8L 3s (3/1-equivalent)

Todo: rewrite Rewrite according to style guide

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
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¢)

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 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