8L 3s (3/1-equivalent)

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Todo: cleanup
Russian obikhod (LLsLLs(LLs)+LL) and 8L 3s <3/1> (LLLsLLLsLLs) are unrelated step patterns, so references to the former should be (re)moved.

↖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
Brightest mode LLLsLLLsLLs
Equave (cents) 3/1 (1902.0¢)
Period (cents) 3/1 (1902.0¢)
Generator ranges
Bright 4\11edt (691.6¢) to 3\8edt (713.2¢)
Dark 5\8edt (1188.7¢) to 7\11edt (1210.3¢)
Related scales
Parent 3L 5s⟨3/1⟩
Sister 3L 8s⟨3/1⟩
Daughters 11L 8s⟨3/1⟩, 8L 11s⟨3/1⟩
Equal tunings
Equalized (L:s = 1:1) 4\11edt (691.6¢)
Supersoft (L:s = 4:3) 15\41edt (695.8¢)
Soft (L:s = 3:2) 11\30edt (697.4¢)
Semisoft (L:s = 5:3) 18\49edt (698.7¢)
Basic (L:s = 2:1) 7\19edt (700.7¢)
Semihard (L:s = 5:2) 17\46edt (702.9¢)
Hard (L:s = 3:1) 10\27edt (704.4¢)
Superhard (L:s = 4:1) 13\35edt (706.4¢)
Collapsed (L:s = 1:0) 3\8edt (713.2¢)

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¢.

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 8L 3s (3/1-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).

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

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

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

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[1] 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¢.

# 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
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
The chromatic 19-note MOS (either 8L 11s, 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:

Degree Size in ~19edt (basic) Size in ~27edt (hard) Size in ~30edt (soft) Note name on G #Gens up
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
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
major 3rd 4\19, 400.00 6\27, 423.53 (422.22) 6\30, 378.95 (380.00) J 6
natural 4th 5\19, 500.00 7\27, 494.12 (493.59) 8\30, 505,26 (506.67) K -2
augmented 4th 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
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
major 6th 9\19, 900.00 13\27, 917.65 (914.81) 14\30, 884.21 (886.67) A 4
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
perfect octave 12\19, 1200.00 17\27, 1200.00 (1196.30) 19\30, 1200.00 (1203.33) C -1
augmented octave 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
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
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
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 superpythagorean diatonic tunings:

  • The large step is near the Pythagorean 9/8 whole tone, somewhere between as in 12edo and as in 17edo.
  • 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.

~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:

Degree Size in ~19edt (basic) Size in ~27edt (hard) Size in ~46edt (semihard) Note name on G #Gens up
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
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
major 3rd 4\19, 400.00 6\27, 423.53 (422.22) 10\46, 413.79 (413.04) J 6
natural 4th 5\19, 500.00 7\27, 494.12 (493.59) 12\46, 496.55 (495.65) K -2
augmented 4th 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
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
major 6th 9\19, 900.00 13\27, 917.65 (914.81) 22\46, 910.345 (908.70) A 4
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
perfect octave 12\19, 1200.00 17\27, 1200.00 (1196.30) 29\46, 1200.00 (1197.83) C -1
augmented octave 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
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
major 10th 16\19, 1600.00 23\27, 1623.53 (1618.52) 39\46, 1613.79 (1610.87) E 5
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 12edo.
  • The major 3rd (made of two large steps) is a near-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).

~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):

Degree Size in ~30edt (soft) ~49edt (semisoft) Note name on G Approximate ratios #Gens up
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
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
major 3rd 6\30, 378.95 (380.00) 10\49, 387.10 (387.755) J 5/4 6
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
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
major 6th 14\30, 884.21 (886.67) 23\49, 890.32 (891.84) A 5/3 4
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
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
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
major 10th 25\30, 1578.95 (1583.33) 41\49, 1587.10 (1590.80) E 5/2 5
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 flattone temperament.

The sizes of the generator, large step and small step of Obikhodic are as follows in various tunings in this range.

~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.

Degree Size in ~41edt (supersoft) Note name on G Approximate ratios #Gens up
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
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
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
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
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
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
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
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
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
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
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.

Degree Size in ~35edt Note name on G Approximate Ratios* #Gens up
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
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
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
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
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
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
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
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
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
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
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

Ultrapythagorean Obikhodic is a rank-2 temperament in the 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 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.

~59edt ~83edt ~91edt ~99edt Optimal (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:

Degree Size in ~59edt Size in ~83edt Size in ~91edt Size in ~99edt Size in PHTE tuning Note name on D 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.

Mode UDP Name
LLLsLLLsLLs 10|0 (Great) Lydian #8 (Tanagran)
LLLsLLsLLLs 9|1 (Great) Lydian
LLsLLLsLLLs 8|2 (Great) Lydian b4, Ionian #11 (Distomian)
LLsLLLsLLsL 7|3 (Great) Ionian
LLsLLsLLLsL 6|4 (Great) Mixolydian
LsLLLsLLLsL 5|5 (Great) Mixolydian b3, Dorian #10 (Livadeian)
LsLLLsLLsLL 4|6 (Great) Dorian
LsLLsLLLsLL 3|7 (Great) Aeolian
sLLLsLLLsLL 2|8 (Great) Aeolian b2, Phrygian #9 (Theban)
sLLLsLLsLLL 1|9 (Great) Phrygian
sLLsLLLsLLL 0|10 (Great) Locrian

Cyclic Permutation order

Spelling Mode UDP Name
GHJKLABCDEFG LLsLLLsLLLs 8|2 (Great) Distomian
HJKLABCDEFGH LsLLLsLLLsL 5|5 (Great) Livadeian
JKLABCDEFGHJ sLLLsLLLsLL 2|8 (Great) Theban
KLABCDEFGHJK LLLsLLLsLLs 10|0 (Great) Tanagran
LABCDEFGHJKL LLsLLLsLLsL 7|3 (Great) Ionian
ABCDEFGHJKLA LsLLLsLLsLL 4|6 (Great) Dorian
BCDEFGHJKLAB sLLLsLLsLLL 1|9 (Great) Phrygian
CDEFGHJKLABC LLLsLLsLLLs 9|1 (Great) Lydian
DEFGHJKLABCD LLsLLsLLLsL 6|4 (Great) Mixolydian
EFGHJKLABCDE LsLLsLLLsLL 3|7 (Great) Aeolian
FGHJKLABCDEF sLLsLLLsLLL 0|10 (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 -> Theban -> Tanagran -> Ionian cyclic sequence will resemble the geography of ancient Greece.

Scale tree

Generator Normalized ed19\12
3\8 720 712.5
19\51 712.5 707.843
35\94 711.864 707.447
16\43 711.111 706.977
29\78 710.204 706.41
13\35 709.091 705.714
36\97 708.197 705.155
23\62 707.692 704.839
33\89 707.143 704.494
43\116 706.849 704.31
10\27 705.882 703.704
27\73 704.348 702.738
17\46 703.448 702.174
24\65 702.439 701.5385
31\84 701.887 701.1905
7\19 700 700
60\163 699.029 699.3865
53\144 698.901 699.306
46\125 698.734 699.2
39\106 698.5075 699.057
32\87 698.182 698.852
25\68 697.674 698.529
18\49 696.77 697.959
29\79 696 697.468
40\109 695.652 697.248
11\30 694.737 696.667
26\71 693.333 695.775
15\41 692.308 695.122
34\93 691.525 694.623
19\52 690.909 694.231
42\115 690.411 693.913
23\63 690 693.651
4\11 685.714 690.909

References

  1. For relative cents