8L 3s (3/1-equivalent)
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.
| ↖ 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⟩ ↘ |
sLLsLLLsLLL
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.[1]
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.
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.
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[2] 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, 700.70 | 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 only harmonic entropyminimum of the oneirotonic spectrum.
In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. 23edo, 28edoand 33edo can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. 38edo & 43edo 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 48edo is where it really comes into its own in terms of harmonies, providing not only an excellent 3/2, but also 7/4 and archipelago harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.
Beyond that, it's a question of which intervals you want to favor. 53edo has an essentially perfect 3/2, 58edo gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while 63edo does the same for the basic 4:6:7 triad. You could in theory go up to 83edo if you want to favor the 7/4 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 oneirotonic are as follows in various buzzard tunings.
| 38edo | 53edo | 63edo | Optimal (POTE) Buzzard tuning | JI intervals represented (2.3.5.7.13 subgroup) | |
|---|---|---|---|---|---|
| generator (g) | 15\38, 473.68 | 21\53, 475.47 | 25\63, 476.19 | 475.69 | 4/3 21/16 |
| L (3g - octave) | 7/38, 221.04 | 10/53, 226.41 | 12/63, 228.57 | 227.07 | 8/7 |
| s (-5g + 2 octaves) | 1/38, 31.57 | 1/53 22.64 | 1/63 19.05 | 21.55 | 50/49 81/80 91/90 |
Intervals
Sortable table of intervals in the Dylathian mode and their Buzzard interpretations:
| Degree | Size in 38edo | Size in 53edo | Size in 63edo | Size in POTE tuning | Note name on Q | Approximate ratios | #Gens up |
|---|---|---|---|---|---|---|---|
| 1 | 0\38, 0.00 | 0\53, 0.00 | 0\63, 0.00 | 0.00 | Q | 1/1 | 0 |
| 2 | 7\38, 221.05 | 10\53, 226.42 | 12\63, 228.57 | 227.07 | J | 8/7 | +3 |
| 3 | 14\38, 442.10 | 20\53, 452.83 | 24\63, 457.14 | 453.81 | K | 13/10, 9/7 | +6 |
| 4 | 15\38, 473.68 | 21\53, 475.47 | 25\63, 476.19 | 475.63 | L | 21/16 | +1 |
| 5 | 22\38, 694.73 | 31\53, 701.89 | 37\63, 704.76 | 702.54 | M | 3/2 | +4 |
| 6 | 29\38, 915.78 | 41\53, 928.30 | 49\63, 933.33 | 929.45 | N | 12/7, 22/13 | +7 |
| 7 | 30\38, 947.36 | 42\53, 950.94 | 50\63, 952.38 | 951.27 | O | 26/15 | +2 |
| 8 | 37\38, 1168.42 | 52\53, 1177.36 | 62\63, 1180.95 | 1178.18 | P | 98/50, 160/81 | +5 |
Modes
Oneirotonic modes are named after cities in the Dreamlands.
| Mode | UDP | Name |
| LLsLLsLs | 7|0 | Dylathian (də-LA(H)TH-iən) |
| LLsLsLLs | 6|1 | Illarnekian (ill-ar-NEK-iən) |
| LsLLsLLs | 5|2 | Celephaïsian (kel-ə-FAY-zhən) |
| LsLLsLsL | 4|3 | Ultharian (ul-THA(I)R-iən) |
| LsLsLLsL | 3|4 | Mnarian (mə-NA(I)R-iən) |
| sLLsLLsL | 2|5 | Kadathian (kə-DA(H)TH-iən) |
| sLLsLsLL | 1|6 | Hlanithian (lə-NITH-iən) |
| sLsLLsLL | 0|7 | Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn" |
- ↑ John Anthony McGuckin, The Encyclopedia of Eastern Orthodox Christianity, 2010, p406. Quote: "During the Soviet period, Russian obikhod-style choral polyphony all but eradicated the received chant traditions of Georgia, Armenia, and Carpatho-Russia, but currently there is a trend to revive the Znamenny, Iberian, and Ruthenian chant ..."
- ↑ For relative cents