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⟩ ↘ |
┌╥╥╥┬╥╥╥┬╥╥┬┐ │║║║│║║║│║║││ │││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┘
sLLsLLLsLLL
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
- ↑ For relative cents