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

Relationship to the Obikhod

The Obikhod (Обиход церковного пения) is a collection of polyphonic Russian Orthodox liturgical chants forming a major tradition of Russian liturgical music; it includes both liturgical texts and psalm settings.

The original Obikhod, the book of rites of the monastery of Volokolamsk, was composed about 1575. Among its subjects were traditional liturgical chants. The Obikhod was originally monodic but later developed polyphony. In 1772 the Obikhod was the first compilation of music printed in Russia, in Moscow. The common version was extensively revised and standardized by composer Nikolai Rimsky-Korsakov; this version was published as the 1909 edition of the Obikhod, the last before the Russian Revolution.

The Obikhod style, and the 1909 edition, was predominately used by the Russian Orthodox Church during the decades of Soviet Union rule in the 20th century, displacing both traditional Russian styles, such as the Ruthenian Prostopinije style, and also the chant traditions of Georgia, Armenia, and Carpatho-Russia.

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. A pathological trait the pitch set exhibits is that normalization to edo collapses the range for the dark generator to the octave.

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 (perfect twelfth equivalent) step size ratios and step ratio ranges.

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:

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:

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:

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

The table of Obikhodic intervals below takes the fifth as the generator.

# 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, Kf	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, 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	2\30, 126.32	Hf	-8
major 2nd	2\19, 200.00	3\27, 211.76	3\30, 189.47	H	3
minor 3rd	3\19, 300.00	4\27, 282.35	5\30, 315.79	Jf	-5
major 3rd	4\19, 400.00	6\27, 423.53	6\30, 378.95	J	6
natural 4th	5\19, 500.00	7\27, 494.12	8\30, 505,26	K, Kf	-2
augmented 4th	6\19, 600.00	9\27, 635.29	9\30, 568.42	K, K#	9
diminished 5th		8\27, 564.71	10\30, 631.58	Lf	-10
perfect 5th	7\19, 700.00	10\27, 705.88	11\30, 694.74	L	1
minor 6th	8\19, 800.00	11\27, 776.47	13\30, 821.05	Af	-7
major 6th	9\19, 900.00	13\27, 917.65	14\30, 884.21	A	4
minor 7th	10\19, 1000.00	14\27, 988.235	16\30, 1010.53	Bf	-4
major 7th	11\19, 1100.00	16\27, 1129.42	17\30, 1073.68	B	7
perfect octave	12\19, 1200.00	17\27, 1200.00	19\30, 1200.00	C	-1
augmented octave	13\19, 1300.00	19\27, 1341.18	20\30, 1263.16	C#	10
minor 9th		18\27, 1270.59	21\30, 1326.32	Df	-9
major 9th	14\19, 1400.00	20\27, 1411.76	22\30, 1389.47	D	2
minor 10th	15\19, 1500.00	21\27, 1482.35	24\30, 1515.79	Ef	-6
major 10th	16\19, 1600.00	23\27, 1623.53	25\30, 1578.95	E	5
natural 11th	17\19, 1700.00	24\27, 1694.12	27\30, 1705.26	Ff	-3
augmented 11th	18\19, 1800.00	26\27, 1835.29	28\30, 1768.42	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	17\46, 703.45
L (3g - ~tritave)	2\19, 200.00	3\27, 211.765	5\46, 206.90
s (-8g + 3 ~tritaves)	1\19, 100.00	1\27, 70.59	2\46, 82.76

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	2\46, 82.76	Hf	-8
major 2nd	2\19, 200.00	3\27, 211.76	5\46, 206.90	H	3
minor 3rd	3\19, 300.00	4\27, 282.35	7\46, 289.655	Jf	-5
major 3rd	4\19, 400.00	6\27, 423.53	10\46, 413.79	J	6
natural 4th	5\19, 500.00	7\27, 494.12	12\46, 496.55	K, Kf	-2
augmented 4th	6\19, 600.00	9\27, 635.29	15\46, 620.69	K, K#	9
diminished 5th		8\27, 564.71	14\46, 579.31	Lf	-10
perfect 5th	7\19, 700.00	10\27, 705.88	17\46, 703.45	L	1
minor 6th	8\19, 800.00	11\27, 776.47	19\46, 786.21	Af	-7
major 6th	9\19, 900.00	13\27, 917.65	22\46, 910.34	A	4
minor 7th	10\19, 1000.00	14\27, 988.235	24\46, 993.10	Bf	-4
major 7th	11\19, 1100.00	16\27, 1129.42	27\46, 1117.24	B	7
perfect octave	12\19, 1200.00	17\27, 1200.00	29\46, 1200.00	C	-1
augmented octave	13\19, 1300.00	19\27, 1341.18	32\46, 1324.14	C#	10
minor 9th		18\27, 1270.59	31\46, 1282.76	Df	-9
major 9th	14\19, 1400.00	20\27, 1411.76	34\46, 1406.90	D	2
minor 10th	15\19, 1500.00	21\27, 1482.35	36\46, 1489.66	Ef	-6
major 10th	16\19, 1600.00	23\27, 1623.53	39\46, 1613.79	E	5
natural 11th	17\19, 1700.00	24\27, 1694.12	41\46, 1696.55	Ff	-3
augmented 11th	18\19, 1800.00	26\27, 1835.29	44\46, 1820.69	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 Obikhodic are as follows in various hyposoft Obikhod tunings (~19edt not shown).

	~30edt (soft)	~49edt (semisoft)
generator (g)	11\30, 694.74	18\49, 696.77
L (3g - ~tritave)	3\30, 189.47	5\49, 193.55
s (-8g + 3 ~tritaves)	2\30, 126.32	3\49, 116.13

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	3\49, 116.13	Hf	16/15	-8
major 2nd	3\30, 189.47	5\49, 193.55	H	10/9, 9/8	3
minor 3rd	5\30, 315.79	8\49, 309.68	Jf	6/5	-5
major 3rd	6\30, 378.95	10\49, 387.10	J	5/4	6
natural 4th	8\30, 505,26	13\49, 503.23	K, Kf	4/3	-2
augmented 4th	9\30, 568.42	15\49, 580.65	K, K#	7/5	9
diminished 5th	10\30, 631.58	16\49, 619.35	Lf	10/7	-10
perfect 5th	11\30, 694.74	18\49, 696.77	L	3/2	1
minor 6th	13\30, 821.05	21\49, 812.90	Af	8/5	-7
major 6th	14\30, 884.21	23\49, 890.32	A	5/3	4
minor 7th	16\30, 1010.53	26\49, 1006.45	Bf	16/9, 9/5	-4
major 7th	17\30, 1073.68	28\49, 1083.87	B	15/8	7
perfect octave	19\30, 1200.00	31\49, 1200.00	C	2/1	-1
augmented octave	20\30, 1263.16	33\49, 1277.42	C#	25/24	10
minor 9th	21\30, 1326.32	34\49, 1316.13	Df	15/7	-9
major 9th	22\30, 1389.47	36\49, 1393.55	D	20/9, 9/4	2
minor 10th	24\30, 1515.79	39\49, 1508.68	Ef	12/5	-6
major 10th	25\30, 1578.95	41\49, 1587.10	E	5/2	5
natural 11th	27\30, 1705.26	44\49, 1703.23	Ff	8/3	-3
augmented 11th	28\30, 1768.42	46\49, 1780.65	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	19\52, 690.91
L (3g - ~tritave)	4\41, 184.62	5\52, 181.81
s (-8g + 3 ~tritaves)	3\41, 138.46	4\52, 145.455

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	G#	33/32, 49/48, 36/35, 25/24	11
diminished 2nd	2\41, 92.31	Hff	21/20, 22/21, 26/25	-19
minor 2nd	3\41, 138.46	Hf	12/11, 13/12, 14/13, 16/15	-8
major 2nd	4\41, 184.62	H	9/8, 10/9, 11/10	3
augmented 2nd	5\41, 230.77	H#	8/7, 15/13	14
diminished 3rd	6\41, 276.92	Jff	7/6, 13/11, 33/28	-16
minor 3rd	7\41, 323.08	Jf	135/112, 6/5	-5
major 3rd	8\41, 369.23	J	5/4, 11/9, 16/13	6
augmented 3rd	9\41, 415.38	J#	9/7, 14/11, 33/26	17
diminished 4th	10\41, 461.54	Kf, Kff	21/16, 13/10	-13
natural 4th	11\41, 507.69	K, Kf	75/56, 4/3	-2
augmented 4th	12\41, 553.85	K, K#	11/8, 18/13	9
doubly augmented 4th, doubly diminished 5th	13\41, 600.00	K#, Kx, Lff	7/5, 10/7	20,-21
diminished 5th	14\41, 646.15	Lf	16/11, 13/9	-10
perfect 5th	15\41, 692.31	L	112/75, 3/2	1
augmented 5th	16\41, 738.46	L#	32/21, 20/13	12
diminished 6th	17\41, 784.62	Aff	11/7, 14/9	-18
minor 6th	18\41, 830.77	Af	13/8, 8/5	-7
major 6th	19\41, 876.92	A	5/3, 224/135	4
augmented 6th	20\41, 923.08	A#	12/7, 22/13	15
diminished 7th	21\41, 969.23	Bff	7/4, 26/15	-15
minor 7th	22\41, 1015.38	Bf	9/5, 16/9, 20/11	-4
major 7th	23\41, 1061.54	B	11/6, 13/7, 15/8, 24/13	7
augmented 7th	24\41, 1107.69	B#	21/11, 25/13, 40/21	18
diminished octave	25\41, 1153.85	Cf	64/33, 96/49, 35/18, 48/25	-12
perfect octave	26\41, 1200.00	C	2/1	-1
augmented octave	27\41, 1246.15	C#	33/16, 49/24, 72/35, 25/12	10
doubly augmented octave, diminished 9th	28\41, 1292.31	Cx, Dff	21/10, 44/21, 52/25	21,-20
minor 9th	29\41, 1338.46	Df	24/11, 13/6, 28/13, 32/15	-9
major 9th	30\41, 1384.62	D	9/4, 20/9, 11/5	2
augmented 9th	31\41, 1430.77	D#	16/7, 30/13	13
diminished 10th	32\41, 1476.92	Eff	7/3, 26/11, 33/14	-17
minor 10th	33\41, 1523.08	Ef	135/56, 12/5	-6
major 10th	34\41, 1569.23	E	5/2, 22/9, 32/13	5
augmented 10th	35\41, 1615.38	E#	18/7, 28/11, 33/13	16
diminished 11th	36\41, 1661.54	Ff	21/8, 13/5	-14
natural 11th	37\41, 1709.69	F	75/28, 8/3	-3
augmented 11th	38\41, 1753.85	F#	11/4, 36/13	8
doubly augmented 11th, doubly diminished 12th	39\41, 1800.00	Fx, Gff	14/5, 20/7	19
diminished 12th	40\41, 1846.15	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	G#	12/11, 11/10, 10/9	11
minor 2nd	1\35, 54.55	Hf	36/35, 34/33, 33/32, 32/31	-8
major 2nd	4\35, 218.18	H	9/8, 17/15, 8/7	3
augmented 2nd	7\35, 381.82	H#	5/4, 96/77	14
diminished 3rd	2\35, 109.09	Jff	18/17, 17/16, 16/15, 15/14	-16
minor 3rd	5\35, 272.73	Jf	20/17, 7/6	-5
major 3rd	8\35, 436.36	J	14/11, 9/7, 22/17	6
augmented 3rd	11\35, 600.00	J#	7/5, 24/17, 17/12, 10/7	17
diminished 4th	6\35, 327.27	Kf, Kff	6/5, 17/14, 11/9	-13
natural 4th	9\35, 490.91	K, Kf	4/3	-2
augmented 4th	12\35, 654.55	K, K#	16/11, 22/15	9
diminished 5th	10\35, 545.45	Lf	15/11, 11/8	-10
perfect 5th	13\35, 709.09	L	3/2	1
augmented 5th	16\35, 872.73	L#	18/11, 28/17, 5/3	12
diminished 6th	11\35, 600.00	Aff	7/5, 24/17, 17/12, 10/7	-18
minor 6th	14\35, 763.64	Af	17/11, 14/9, 11/7	-7
major 6th	17\35, 927.27	A	17/10, 12/7	4
augmented 6th	20\35, 1090.91	A#	28/15, 15/8, 32/17, 17/9	15
diminished 7th	15\35, 818.18	Bff	8/5, 77/48	-15
minor 7th	18/35, 981.82	Bf	7/4, 30/17, 16/9	-4
major 7th	21\35, 1145.45	B	31/16, 64/33, 33/17, 35/18	7
augmented 7th	24\35, 1309.09	B#	36/17, 17/8, 32/15, 15/7	18
diminished octave	19\22, 1036.36	Cf	9/5, 11/6, 20/11	-12
perfect octave	22\35, 1200.00	C	2/1	-1
augmented octave	25\35, 1363.64	C#	24/11, 11/5, 20/9	10
minor 9th	23\35, 1254.55	Df	72/35, 68/33, 33/16, 64/31	-9
major 9th	26\35, 1418.18	D	9/4, 34/15, 16/7	2
augmented 9th	29\35, 1581.81	D#	5/2, 192/77	13
diminished 10th	24\35, 1309.09	Eff	36/17, 17/8, 32/15, 15/7	-17
minor 10th	27\35, 1472.72	Ef	40/17, 7/3	-6
major 10th	30\35, 1636.36	E	28/11, 18/7, 44/17	5
augmented 10th	33\35, 1800.00	E#	14/5, 48/17, 17/6, 20/7	16
diminished 11th	28\35, 1527.27	Ff	12/5, 17/7, 22/9	-14
natural 11th	31\35, 1690.91	F	8/3	-3
augmented 11th	34\35, 1854.55	F#	32/11, 44/15	8
diminished 12th	32\35, 1745.45	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, 715.38	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	Note name on H	Approximate ratios	#Gens up
1	0\59, 0.00	0\83, 0.00	0\91, 0.00	0\99, 0.00	0.00	D	H	1/1	0
2	7\59, 227.03	10\83, 230.77	11\91, 231.58	12\99, 232.26	230.55	E	J	8/7	+3
3	14\59, 454.05	20\83, 461.54	22\91, 463.16	24\99, 464.52	461.10	F	K	13/10, 9/7	+6
4	15\59, 486.49	21\83, 484.62	23\91, 484.21	25\99, 483.87	482.06	G	L	4/3	-2
5	22\59, 713.51	31\83, 715.38	34\91, 715.79	37\99, 716.13	712.61	H	A	3/2	+1
6	29\59, 940.54	41\83, 946.15	45\91, 947.36	49\99, 948.39	943.16	J	B	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	C	7/4	-4
8	37\59, 1200.00	52\83, 1200.00	57\91, 1200.00	62\99, 1200.00	1194.67	L	D	2/1	-1
9	44\59, 1427.03	62\83, 1430.77	68\91, 1431.58	74\99, 1432.26	1425.22	A	E	16/7	+2
10	51\59, 1654.05	72\83, 1661.54	79\91, 1663.16	86\99, 1664.52	1655.77	B	F	13/5, 18/7	+5
11	52\59, 1686.49	73\83, 1684.62	80\91, 1684.21	87\99, 1683.87	1676.32	C	G	4/3	-3

Modes

Obikhodic modes are named after the Church modes, but with a “Great” prefix.

UDP	Mode	Name
10|0	LLLsLLLsLLs	(Great) Lydian #8 (Tanagran)
9|1	LLLsLLsLLLs	(Great) Lydian
8|2	LLsLLLsLLLs	(Great) Lydian f4, Ionian #11 (Distomian)
7|3	LLsLLLsLLsL	(Great) Ionian
6|4	LLsLLsLLLsL	(Great) Mixolydian
5|5	LsLLLsLLLsL	(Great) Mixolydian f3, Dorian #10 (Livadeian)
4|6	LsLLLsLLsLL	(Great) Dorian
3|7	LsLLsLLLsLL	(Great) Aeolian
2|8	sLLLsLLLsLL	(Great) Aeolian f2, Phrygian #9 (Theban)
1|9	sLLLsLLsLLL	(Great) Phrygian
0|10	sLLsLLLsLLL	(Great) Locrian

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.

Mode	UDP 1	UDP 2	Name 1	Name 2
LLsLLsLLsLL	6|4 b10	4|6 #3	(Great) Mixolydian f10	(Great) Dorian #3
LLsLLsLLLLs	8|2 b7	6|4 #11	(Great) Lydian f4 f7, Ionian f7 #11 (Distomian Dominant)	(Great) Mixolydian #11
LLsLLLLsLLs	10|0 b4	8|2 #8	(Great) Lydian f4 #8 (Tanagran f4)	(Great) Lydian f4 #8, Ionian #8 #11 (Distomian #8)
LLLLsLLsLLs	1|9 *b1	10|0 #5	(Great) Phrygian *f1	(Great) Lydian #5 #8 (Tanagran #5)
LsLLsLLsLLL	3|7 b9	1|9 #2	(Great) Aeolian f9	(Great) Phrygian #2
LsLLsLLLLsL	5|5 b6	3|7 #10	(Great) Mixolydian f3 f6, Dorian f6 #10 (Livadeian f6)	(Great) Aeolian #10
sLLLLsLLsLL	7|3 b3	5|5 #7	(Great) Ionian f3	(Great) Mixolydian f3 #7, Dorian #7 #10 (Livadeian #7)
LLLsLLsLLsL	9|1 b11	7|3 #4	(Great) Lydian f11	(Great) Ionian #4
sLLsLLsLLLL	0|10 b8	9|1 *#1	(Great) Locrian f8	(Great) Lydian *#1
sLLsLLLLsLL	2|8 b5	0|10 #9	(Great) Aeolian f2 f5, Phrygian f5 #9 (Theban f5)	(Great) Locrian #9
LsLLLLsLLsL	4|6 b2	2|8 #6	(Great) Dorian f2	(Great) Aeolian f2 #6, Phrygian #6 #9 (Theban #6)

Cyclic Permutation order

Spelling 1	Spelling 2	Mode	UDP	Name
GHJKLABCDEFG	LABCDEFGHJKL	LLsLLLsLLLs	8|2	(Great) Distomian
HJKLABCDEFGH	ABCDEFGHJKLA	LsLLLsLLLsL	5|5	(Great) Livadeian
JKLABCDEFGHJ	BCDEFGHJKLAB	sLLLsLLLsLL	2|8	(Great) Theban
KLABCDEFGHJK	CDEFGHJKLABC	LLLsLLLsLLs	10|0	(Great) Tanagran
LABCDEFGHJKL	DEFGHJKLABCD	LLsLLLsLLsL	7|3	(Great) Ionian
ABCDEFGHJKLA	EFGHJKLABCDE	LsLLLsLLsLL	4|6	(Great) Dorian
BCDEFGHJKLAB	FGHJKLABCDEF	sLLLsLLsLLL	1|9	(Great) Phrygian
CDEFGHJKLABC	GHJKLABCDEFG	LLLsLLsLLLs	9|1	(Great) Lydian
DEFGHJKLABCD	HJKLABCDEFGH	LLsLLsLLLsL	6|4	(Great) Mixolydian
EFGHJKLABCDE	JKLABCDEFGHJ	LsLLsLLLsLL	3|7	(Great) Aeolian
FGHJKLABCDEF	KLABCDEFGHJK	sLLsLLLsLLL	0|10	(Great) Locrian

Spelling 1	Spelling 2	Mode	UDP	Name
GHJKLABfCDEFG	LABCDEFfGHJKL	LLsLLsLLLLs	8|2 b7	(Great) Distomian Dominant
HJKLABfCDEFGH	ABCDEFfGHJKLA	LsLLLLsLLsL	5|5 b6	(Great) Livadeian f6
JKLABfCDEFGHJ	BCDEFfGHJKLAB	sLLsLLLLsLL	2|8 b5	(Great) Theban f5
KLABfCDEFGHJK	CDEFfGHJKLABC	LLsLLLLsLLs	10|0 b4	(Great) Tanagran f4
LABfCDEFGHJKL	DEFfGHJKLABCD	LsLLLLsLLsL	7|3 b3	(Great) Ionian f3
ABfCDEFGHJKLA	EFfGHJKLABCDE	sLLLLsLLsLL	4|6 b2	(Great) Dorian f2
BfCDEFGHJKLABf	FfGHJKLABCDEFf	LsLLsLLsLLL	1|9 *b1	(Great) Phrygian *f1
CDEFGHJKLABfC	GHJKLABCDEFfG	LLLsLLsLLsL	9|1 b11	(Great) Lydian f11
DEFGHJKLABfCD	HJKLABCDEFfGH	LLsLLsLLsLL	6|4 b10	(Great) Mixolydian f10
EFGHJKLABfCDE	JKLABCDEFfGHJ	LsLLsLLsLLL	3|7 b9	(Great) Aeolian f9
FGHJKLABfCDEF	KLABCDEFfGHJK	sLLsLLsLLLL	0|10 b8	(Great) Locrian f8

Spelling 1	Spelling 2	Mode	UDP	Name
GHJKLABC#DEFG	LABCDEFG#HJKL	LLsLLsLLLLs	8|2 #8	(Great) Distomian #8
HJKLABC#DEFGH	ABCDEFG#HJKLA	LsLLLLsLLsL	5|5 #7	(Great) Livadeian #7
JKLABC#DEFGHJ	BCDEFG#HJKLAB	sLLLLsLLsLL	2|8 #6	(Great) Theban #6
KLABC#DEFGHJK	CDEFG#HJKLABC	LLLLsLLsLLs	10|0 #5	(Great) Tanagran #5
LABC#DEFGHJKL	DEFG#HJKLABCD	LLLsLLsLLsL	7|3 #4	(Great) Ionian #4
ABC#DEFGHJKLA	EFG#HJKLABCDE	LLsLLsLLsLL	4|6 #3	(Great) Dorian #3
BC#DEFGHJKLAB	FG#HJKLABCDEF	LsLLsLLsLLL	1|9 #2	(Great) Phrygian #2
C#DEFGHJKLABC#	G#HJKLABCDEFG#	sLLsLLsLLLL	9|1 *#1	(Great) Lydian *#1
DEFGHJKLABC#D	HJKLABCDEFG#H	LLsLLsLLLLs	6|4 #11	(Great) Mixolydian #11
EFGHJKLABC#DE	JKLABCDEFG#HJ	LsLLsLLLLsL	3|7 #10	(Great) Aeolian #10
FGHJKLABC#DEF	KLABCDEFG#HJK	sLLsLLLLsLL	0|10 #9	(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 -> Theban -> Tanagran -> Ionian cyclic sequence will resemble the geography of ancient Greece.

Scale tree

Generator	Normalized	Large step	Small step
3\8	720.000	1\5, 240.000	0
19\51	712.500	6\51, 225.000	1\51, 37.500
54\145	712.088	17\145, 224.175	3\145, 39.560
35\94	711.864	11\94, 223.729	2\94, 40.678
16\43	711.111	5\43, 222.222	1\43, 44.444
45\121	710.526	14\121, 221.053	3\121, 47.368
29\78	710.204	9\78, 220.408	2\78, 48.980
42\113	709.859	13\113, 219.718	3\113, 50.704
13\35	709.091	4\35, 218.182	1\35, 54.545
49\132	708.434	15\132, 216.867	4\132, 57.831
36\97	708.197	11\97, 216.393	3\97, 59.016
23\62	707.692	7\62, 215.385	2\62, 61.538
33\89	707.143	10\89, 214.286	3\89, 64.286
43\116	706.849	13\116, 213.699	4\116, 65.753
53\143	706.667	16\143, 213.333	5\143, 66.667
63\170	706.542	19\170, 213.084	6\170, 67.290
73\197	706.452	22\197, 212.903	7\197, 67.742
10\27	705.882	3\27, 211.765	1\27, 70.588
47\127	705.000	14\127, 210.000	5\127, 75.000
37\100	704.762	11\100, 209.524	4\100, 76.190
27\73	704.348	8\73, 208.696	3\27, 78.261
17\46	703.448	5\46, 206.897	2\46, 82.759
41\111	702.857	12\111, 205.714	5\111, 85.714
24\65	702.439	7\65, 204.878	3\65, 87.805
31\84	701.887	9\84, 203.774	4\84, 90.566
…	…	…	…
59\160	700.990	17\160, 201.980	8\160, 95.050
7\19	700.000	2\19, 200.000	1\19, 100.000
123\334	699.526	35\334, 199.052	18\334, 102.370
…	…	…	…
32\87	698.182	9\87, 196.364	5\87, 109.091
25\68	697.674	7\68, 195.349	4\68, 111.628
43\117	697.297	12\117, 194.594	7\117, 113.514
18\49	696.774	5\49, 193.548	3\49, 116.129
29\79	696.000	8\79, 192.000	5\79, 120.000
40\109	695.652	11\109, 191.304	7\109, 121.739
51\139	695.455	14\139, 190.909	9\139, 122.727
11\30	694.737	3\30, 189.474	2\30, 126.316
59\161	694.118	16\161, 188.235	11\161, 129.412
48\131	693.976	13\131, 187.952	9\131, 130.120
37\101	693.750	10\101, 187.500	7\101, 131.250
26\71	693.333	7\71, 186.667	5\71, 133.333
41\112	692.958	11\112, 185.915	8\112, 135.211
15\41	692.308	4\41, 184.615	3\41, 138.462
34\93	691.525	9\93, 183.051	7\93, 142.373
53\145	691.304	14\145, 182.609	11\145, 143.478
19\52	690.909	5\52, 181.818	4\52, 145.455
23\63	690.000	6\63, 180.000	5\63, 150.000
4\11	685.714	1\11, 171.429	1\11, 171.429

See also

8L 3s (3/1-equivalent)

16L 6s (80/9-equivalent)

16L 6s (352/39-equivalent)

16L 6s (64/7-equivalent)