 3s⟨3/1⟩, 8L 14s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 4\11 (691.6 ¢)
Supersoft (L:s = 4:3) 15\41 (695.8 ¢)
Soft (L:s = 3:2) 11\30 (697.4 ¢)
Semisoft (L:s = 5:3) 18\49 (698.7 ¢)
Basic (L:s = 2:1) 7\19 (700.7 ¢)
Semihard (L:s = 5:2) 17\46 (702.9 ¢)
Hard (L:s = 3:1) 10\27 (704.4 ¢)
Superhard (L:s = 4:1) 13\35 (706.4 ¢)
Collapsed (L:s = 1:0) 3\8 (713.2 ¢)
View • Talk • Edit

8L 3s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 8 large steps and 3 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 691.6 ¢ to 713.2 ¢, or from 1188.7 ¢ to 1210.3 ¢. 8L 3s⟨3/1⟩ scale pattern includes the well-known 5L 2s pattern within it.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 8L 3s⟨3/1⟩
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Minor 1-mosstep m1ms s 0.0 ¢ to 172.9 ¢
Major 1-mosstep M1ms L 172.9 ¢ to 237.7 ¢
2-mosstep Minor 2-mosstep m2ms L + s 237.7 ¢ to 345.8 ¢
Major 2-mosstep M2ms 2L 345.8 ¢ to 475.5 ¢
3-mosstep Minor 3-mosstep m3ms 2L + s 475.5 ¢ to 518.7 ¢
Major 3-mosstep M3ms 3L 518.7 ¢ to 713.2 ¢
4-mosstep Diminished 4-mosstep d4ms 2L + 2s 475.5 ¢ to 691.6 ¢
Perfect 4-mosstep P4ms 3L + s 691.6 ¢ to 713.2 ¢
5-mosstep Minor 5-mosstep m5ms 3L + 2s 713.2 ¢ to 864.5 ¢
Major 5-mosstep M5ms 4L + s 864.5 ¢ to 951.0 ¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 951.0 ¢ to 1037.4 ¢
Major 6-mosstep M6ms 5L + s 1037.4 ¢ to 1188.7 ¢
7-mosstep Perfect 7-mosstep P7ms 5L + 2s 1188.7 ¢ to 1210.3 ¢
Augmented 7-mosstep A7ms 6L + s 1210.3 ¢ to 1426.5 ¢
8-mosstep Minor 8-mosstep m8ms 5L + 3s 1188.7 ¢ to 1383.2 ¢
Major 8-mosstep M8ms 6L + 2s 1383.2 ¢ to 1426.5 ¢
9-mosstep Minor 9-mosstep m9ms 6L + 3s 1426.5 ¢ to 1556.1 ¢
Major 9-mosstep M9ms 7L + 2s 1556.1 ¢ to 1664.2 ¢
10-mosstep Minor 10-mosstep m10ms 7L + 3s 1664.2 ¢ to 1729.1 ¢
Major 10-mosstep M10ms 8L + 2s 1729.1 ¢ to 1902.0 ¢
11-mosstep Perfect 11-mosstep P11ms 8L + 3s 1902.0 ¢

Generator chain

Generator chain of 8L 3s⟨3/1⟩
Bright gens Scale degree Abbrev.
18 Augmented 6-mosdegree A6md
17 Augmented 2-mosdegree A2md
16 Augmented 9-mosdegree A9md
15 Augmented 5-mosdegree A5md
14 Augmented 1-mosdegree A1md
13 Augmented 8-mosdegree A8md
12 Augmented 4-mosdegree A4md
11 Augmented 0-mosdegree A0md
10 Augmented 7-mosdegree A7md
9 Major 3-mosdegree M3md
8 Major 10-mosdegree M10md
7 Major 6-mosdegree M6md
6 Major 2-mosdegree M2md
5 Major 9-mosdegree M9md
4 Major 5-mosdegree M5md
3 Major 1-mosdegree M1md
2 Major 8-mosdegree M8md
1 Perfect 4-mosdegree P4md
0 Perfect 0-mosdegree
Perfect 11-mosdegree		 P0md
P11md
−1 Perfect 7-mosdegree P7md
−2 Minor 3-mosdegree m3md
−3 Minor 10-mosdegree m10md
−4 Minor 6-mosdegree m6md
−5 Minor 2-mosdegree m2md
−6 Minor 9-mosdegree m9md
−7 Minor 5-mosdegree m5md
−8 Minor 1-mosdegree m1md
−9 Minor 8-mosdegree m8md
−10 Diminished 4-mosdegree d4md
−11 Diminished 11-mosdegree d11md
−12 Diminished 7-mosdegree d7md
−13 Diminished 3-mosdegree d3md
−14 Diminished 10-mosdegree d10md
−15 Diminished 6-mosdegree d6md
−16 Diminished 2-mosdegree d2md
−17 Diminished 9-mosdegree d9md
−18 Diminished 5-mosdegree d5md

Modes

Scale degrees of the modes of 8L 3s⟨3/1⟩
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11
10|0 1 LLLsLLLsLLs Perf. Maj. Maj. Maj. Perf. Maj. Maj. Aug. Maj. Maj. Maj. Perf.
9|1 5 LLLsLLsLLLs Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
8|2 9 LLsLLLsLLLs Perf. Maj. Maj. Min. Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
7|3 2 LLsLLLsLLsL Perf. Maj. Maj. Min. Perf. Maj. Maj. Perf. Maj. Maj. Min. Perf.
6|4 6 LLsLLsLLLsL Perf. Maj. Maj. Min. Perf. Maj. Min. Perf. Maj. Maj. Min. Perf.
5|5 10 LsLLLsLLLsL Perf. Maj. Min. Min. Perf. Maj. Min. Perf. Maj. Maj. Min. Perf.
4|6 3 LsLLLsLLsLL Perf. Maj. Min. Min. Perf. Maj. Min. Perf. Maj. Min. Min. Perf.
3|7 7 LsLLsLLLsLL Perf. Maj. Min. Min. Perf. Min. Min. Perf. Maj. Min. Min. Perf.
2|8 11 sLLLsLLLsLL Perf. Min. Min. Min. Perf. Min. Min. Perf. Maj. Min. Min. Perf.
1|9 4 sLLLsLLsLLL Perf. Min. Min. Min. Perf. Min. Min. Perf. Min. Min. Min. Perf.
0|10 8 sLLsLLLsLLL Perf. Min. Min. Min. Dim. Min. Min. Perf. Min. Min. Min. Perf.

Theory

By dividing the 5L 2s of LLsLLLs into A=LLs and B=LLLs, and combining them as ABBABB..., it becomes 8L 3s⟨3/1⟩. This scale has octaves that are too frequent for the listener to feel a tritave equivalence. The range of possible dark generators will likely feel sufficiently pseudo-octave. Similar to Angel, it would be good to utilize a finite-length chain of octaves and make use of existing diatonic music theory.

Low harmonic entropy scales

  • Pythagorean tuning (period = 3/1, generator = 3/2): L/s = 2.260
  • Tritave-equivalent meantone tunings:

Tuning ranges

Simple tunings

Simple Tunings of 8L 3s⟨3/1⟩
Scale degree Abbrev. Basic (2:1)
19edt 		Hard (3:1)
27edt 		Soft (3:2)
30edt
Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\19 0.0 0\27 0.0 0\30 0.0
Minor 1-mosdegree m1md 1\19 100.1 1\27 70.4 2\30 126.8
Major 1-mosdegree M1md 2\19 200.2 3\27 211.3 3\30 190.2
Minor 2-mosdegree m2md 3\19 300.3 4\27 281.8 5\30 317.0
Major 2-mosdegree M2md 4\19 400.4 6\27 422.7 6\30 380.4
Minor 3-mosdegree m3md 5\19 500.5 7\27 493.1 8\30 507.2
Major 3-mosdegree M3md 6\19 600.6 9\27 634.0 9\30 570.6
Diminished 4-mosdegree d4md 6\19 600.6 8\27 563.5 10\30 634.0
Perfect 4-mosdegree P4md 7\19 700.7 10\27 704.4 11\30 697.4
Minor 5-mosdegree m5md 8\19 800.8 11\27 774.9 13\30 824.2
Major 5-mosdegree M5md 9\19 900.9 13\27 915.8 14\30 887.6
Minor 6-mosdegree m6md 10\19 1001.0 14\27 986.2 16\30 1014.4
Major 6-mosdegree M6md 11\19 1101.1 16\27 1127.1 17\30 1077.8
Perfect 7-mosdegree P7md 12\19 1201.2 17\27 1197.5 19\30 1204.6
Augmented 7-mosdegree A7md 13\19 1301.3 19\27 1338.4 20\30 1268.0
Minor 8-mosdegree m8md 13\19 1301.3 18\27 1268.0 21\30 1331.4
Major 8-mosdegree M8md 14\19 1401.4 20\27 1408.9 22\30 1394.8
Minor 9-mosdegree m9md 15\19 1501.5 21\27 1479.3 24\30 1521.6
Major 9-mosdegree M9md 16\19 1601.6 23\27 1620.2 25\30 1585.0
Minor 10-mosdegree m10md 17\19 1701.7 24\27 1690.6 27\30 1711.8
Major 10-mosdegree M10md 18\19 1801.9 26\27 1831.5 28\30 1775.2
Perfect 11-mosdegree P11md 19\19 1902.0 27\27 1902.0 30\30 1902.0

Soft-of-basic tunings

Tunings of 8L 3s⟨3/1⟩
Scale degree Abbrev. 5:4
52edt 		Supersoft (4:3)
41edt 		Soft (3:2)
30edt 		Semisoft (5:3)
49edt
Steps ¢ Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\52 0.0 0\41 0.0 0\30 0.0 0\49 0.0
Minor 1-mosdegree m1md 4\52 146.3 3\41 139.2 2\30 126.8 3\49 116.4
Major 1-mosdegree M1md 5\52 182.9 4\41 185.6 3\30 190.2 5\49 194.1
Minor 2-mosdegree m2md 9\52 329.2 7\41 324.7 5\30 317.0 8\49 310.5
Major 2-mosdegree M2md 10\52 365.8 8\41 371.1 6\30 380.4 10\49 388.2
Minor 3-mosdegree m3md 14\52 512.1 11\41 510.3 8\30 507.2 13\49 504.6
Major 3-mosdegree M3md 15\52 548.6 12\41 556.7 9\30 570.6 15\49 582.2
Diminished 4-mosdegree d4md 18\52 658.4 14\41 649.4 10\30 634.0 16\49 621.0
Perfect 4-mosdegree P4md 19\52 694.9 15\41 695.8 11\30 697.4 18\49 698.7
Minor 5-mosdegree m5md 23\52 841.2 18\41 835.0 13\30 824.2 21\49 815.1
Major 5-mosdegree M5md 24\52 877.8 19\41 881.4 14\30 887.6 23\49 892.8
Minor 6-mosdegree m6md 28\52 1024.1 22\41 1020.6 16\30 1014.4 26\49 1009.2
Major 6-mosdegree M6md 29\52 1060.7 23\41 1067.0 17\30 1077.8 28\49 1086.8
Perfect 7-mosdegree P7md 33\52 1207.0 26\41 1206.1 19\30 1204.6 31\49 1203.3
Augmented 7-mosdegree A7md 34\52 1243.6 27\41 1252.5 20\30 1268.0 33\49 1280.9
Minor 8-mosdegree m8md 37\52 1353.3 29\41 1345.3 21\30 1331.4 34\49 1319.7
Major 8-mosdegree M8md 38\52 1389.9 30\41 1391.7 22\30 1394.8 36\49 1397.4
Minor 9-mosdegree m9md 42\52 1536.2 33\41 1530.8 24\30 1521.6 39\49 1513.8
Major 9-mosdegree M9md 43\52 1572.8 34\41 1577.2 25\30 1585.0 41\49 1591.4
Minor 10-mosdegree m10md 47\52 1719.1 37\41 1716.4 27\30 1711.8 44\49 1707.9
Major 10-mosdegree M10md 48\52 1755.7 38\41 1762.8 28\30 1775.2 46\49 1785.5
Perfect 11-mosdegree P11md 52\52 1902.0 41\41 1902.0 30\30 1902.0 49\49 1902.0

Hard-of-basic tunings

Tunings of 8L 3s⟨3/1⟩
Scale degree Abbrev. Semihard (5:2)
46edt 		Hard (3:1)
27edt 		Superhard (4:1)
35edt 		5:1
43edt
Steps ¢ Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\46 0.0 0\27 0.0 0\35 0.0 0\43 0.0
Minor 1-mosdegree m1md 2\46 82.7 1\27 70.4 1\35 54.3 1\43 44.2
Major 1-mosdegree M1md 5\46 206.7 3\27 211.3 4\35 217.4 5\43 221.2
Minor 2-mosdegree m2md 7\46 289.4 4\27 281.8 5\35 271.7 6\43 265.4
Major 2-mosdegree M2md 10\46 413.5 6\27 422.7 8\35 434.7 10\43 442.3
Minor 3-mosdegree m3md 12\46 496.2 7\27 493.1 9\35 489.1 11\43 486.5
Major 3-mosdegree M3md 15\46 620.2 9\27 634.0 12\35 652.1 15\43 663.5
Diminished 4-mosdegree d4md 14\46 578.9 8\27 563.5 10\35 543.4 12\43 530.8
Perfect 4-mosdegree P4md 17\46 702.9 10\27 704.4 13\35 706.4 16\43 707.7
Minor 5-mosdegree m5md 19\46 785.6 11\27 774.9 14\35 760.8 17\43 751.9
Major 5-mosdegree M5md 22\46 909.6 13\27 915.8 17\35 923.8 21\43 928.9
Minor 6-mosdegree m6md 24\46 992.3 14\27 986.2 18\35 978.1 22\43 973.1
Major 6-mosdegree M6md 27\46 1116.4 16\27 1127.1 21\35 1141.2 26\43 1150.0
Perfect 7-mosdegree P7md 29\46 1199.1 17\27 1197.5 22\35 1195.5 27\43 1194.3
Augmented 7-mosdegree A7md 32\46 1323.1 19\27 1338.4 25\35 1358.5 31\43 1371.2
Minor 8-mosdegree m8md 31\46 1281.8 18\27 1268.0 23\35 1249.9 28\43 1238.5
Major 8-mosdegree M8md 34\46 1405.8 20\27 1408.9 26\35 1412.9 32\43 1415.4
Minor 9-mosdegree m9md 36\46 1488.5 21\27 1479.3 27\35 1467.2 33\43 1459.6
Major 9-mosdegree M9md 39\46 1612.5 23\27 1620.2 30\35 1630.2 37\43 1636.6
Minor 10-mosdegree m10md 41\46 1695.2 24\27 1690.6 31\35 1684.6 38\43 1680.8
Major 10-mosdegree M10md 44\46 1819.3 26\27 1831.5 34\35 1847.6 42\43 1857.7
Perfect 11-mosdegree P11md 46\46 1902.0 27\27 1902.0 35\35 1902.0 43\43 1902.0

Scale tree

Scale tree and tuning spectrum of 8L 3s⟨3/1⟩
Generator(edt) Cents Step ratio Comments
Bright Dark L:s Hardness
4\11 691.620 1210.335 1:1 1.000 Equalized 8L 3s⟨3/1⟩
23\63 694.365 1207.590 6:5 1.200
19\52 694.945 1207.010 5:4 1.250
34\93 695.338 1206.617 9:7 1.286
15\41 695.837 1206.118 4:3 1.333 Supersoft 8L 3s⟨3/1⟩
41\112 696.251 1205.704 11:8 1.375
26\71 696.491 1205.464 7:5 1.400
37\101 696.756 1205.199 10:7 1.429
11\30 697.384 1204.572 3:2 1.500 Soft 8L 3s⟨3/1⟩
40\109 697.965 1203.990 11:7 1.571
29\79 698.186 1203.769 8:5 1.600
47\128 698.374 1203.581 13:8 1.625
18\49 698.677 1203.278 5:3 1.667 Semisoft 8L 3s⟨3/1⟩
43\117 699.009 1202.946 12:7 1.714
25\68 699.248 1202.707 7:4 1.750
32\87 699.570 1202.385 9:5 1.800
7\19 700.720 1201.235 2:1 2.000 Basic 8L 3s⟨3/1⟩
Scales with tunings softer than this are proper
31\84 701.912 1200.043 9:4 2.250 Pythagorean tuning is around here
24\65 702.260 1199.695 7:3 2.333
41\111 702.524 1199.431 12:5 2.400
17\46 702.896 1199.059 5:2 2.500 Semihard 8L 3s⟨3/1⟩
44\119 703.244 1198.711 13:5 2.600
27\73 703.463 1198.492 8:3 2.667
37\100 703.723 1198.232 11:4 2.750
10\27 704.428 1197.527 3:1 3.000 Hard 8L 3s⟨3/1⟩
33\89 705.219 1196.736 10:3 3.333
23\62 705.564 1196.391 7:2 3.500
36\97 705.880 1196.075 11:3 3.667
13\35 706.440 1195.515 4:1 4.000 Superhard 8L 3s⟨3/1⟩
29\78 707.137 1194.818 9:2 4.500
16\43 707.704 1194.251 5:1 5.000
19\51 708.571 1193.384 6:1 6.000
3\8 713.233 1188.722 1:0 → ∞ Collapsed 8L 3s⟨3/1⟩
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