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

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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).
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.<ref>[https://en.wikipedia.org/wiki/Obikhod Obikhod - Wikipedia]. ''en.wikipedia.org''. Retrieved July 28, 2021.</ref>
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.<ref>[https://en.wikipedia.org/wiki/Obikhod Obikhod - Wikipedia]. ''en.wikipedia.org''. Retrieved July 28, 2021.</ref> A pathological trait the pitch set exhibits is that normalization to [[edo]] collapses the range for the [[dark]] [[generator]] to the octave.
==Standing assumptions==
==Standing assumptions==
The tempered generalized Livenka accordion is used in this article to refer to tunings of the pitch set.
The tempered generalized Livenka accordion is used in this article to refer to tunings of the pitch set.
Line 15: Line 15:
The [[TAMNAMS]] system is used in this article to refer to 8L 3s (perfect twelfth equivalent) step size ratios and step ratio ranges.
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), #/f = up/down by chroma (mnemonic f = F molle in Latin).
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:
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 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:
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 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:
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
'''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==
==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<ref>For relative cents</ref> 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¢.
The table of Obikhodic intervals below takes the fifth as the generator.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 67: Line 73:
|6L+2s
|6L+2s
| -2
| -2
|K
|K, Kf
|natural 4th
|natural 4th
|2L+1s
|2L+1s
Line 126: Line 132:
|-
|-
|9
|9
|K#
|K, K#
|augmented 4th
|augmented 4th
|3L
|3L
Line 152: Line 158:
|7L+4s
|7L+4s
|-
|-
| colspan="8" style="text-align:center" |The chromatic 19-note MOS (either [[8L 11s (tritave equivalent)|8L 11s]], [[11L 8s (tritave equivalent)|11L 8s]], or [[19edt]]) also has the following intervals (from some root):
| colspan="8" style="text-align:center" |The chromatic 19-note MOS (either [[8L 11s (perfect twelfth equivalent)|8L 11s]], [[11L 8s (perfect twelfth equivalent)|11L 8s]], or [[19edt]]) also has the following intervals (from some root):
|-
|-
|12
|12
Line 238: Line 244:
|minor 2nd
|minor 2nd
|1\19, 100.00
|1\19, 100.00
|1\27, 70.59 (70.37)
|1\27, 70.59
|2\30, 126.32 (126.67)
|2\30, 126.32  
|Hf
|Hf
| -8
| -8
Line 245: Line 251:
|major 2nd
|major 2nd
|2\19, 200.00
|2\19, 200.00
|3\27, 211.765 (211.11)
|3\27, 211.76
|3\30, 189.47 (190.00)
|3\30, 189.47
|H
|H
|3
|3
Line 252: Line 258:
|minor 3rd
|minor 3rd
|3\19, 300.00
|3\19, 300.00
|4\27, 282.35 (281.48)
|4\27, 282.35
|5\30, 315.79 (316.67)
|5\30, 315.79
|Jf
|Jf
| -5
| -5
Line 259: Line 265:
|major 3rd
|major 3rd
|4\19, 400.00
|4\19, 400.00
|6\27, 423.53 (422.22)
|6\27, 423.53
|6\30, 378.95 (380.00)
|6\30, 378.95
|J
|J
|6
|6
Line 266: Line 272:
|natural 4th
|natural 4th
|5\19, 500.00
|5\19, 500.00
|7\27, 494.12 (493.59)
|7\27, 494.12
|8\30, 505,26 (506.67)
|8\30, 505,26
|K
|K, Kf
| -2
| -2
|-
|-
|augmented 4th
|augmented 4th
| rowspan="2" |6\19, 600.00
| rowspan="2" |6\19, 600.00
|9\27, 635.29 (633.33)
|9\27, 635.29
|9\30, 568.42 (570.00)
|9\30, 568.42
|K#
|K, K#
|9
|9
|-
|-
|diminished 5th
|diminished 5th
|8\27, 564.71 (562.96)
|8\27, 564.71
|10\30, 631.58 (633.33)
|10\30, 631.58
|Lf
|Lf
| -10
| -10
Line 286: Line 292:
|perfect 5th
|perfect 5th
|7\19, 700.00
|7\19, 700.00
|10\27, 705.88 (703.70)
|10\27, 705.88
|11\30, 694.74 (696.67)
|11\30, 694.74
|L
|L
|1
|1
Line 293: Line 299:
|minor 6th
|minor 6th
|8\19, 800.00
|8\19, 800.00
|11\27, 776.47 (774.07)
|11\27, 776.47
|13\30, 821.05 (823.33)
|13\30, 821.05
|Af
|Af
| -7
| -7
Line 300: Line 306:
|major 6th
|major 6th
|9\19, 900.00
|9\19, 900.00
|13\27, 917.65 (914.81)
|13\27, 917.65
|14\30, 884.21 (886.67)
|14\30, 884.21
|A
|A
|4
|4
Line 307: Line 313:
|minor 7th
|minor 7th
|10\19, 1000.00
|10\19, 1000.00
|14\27, 988.235 (985.19)
|14\27, 988.235
|16\30, 1010.53 (1013.33)
|16\30, 1010.53
|Bf
|Bf
| -4
| -4
Line 314: Line 320:
|major 7th
|major 7th
|11\19, 1100.00
|11\19, 1100.00
|16\27, 1129.42 (1125.93)
|16\27, 1129.42
|17\30, 1073.68 (1076.67)
|17\30, 1073.68
|B
|B
|7
|7
Line 321: Line 327:
|perfect octave
|perfect octave
|12\19, 1200.00
|12\19, 1200.00
|17\27, 1200.00 (1196.30)
|17\27, 1200.00
|19\30, 1200.00 (1203.33)
|19\30, 1200.00
|C
|C
| -1
| -1
Line 328: Line 334:
|augmented octave
|augmented octave
| rowspan="2" |13\19, 1300.00
| rowspan="2" |13\19, 1300.00
|19\27, 1341.18 (1337.04)
|19\27, 1341.18
|20\30, 1263.16 (1266.67)
|20\30, 1263.16
|C#
|C#
|10
|10
|-
|-
|minor 9th
|minor 9th
|18\27, 1270.59 (1266.67)
|18\27, 1270.59
|21\30, 1326.32 (1330.00)
|21\30, 1326.32
|Df
|Df
| -9
| -9
Line 341: Line 347:
|major 9th
|major 9th
|14\19, 1400.00
|14\19, 1400.00
|20\27, 1411.765 (1406.07)
|20\27, 1411.76
|22\30, 1389.47 (1393.33)
|22\30, 1389.47
|D
|D
|2
|2
Line 348: Line 354:
|minor 10th
|minor 10th
|15\19, 1500.00
|15\19, 1500.00
|21\27, 1482.35 (1477.78)
|21\27, 1482.35
|24\30, 1515.79 (1520.00)
|24\30, 1515.79
|Ef
|Ef
| -6
| -6
Line 355: Line 361:
|major 10th
|major 10th
|16\19, 1600.00
|16\19, 1600.00
|23\27, 1623.53 (1618.52)
|23\27, 1623.53
|25\30, 1578.95 (1583.33)
|25\30, 1578.95
|E
|E
|5
|5
Line 362: Line 368:
|natural 11th
|natural 11th
|17\19, 1700.00
|17\19, 1700.00
|24\27, 1694.12 (1688.89)
|24\27, 1694.12
|27\30, 1705.26 (1710.00)
|27\30, 1705.26
|Ff
|Ff
| -3
| -3
Line 369: Line 375:
|augmented 11th
|augmented 11th
|18\19, 1800.00
|18\19, 1800.00
|26\27, 1835.29 (1829.63)
|26\27, 1835.29  
|28\30, 1768.42 (1773.33)
|28\30, 1768.42
|F
|F
|8
|8
Line 392: Line 398:
|generator (g)
|generator (g)
|7\19, 700.00
|7\19, 700.00
|10\27, 705.88 (703.70)
|10\27, 705.88
|17\46, 703.45 (702.17)
|17\46, 703.45
|-
|-
|L (3g - ~tritave)
|L (3g - ~tritave)
|2\19, 200.00
|2\19, 200.00
|3\27, 211.765 (211.11)
|3\27, 211.765
|5\46, 206.90 (206.52)
|5\46, 206.90
|-
|-
|s (-8g + 3 ~tritaves)
|s (-8g + 3 ~tritaves)
|1\19, 100.00
|1\19, 100.00
|1\27, 70.59 (70.37)
|1\27, 70.59
|2\46, 82.76 (82.61)
|2\46, 82.76
|}
|}
====Intervals====
====Intervals====
Line 425: Line 431:
|minor 2nd
|minor 2nd
|1\19, 100.00
|1\19, 100.00
|1\27, 70.59 (70.37)
|1\27, 70.59
|2\46, 82.76 (82.61)
|2\46, 82.76
|Hf
|Hf
| -8
| -8
Line 432: Line 438:
|major 2nd
|major 2nd
|2\19, 200.00
|2\19, 200.00
|3\27, 211.765 (211.11)
|3\27, 211.76
|5\46, 206.90 (206.52)
|5\46, 206.90
|H
|H
|3
|3
Line 439: Line 445:
|minor 3rd
|minor 3rd
|3\19, 300.00
|3\19, 300.00
|4\27, 282.35 (281.48)
|4\27, 282.35
|7\46, 289.655 (289.13)
|7\46, 289.655
|Jf
|Jf
| -5
| -5
Line 446: Line 452:
|major 3rd
|major 3rd
|4\19, 400.00
|4\19, 400.00
|6\27, 423.53 (422.22)
|6\27, 423.53
|10\46, 413.79 (413.04)
|10\46, 413.79
|J
|J
|6
|6
Line 453: Line 459:
|natural 4th
|natural 4th
|5\19, 500.00
|5\19, 500.00
|7\27, 494.12 (493.59)
|7\27, 494.12
|12\46, 496.55 (495.65)
|12\46, 496.55
|K
|K, Kf
| -2
| -2
|-
|-
|augmented 4th
|augmented 4th
| rowspan="2" |6\19, 600.00
| rowspan="2" |6\19, 600.00
|9\27, 635.29 (633.33)
|9\27, 635.29
|15\46, 620.69 (619.565)
|15\46, 620.69
|K#
|K, K#
|9
|9
|-
|-
|diminished 5th
|diminished 5th
|8\27, 564.71 (562.96)
|8\27, 564.71
|14\46, 579.31 (578.26)
|14\46, 579.31
|Lf
|Lf
| -10
| -10
Line 473: Line 479:
|perfect 5th
|perfect 5th
|7\19, 700.00
|7\19, 700.00
|10\27, 705.88 (703.70)
|10\27, 705.88
|17\46, 703.45 (702.17)
|17\46, 703.45
|L
|L
|1
|1
Line 480: Line 486:
|minor 6th
|minor 6th
|8\19, 800.00
|8\19, 800.00
|11\27, 776.47 (774.07)
|11\27, 776.47
|19\46, 786.21 (784.78)
|19\46, 786.21
|Af
|Af
| -7
| -7
Line 487: Line 493:
|major 6th
|major 6th
|9\19, 900.00
|9\19, 900.00
|13\27, 917.65 (914.81)
|13\27, 917.65
|22\46, 910.345 (908.70)
|22\46, 910.34
|A
|A
|4
|4
Line 494: Line 500:
|minor 7th
|minor 7th
|10\19, 1000.00
|10\19, 1000.00
|14\27, 988.235 (985.19)
|14\27, 988.235
|24\46, 993.10 (991.30)
|24\46, 993.10
|Bf
|Bf
| -4
| -4
Line 501: Line 507:
|major 7th
|major 7th
|11\19, 1100.00
|11\19, 1100.00
|16\27, 1129.42 (1125.93)
|16\27, 1129.42
|27\46, 1117.24 (1115.22)
|27\46, 1117.24
|B
|B
|7
|7
Line 508: Line 514:
|perfect octave
|perfect octave
|12\19, 1200.00
|12\19, 1200.00
|17\27, 1200.00 (1196.30)
|17\27, 1200.00
|29\46, 1200.00 (1197.83)
|29\46, 1200.00
|C
|C
| -1
| -1
Line 515: Line 521:
|augmented octave
|augmented octave
| rowspan="2" |13\19, 1300.00
| rowspan="2" |13\19, 1300.00
|19\27, 1341.18 (1337.04)
|19\27, 1341.18
|32\46, 1324.14 (1321.74)
|32\46, 1324.14
|C#
|C#
|10
|10
|-
|-
|minor 9th
|minor 9th
|18\27, 1270.59 (1266.67)
|18\27, 1270.59
|31\46, 1282.76 (1280.435)
|31\46, 1282.76
|Df
|Df
| -9
| -9
Line 528: Line 534:
|major 9th
|major 9th
|14\19, 1400.00
|14\19, 1400.00
|20\27, 1411.765 (1406.07)
|20\27, 1411.76
|34\46, 1406.90 (1404.35)
|34\46, 1406.90
|D
|D
|2
|2
Line 535: Line 541:
|minor 10th
|minor 10th
|15\19, 1500.00
|15\19, 1500.00
|21\27, 1482.35 (1477.78)
|21\27, 1482.35
|36\46, 1489.655 (1486.96)
|36\46, 1489.66
|Ef
|Ef
| -6
| -6
Line 542: Line 548:
|major 10th
|major 10th
|16\19, 1600.00
|16\19, 1600.00
|23\27, 1623.53 (1618.52)
|23\27, 1623.53
|39\46, 1613.79 (1610.87)
|39\46, 1613.79
|E
|E
|5
|5
Line 549: Line 555:
|natural 11th
|natural 11th
|17\19, 1700.00
|17\19, 1700.00
|24\27, 1694.12 (1688.89)
|24\27, 1694.12
|41\46, 1696.55 (1693.48)
|41\46, 1696.55
|Ff
|Ff
| -3
| -3
Line 556: Line 562:
|augmented 11th
|augmented 11th
|18\19, 1800.00
|18\19, 1800.00
|26\27, 1835.29 (1829.63)
|26\27, 1835.29  
|44\46, 1820.69 (1817.39)
|44\46, 1820.69
|F
|F
|8
|8
Line 565: Line 571:
*The large step is between near the meantone and near the Pythagorean 9/8 whole tone, somewhere between as in [[19edo]] and as in [[17edo|12edo]].
*The large step is between near the meantone and near the Pythagorean 9/8 whole tone, somewhere between as in [[19edo]] and as in [[17edo|12edo]].
*The major 3rd (made of two large steps) is a near-[[Just intonation|just]] to near-[[Pythagorean]] major third.
*The major 3rd (made of two large steps) is a near-[[Just intonation|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).
The sizes of the generator, large step and small step of Obikhodic are as follows in various hyposoft Obikhod tunings (~19edt not shown).
{| class="wikitable right-2 right-3 right-4 right-5"
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
Line 573: Line 579:
|-
|-
|generator (g)
|generator (g)
|11\30, 694.74 (696.67)
|11\30, 694.74
|18\49, 696.77 (697.96)
|18\49, 696.77
|-
|-
|L (3g - ~tritave)
|L (3g - ~tritave)
|3\30, 189.47 (190.00)
|3\30, 189.47
|5\49, 193.55 (193.87)
|5\49, 193.55  
|-
|-
|s (-8g + 3 ~tritaves)
|s (-8g + 3 ~tritaves)
|2\30, 126.32 (126.67)
|2\30, 126.32  
|3\49, 116.13 (116.33)
|3\49, 116.13
|}
|}
====Intervals====
====Intervals====
Line 603: Line 609:
|-
|-
|minor 2nd
|minor 2nd
|2\30, 126.32 (126.67)
|2\30, 126.32  
|3\49, 116.13 (116.33)
|3\49, 116.13
|Hf
|Hf
|16/15
|16/15
Line 610: Line 616:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 2nd
|major 2nd
|3\30, 189.47 (190.00)
|3\30, 189.47
|5\49, 193.55 (193.87)
|5\49, 193.55  
|H
|H
|10/9, 9/8
|10/9, 9/8
Line 617: Line 623:
|-
|-
|minor 3rd
|minor 3rd
|5\30, 315.79 (316.67)
|5\30, 315.79
|8\49, 309.68 (310.20)
|8\49, 309.68
|Jf
|Jf
|6/5
|6/5
Line 624: Line 630:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|6\30, 378.95 (380.00)
|6\30, 378.95
|10\49, 387.10 (387.755)
|10\49, 387.10
|J
|J
|5/4
|5/4
Line 631: Line 637:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|natural 4th
|natural 4th
|8\30, 505,26 (506.67)
|8\30, 505,26
|13\49, 503.23 (504.08)
|13\49, 503.23
|K
|K, Kf
|4/3
|4/3
| -2
| -2
|-
|-
|augmented 4th
|augmented 4th
|9\30, 568.42 (570.00)
|9\30, 568.42
|15\49, 580.645 (581.63)
|15\49, 580.65
|K#
|K, K#
|7/5
|7/5
|9
|9
|-
|-
|diminished 5th
|diminished 5th
|10\30, 631.58 (633.33)
|10\30, 631.58
|16\49, 619.355 (620.41)
|16\49, 619.35
|Lf
|Lf
|10/7
|10/7
Line 652: Line 658:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect 5th
|perfect 5th
|11\30, 694.74 (696.67)
|11\30, 694.74
|18\49, 696.77 (697.96)
|18\49, 696.77
|L
|L
|3/2
|3/2
Line 659: Line 665:
|-
|-
|minor 6th
|minor 6th
|13\30, 821.05 (823.33)
|13\30, 821.05
|21\49, 812.90 (814.29)
|21\49, 812.90
|Af
|Af
|8/5
|8/5
Line 666: Line 672:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6th
|major 6th
|14\30, 884.21 (886.67)
|14\30, 884.21
|23\49, 890.32 (891.84)
|23\49, 890.32
|A
|A
|5/3
|5/3
Line 673: Line 679:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|16\30, 1010.53 (1013.33)
|16\30, 1010.53
|26\49, 1006.45 (1008.16)
|26\49, 1006.45
|Bf
|Bf
|16/9, 9/5
|16/9, 9/5
Line 680: Line 686:
|-
|-
|major 7th
|major 7th
|17\30, 1073.68 (1076.67)
|17\30, 1073.68
|28\49, 1083.87 (1085.71)
|28\49, 1083.87
|B
|B
|15/8
|15/8
Line 687: Line 693:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect octave
|perfect octave
|19\30, 1200.00 (1203.33)
|19\30, 1200.00
|31\49, 1200.00 (1202.04)
|31\49, 1200.00
|C
|C
|2/1
|2/1
Line 694: Line 700:
|-
|-
|augmented octave
|augmented octave
|20\30, 1263.16 (1266.67)
|20\30, 1263.16
|33\49, 1277.42 (1279.59)
|33\49, 1277.42
|C#
|C#
|25/24
|25/24
Line 701: Line 707:
|-
|-
|minor 9th
|minor 9th
|21\30, 1326.32 (1330.00)
|21\30, 1326.32
|34\49, 1316.13 (1318.37)
|34\49, 1316.13
|Df
|Df
|15/7
|15/7
Line 708: Line 714:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 9th
|major 9th
|22\30, 1389.47 (1393.33)
|22\30, 1389.47
|36\49, 1393.55 (1395.92)
|36\49, 1393.55
|D
|D
|20/9, 9/4
|20/9, 9/4
Line 715: Line 721:
|-
|-
|minor 10th
|minor 10th
|24\30, 1515.79 (1520.00)
|24\30, 1515.79
|39\49, 1508.68 (1512.245)
|39\49, 1508.68
|Ef
|Ef
|12/5
|12/5
Line 722: Line 728:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 10th
|major 10th
|25\30, 1578.95 (1583.33)
|25\30, 1578.95
|41\49, 1587.10 (1590.80)
|41\49, 1587.10
|E
|E
|5/2
|5/2
Line 729: Line 735:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|natural 11th
|natural 11th
|27\30, 1705.26 (1710.00)
|27\30, 1705.26
|44\49, 1703.23 (1706.13j
|44\49, 1703.23
|Ff
|Ff
|8/3
|8/3
Line 736: Line 742:
|-
|-
|augmented 11th
|augmented 11th
|28\30, 1768.42 (1773.33)
|28\30, 1768.42
|46\49, 1780.645 (1783.67)
|46\49, 1780.65
|F
|F
|14/5
|14/5
Line 753: Line 759:
|-
|-
|generator (g)
|generator (g)
|15\41, 692.31 (695.12)
|15\41, 692.31
|19\52, 690.91 (694.23)
|19\52, 690.91
|-
|-
|L (3g - ~tritave)
|L (3g - ~tritave)
|4\41, 184.615 (185.37)
|4\41, 184.62
|5\52, 181.81 (182.69)
|5\52, 181.81
|-
|-
|s (-8g + 3 ~tritaves)
|s (-8g + 3 ~tritaves)
|3\41, 138.46 (139.02)
|3\41, 138.46  
|4\52, 145.455 (146.15)
|4\52, 145.455
|}
|}
====Intervals====
====Intervals====
Line 781: Line 787:
|-
|-
|chroma
|chroma
|1\41, 46.15 (46.34)
|1\41, 46.15
|G#
|G#
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
Line 787: Line 793:
|-
|-
|diminished 2nd
|diminished 2nd
|2\41, 92.31 (92.68)
|2\41, 92.31
|Hff
|Hff
|[[21/20]], [[22/21]], [[26/25]]
|[[21/20]], [[22/21]], [[26/25]]
Line 793: Line 799:
|-
|-
|minor 2nd
|minor 2nd
|3\41, 138.46 (139.02)
|3\41, 138.46
|Hf
|Hf
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
Line 799: Line 805:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 2nd
|major 2nd
|4\41, 184.615 (185.37)
|4\41, 184.62
|H
|H
|[[9/8]], [[10/9]], [[11/10]]
|[[9/8]], [[10/9]], [[11/10]]
Line 805: Line 811:
|-
|-
|augmented 2nd
|augmented 2nd
|5\41, 230.77 (231.71)
|5\41, 230.77
|H#
|H#
|[[8/7]], [[15/13]]
|[[8/7]], [[15/13]]
Line 811: Line 817:
|-
|-
|diminished 3rd
|diminished 3rd
|6\41, 276.92 (278.05)
|6\41, 276.92
|Jff
|Jff
|[[7/6]], [[13/11]], [[33/28]]
|[[7/6]], [[13/11]], [[33/28]]
Line 817: Line 823:
|-
|-
|minor 3rd
|minor 3rd
|7\41, 323.08 (324.39)
|7\41, 323.08
|Jf
|Jf
|[[135/112]], [[6/5]]
|[[135/112]], [[6/5]]
Line 823: Line 829:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|8\41, 369.23 (370.73)
|8\41, 369.23
|J
|J
|[[5/4]], [[11/9]], [[16/13]]
|[[5/4]], [[11/9]], [[16/13]]
Line 829: Line 835:
|-
|-
|augmented 3rd
|augmented 3rd
|9\41, 415.385 (417.07)
|9\41, 415.38
|J#
|J#
|[[9/7]], [[14/11]], [[33/26]]
|[[9/7]], [[14/11]], [[33/26]]
Line 835: Line 841:
|-
|-
|diminished 4th
|diminished 4th
|10\41, 461.54 (463.415)
|10\41, 461.54
|Kff
|Kf, Kff
|[[21/16]], [[13/10]]
|[[21/16]], [[13/10]]
| -13
| -13
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|natural 4th
|natural 4th
|11\41, 507.69 (509.76)
|11\41, 507.69
|Kf
|K, Kf
|[[75/56]], [[4/3]]
|[[75/56]], [[4/3]]
| -2
| -2
|-
|-
|augmented 4th
|augmented 4th
|12\41, 553.85 (556.10)
|12\41, 553.85
|K
|K, K#
|[[11/8]], [[18/13]]
|[[11/8]], [[18/13]]
|9
|9
|-
|-
|doubly augmented 4th, doubly diminished 5th
|doubly augmented 4th, doubly diminished 5th
|13\41, 600.00 (602.44)
|13\41, 600.00
|K#, Lff
|K#, Kx, Lff
|[[7/5]], [[10/7]]
|[[7/5]], [[10/7]]
|20,-21
|20,-21
|-
|-
|diminished 5th
|diminished 5th
|14\41, 646.15 (648.78)
|14\41, 646.15
|Lf
|Lf
|[[16/11]], [[13/9]]
|[[16/11]], [[13/9]]
Line 865: Line 871:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect 5th
|perfect 5th
|15\41, 692.31 (695.12)
|15\41, 692.31
|L
|L
|[[112/75]], [[3/2]]
|[[112/75]], [[3/2]]
Line 871: Line 877:
|-
|-
|augmented 5th
|augmented 5th
|16\41, 738.46 (741.46)
|16\41, 738.46
|L#
|L#
|[[32/21]], [[20/13]]
|[[32/21]], [[20/13]]
Line 877: Line 883:
|-
|-
|diminished 6th
|diminished 6th
|17\41, 784.615 (787.805)
|17\41, 784.62
|Aff
|Aff
|[[11/7]], [[14/9]]
|[[11/7]], [[14/9]]
Line 883: Line 889:
|-
|-
|minor 6th
|minor 6th
|18\41, 830.77 (834.15)
|18\41, 830.77
|Af
|Af
|[[13/8]], [[8/5]]
|[[13/8]], [[8/5]]
Line 889: Line 895:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6th
|major 6th
|19\41, 876.92 (880.49)
|19\41, 876.92
|A
|A
|[[5/3]], [[224/135]]
|[[5/3]], [[224/135]]
Line 895: Line 901:
|-
|-
|augmented 6th
|augmented 6th
|20\41, 923.08 (926.83)
|20\41, 923.08
|A#
|A#
|[[12/7]], [[22/13]]
|[[12/7]], [[22/13]]
Line 901: Line 907:
|-
|-
|diminished 7th
|diminished 7th
|21\41, 969.23 (973.17)
|21\41, 969.23
|Bff
|Bff
|[[7/4]], [[26/15]]
|[[7/4]], [[26/15]]
Line 907: Line 913:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|22\41, 1015.385 (1019.51)
|22\41, 1015.38
|Bf
|Bf
|[[9/5]], [[16/9]], [[20/11]]
|[[9/5]], [[16/9]], [[20/11]]
Line 913: Line 919:
|-
|-
|major 7th
|major 7th
|23\41, 1061.54 (1065.85)
|23\41, 1061.54
|B
|B
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
Line 919: Line 925:
|-
|-
|augmented 7th
|augmented 7th
|24\41, 1107.69 (1112.195)
|24\41, 1107.69
|B#
|B#
|[[21/11]], [[25/13]], [[40/21]]
|[[21/11]], [[25/13]], [[40/21]]
Line 925: Line 931:
|-
|-
|diminished octave
|diminished octave
|25\41, 1153.85 (1158.54)
|25\41, 1153.85
|Cf
|Cf
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
Line 931: Line 937:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect octave
|perfect octave
|26\41, 1200.00 (1204.88)
|26\41, 1200.00
|C
|C
|2/1
|2/1
Line 937: Line 943:
|-
|-
|augmented octave
|augmented octave
|27\41, 1246.15 (1251.22)
|27\41, 1246.15
|C#
|C#
|33/16, 49/24, 72/35, 25/12
|33/16, 49/24, 72/35, 25/12
Line 943: Line 949:
|-
|-
|doubly augmented octave, diminished 9th
|doubly augmented octave, diminished 9th
|28\41, 1292.31 (1297.56)
|28\41, 1292.31
|Cx, Dff
|Cx, Dff
|21/10, 44/21, 52/25
|21/10, 44/21, 52/25
Line 949: Line 955:
|-
|-
|minor 9th
|minor 9th
|29\41, 1338.46 (1343.90)
|29\41, 1338.46
|Df
|Df
|24/11, 13/6, 28/13, 32/15
|24/11, 13/6, 28/13, 32/15
Line 955: Line 961:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 9th
|major 9th
|30\41, 1384.615 (1390.24)
|30\41, 1384.62
|D
|D
|9/4, 20/9, 11/5
|9/4, 20/9, 11/5
Line 961: Line 967:
|-
|-
|augmented 9th
|augmented 9th
|31\41, 1430.77 (1436.595)
|31\41, 1430.77
|D#
|D#
|16/7, 30/13
|16/7, 30/13
Line 967: Line 973:
|-
|-
|diminished 10th
|diminished 10th
|32\41, 1476.92 (1492.93)
|32\41, 1476.92
|Eff
|Eff
|7/3, 26/11, 33/14
|7/3, 26/11, 33/14
Line 973: Line 979:
|-
|-
|minor 10th
|minor 10th
|33\41, 1523.08 (1529.27)
|33\41, 1523.08
|Ef
|Ef
|135/56, 12/5
|135/56, 12/5
Line 979: Line 985:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 10th
|major 10th
|34\41, 1569.23 (1575.61)
|34\41, 1569.23
|E
|E
|5/2, 22/9, 32/13
|5/2, 22/9, 32/13
Line 985: Line 991:
|-
|-
|augmented 10th
|augmented 10th
|35\41, 1615.385 (1621.95)
|35\41, 1615.38
|E#
|E#
|18/7, 28/11, 33/13
|18/7, 28/11, 33/13
Line 991: Line 997:
|-
|-
|diminished 11th
|diminished 11th
|36\41, 1661.54 (1668.29)
|36\41, 1661.54
|Ff
|Ff
|21/8, 13/5
|21/8, 13/5
Line 997: Line 1,003:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|natural 11th
|natural 11th
|37\41, 1709.69 (1714.63)
|37\41, 1709.69
|F
|F
|75/28, 8/3
|75/28, 8/3
Line 1,003: Line 1,009:
|-
|-
|augmented 11th
|augmented 11th
|38\41, 1753.85 (1760.98)
|38\41, 1753.85
|F#
|F#
|11/4, 36/13
|11/4, 36/13
Line 1,009: Line 1,015:
|-
|-
|doubly augmented 11th, doubly diminished 12th
|doubly augmented 11th, doubly diminished 12th
|39\41, 1800.00 (1807.32)
|39\41, 1800.00
|Fx, Gff
|Fx, Gff
|14/5, 20/7
|14/5, 20/7
Line 1,015: Line 1,021:
|-
|-
|diminished 12th
|diminished 12th
|40\41, 1846.15 (1853.66)
|40\41, 1846.15
|Gf
|Gf
|32/11, 26/9
|32/11, 26/9
Line 1,039: Line 1,045:
|-
|-
|chroma
|chroma
|3\35, 163.64 (162.86)
|3\35, 163.64
|G#
|G#
|[[12/11]], [[11/10]], [[10/9]]
|[[12/11]], [[11/10]], [[10/9]]
Line 1,045: Line 1,051:
|-
|-
|minor 2nd
|minor 2nd
|1\35, 54.545 (54.29)
|1\35, 54.55
|Hf
|Hf
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
Line 1,051: Line 1,057:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 2nd
|major 2nd
|4\35, 218.18 (217.14)
|4\35, 218.18
|H
|H
|[[9/8]], [[17/15]], [[8/7]]
|[[9/8]], [[17/15]], [[8/7]]
Line 1,057: Line 1,063:
|-
|-
|augmented 2nd
|augmented 2nd
|7\35, 381.818 (380.00)
|7\35, 381.82
|H#
|H#
|[[5/4]], [[96/77]]
|[[5/4]], [[96/77]]
Line 1,063: Line 1,069:
|-
|-
|diminished 3rd
|diminished 3rd
|2\35, 109.09 (108.57)
|2\35, 109.09
|Jff
|Jff
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
Line 1,069: Line 1,075:
|-
|-
|minor 3rd
|minor 3rd
|5\35, 272.73 (271.43)
|5\35, 272.73
|Jf
|Jf
|[[20/17]], [[7/6]]
|[[20/17]], [[7/6]]
Line 1,075: Line 1,081:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|8\35, 436.36 (434.29)
|8\35, 436.36
|J
|J
|[[14/11]], [[9/7]], [[22/17]]
|[[14/11]], [[9/7]], [[22/17]]
Line 1,081: Line 1,087:
|-
|-
|augmented 3rd
|augmented 3rd
|11\35, 600.00 (542.86)
|11\35, 600.00
|J#
|J#
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
Line 1,087: Line 1,093:
|-
|-
|diminished 4th
|diminished 4th
|6\35, 327.27 (325.71)
|6\35, 327.27
|Kff
|Kf, Kff
|[[6/5]], [[17/14]], [[11/9]]
|[[6/5]], [[17/14]], [[11/9]]
| -13
| -13
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|natural 4th
|natural 4th
|9\35, 490.91 (488.57)
|9\35, 490.91
|Kf
|K, Kf
|[[4/3]]
|[[4/3]]
| -2
| -2
|-
|-
|augmented 4th
|augmented 4th
|12\35, 654.545 (651.43)
|12\35, 654.55
|K
|K, K#
|[[16/11]], [[22/15]]
|[[16/11]], [[22/15]]
|9
|9
|-
|-
|diminished 5th
|diminished 5th
|10\35, 545.455 (542.86)
|10\35, 545.45
|Lf
|Lf
|[[15/11]], [[11/8]]
|[[15/11]], [[11/8]]
Line 1,111: Line 1,117:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect 5th
|perfect 5th
|13\35, 709.09 (705.71)
|13\35, 709.09
|L
|L
|[[3/2]]
|[[3/2]]
Line 1,117: Line 1,123:
|-
|-
|augmented 5th
|augmented 5th
|16\35, 872.73 (868.57)
|16\35, 872.73
|L#
|L#
|[[18/11]], [[28/17]], [[5/3]]
|[[18/11]], [[28/17]], [[5/3]]
Line 1,123: Line 1,129:
|-
|-
|diminished 6th
|diminished 6th
|11\35, 600.00 (597.14)
|11\35, 600.00
|Aff
|Aff
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
Line 1,129: Line 1,135:
|-
|-
|minor 6th
|minor 6th
|14\35, 763.64 (760.00)
|14\35, 763.64
|Af
|Af
|[[17/11]], [[14/9]], [[11/7]]
|[[17/11]], [[14/9]], [[11/7]]
Line 1,135: Line 1,141:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6th
|major 6th
|17\35, 927.27 (822.86)
|17\35, 927.27
|A
|A
|[[17/10]], [[12/7]]
|[[17/10]], [[12/7]]
Line 1,141: Line 1,147:
|-
|-
|augmented 6th
|augmented 6th
|20\35, 1090.909 (1085.71)
|20\35, 1090.91
|A#
|A#
|[[28/15]], [[15/8]], [[32/17]], [[17/9]]
|[[28/15]], [[15/8]], [[32/17]], [[17/9]]
Line 1,147: Line 1,153:
|-
|-
|diminished 7th
|diminished 7th
|15\35, 818.182 (814.29)
|15\35, 818.18
|Bff
|Bff
|[[8/5]], [[77/48]]
|[[8/5]], [[77/48]]
Line 1,153: Line 1,159:
|-
|-
|minor 7th
|minor 7th
|18/35, 981.82 (977.14)
|18/35, 981.82
|Bf
|Bf
|[[7/4]], [[30/17]], [[16/9]]
|[[7/4]], [[30/17]], [[16/9]]
Line 1,159: Line 1,165:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 7th
|major 7th
|21\35, 1145.455 (1140.00)
|21\35, 1145.45
|B
|B
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
Line 1,165: Line 1,171:
|-
|-
|augmented 7th
|augmented 7th
|24\35, 1309.09 (1302.86)
|24\35, 1309.09
|B#
|B#
|36/17, 17/8, 32/15, 15/7
|36/17, 17/8, 32/15, 15/7
Line 1,171: Line 1,177:
|-
|-
|diminished octave
|diminished octave
|19\22, 1036.36 (1031.43)
|19\22, 1036.36
|Cf
|Cf
|[[9/5]], [[11/6]], [[20/11]]
|[[9/5]], [[11/6]], [[20/11]]
Line 1,177: Line 1,183:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect octave
|perfect octave
|22\35, 1200.00 (1194.29)
|22\35, 1200.00
|C
|C
|[[2/1]]
|[[2/1]]
Line 1,183: Line 1,189:
|-
|-
|augmented octave
|augmented octave
|25\35, 1363.64 (1357.14)
|25\35, 1363.64
|C#
|C#
|24/11, 11/5, 20/9
|24/11, 11/5, 20/9
Line 1,189: Line 1,195:
|-
|-
|minor 9th
|minor 9th
|23\35, 1254.55 (1248.57)
|23\35, 1254.55
|Df
|Df
|72/35, 68/33, 33/16, 64/31
|72/35, 68/33, 33/16, 64/31
Line 1,195: Line 1,201:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 9th
|major 9th
|26\35, 1418.18 (1411.43)
|26\35, 1418.18
|D
|D
|9/4, 34/15, 16/7
|9/4, 34/15, 16/7
Line 1,201: Line 1,207:
|-
|-
|augmented 9th
|augmented 9th
|29\35, 1581.81 (1574.29)
|29\35, 1581.81
|D#
|D#
|5/2, 192/77
|5/2, 192/77
Line 1,207: Line 1,213:
|-
|-
|diminished 10th
|diminished 10th
|24\35, 1309.09 (1302.86)
|24\35, 1309.09
|Eff
|Eff
|36/17, 17/8, 32/15, 15/7
|36/17, 17/8, 32/15, 15/7
Line 1,213: Line 1,219:
|-
|-
|minor 10th
|minor 10th
|27\35, 1472.72 (1465.71)
|27\35, 1472.72
|Ef
|Ef
|40/17, 7/3
|40/17, 7/3
Line 1,219: Line 1,225:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 10th
|major 10th
|30\35, 1636.36 (1628.57)
|30\35, 1636.36
|E
|E
|28/11, 18/7, 44/17
|28/11, 18/7, 44/17
Line 1,225: Line 1,231:
|-
|-
|augmented 10th
|augmented 10th
|33\35, 1800.00 (1791.43)
|33\35, 1800.00
|E#
|E#
|14/5, 48/17, 17/6, 20/7
|14/5, 48/17, 17/6, 20/7
Line 1,231: Line 1,237:
|-
|-
|diminished 11th
|diminished 11th
|28\35, 1527.27 (1520.00)
|28\35, 1527.27
|Ff
|Ff
|12/5, 17/7, 22/9
|12/5, 17/7, 22/9
Line 1,237: Line 1,243:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|natural 11th
|natural 11th
|31\35, 1690.91 (1682.86)
|31\35, 1690.91
|F
|F
|8/3
|8/3
Line 1,243: Line 1,249:
|-
|-
|augmented 11th
|augmented 11th
|34\35, 1854.545 (1845.71)
|34\35, 1854.55
|F#
|F#
|32/11, 44/15
|32/11, 44/15
Line 1,249: Line 1,255:
|-
|-
|diminished 12th
|diminished 12th
|32\35, 1745.455 (1737.14)
|32\35, 1745.45
|Gf
|Gf
|30/11, 11/4
|30/11, 11/4
Line 1,274: Line 1,280:
|generator (g)
|generator (g)
|22\59, 713.51
|22\59, 713.51
|31\83, 715.385
|31\83, 715.38
|34\91, 715.79
|34\91, 715.79
|37\99, 716.13
|37\99, 716.13
Line 1,307: Line 1,313:
!Size in PHTE tuning
!Size in PHTE tuning
!Note name on D
!Note name on D
!Note name on H
! class="unsortable" |Approximate ratios
! class="unsortable" |Approximate ratios
!#Gens up
!#Gens up
Line 1,317: Line 1,324:
|0.00
|0.00
|D
|D
|H
|1/1
|1/1
|0
|0
Line 1,327: Line 1,335:
|230.55
|230.55
|E
|E
|J
|8/7
|8/7
| +3
| +3
Line 1,337: Line 1,346:
|461.10
|461.10
|F
|F
|K
|13/10, 9/7
|13/10, 9/7
| +6
| +6
Line 1,342: Line 1,352:
|4
|4
|15\59, 486.49
|15\59, 486.49
|21\83, 484.615
|21\83, 484.62
|23\91, 484.21
|23\91, 484.21
|25\99, 483.87
|25\99, 483.87
|482.06
|482.06
|G
|G
|L
|4/3
|4/3
| -2
| -2
Line 1,352: Line 1,363:
|5
|5
|22\59, 713.51
|22\59, 713.51
|31\83, 715.385
|31\83, 715.38
|34\91, 715.79
|34\91, 715.79
|37\99, 716.13
|37\99, 716.13
|712.61
|712.61
|H
|H
|A
|3/2
|3/2
| +1
| +1
Line 1,367: Line 1,379:
|943.16
|943.16
|J
|J
|B
|12/7, 26/15
|12/7, 26/15
| +4
| +4
Line 1,377: Line 1,390:
|964.12
|964.12
|K
|K
|C
|7/4
|7/4
| -4
| -4
Line 1,387: Line 1,401:
|1194.67
|1194.67
|L
|L
|D
|2/1
|2/1
| -1
| -1
Line 1,393: Line 1,408:
|44\59, 1427.03
|44\59, 1427.03
|62\83, 1430.77
|62\83, 1430.77
|68\91, 1430.58
|68\91, 1431.58
|74\99, 1432.26
|74\99, 1432.26
|1425.22
|1425.22
|A
|A
|E
|16/7
|16/7
| +2
| +2
Line 1,407: Line 1,423:
|1655.77
|1655.77
|B
|B
|F
|13/5, 18/7
|13/5, 18/7
| +5
| +5
Line 1,413: Line 1,430:
|52\59, 1686.49
|52\59, 1686.49
|73\83,
|73\83,
1684.615
1684.62
|80\91,
|80\91,
1684.21
1684.21
Line 1,419: Line 1,436:
|1676.32
|1676.32
|C
|C
|G
|4/3
|4/3
| -3
| -3
Line 1,426: Line 1,444:
{| class="wikitable"
{| class="wikitable"
|-
|-
| style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
| style="text-align:center;" |'''Mode'''
| style="text-align:center;" |'''Mode'''
| style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
| style="text-align:center;" |'''Name'''
| style="text-align:center;" |'''Name'''
|-
|-
| style="text-align:center;" |<nowiki>10|0</nowiki>
|LLLsLLLsLLs
|LLLsLLLsLLs
| style="text-align:center;" |<nowiki>10|0</nowiki>
|(Great) Lydian #8 (Tanagran)
|(Great) Lydian #8 (Tanagran)
|-
|-
| style="text-align:center;" |<nowiki>9|1</nowiki>
|LLLsLLsLLLs
|LLLsLLsLLLs
| style="text-align:center;" |<nowiki>9|1</nowiki>
|(Great) Lydian
|(Great) Lydian
|-
|-
| style="text-align:center;" |<nowiki>8|2</nowiki>
|LLsLLLsLLLs
|LLsLLLsLLLs
| style="text-align:center;" |<nowiki>8|2</nowiki>
|(Great) Lydian f4, Ionian #11 (Distomian)
|(Great) Lydian b4, Ionian #11 (Distomian)
|-
|-
| style="text-align:center;" |<nowiki>7|3</nowiki>
| |LLsLLLsLLsL
| |LLsLLLsLLsL
| style="text-align:center;" |<nowiki>7|3</nowiki>
| |(Great) Ionian
| |(Great) Ionian
|-
|-
| style="text-align:center;" |<nowiki>6|4</nowiki>
| |LLsLLsLLLsL
| |LLsLLsLLLsL
| style="text-align:center;" |<nowiki>6|4</nowiki>
| |(Great) Mixolydian
| |(Great) Mixolydian
|-
|-
| style="text-align:center;" |<nowiki>5|5</nowiki>
| |LsLLLsLLLsL
| |LsLLLsLLLsL
| style="text-align:center;" |<nowiki>5|5</nowiki>
| |(Great) Mixolydian f3, Dorian #10 (Livadeian)
| |(Great) Mixolydian b3, Dorian #10 (Livadeian)
|-
|-
| style="text-align:center;" |<nowiki>4|6</nowiki>
| |LsLLLsLLsLL
| |LsLLLsLLsLL
| style="text-align:center;" |<nowiki>4|6</nowiki>
| |(Great) Dorian
| |(Great) Dorian
|-
|-
| style="text-align:center;" |<nowiki>3|7</nowiki>
| |LsLLsLLLsLL
| |LsLLsLLLsLL
| style="text-align:center;" |<nowiki>3|7</nowiki>
| |(Great) Aeolian
| |(Great) Aeolian
|-
|-
| style="text-align:center;" |<nowiki>2|8</nowiki>
| |sLLLsLLLsLL
| |sLLLsLLLsLL
| style="text-align:center;" |<nowiki>2|8</nowiki>
| |(Great) Aeolian f2, Phrygian #9 (Theban)
| |(Great) Aeolian b2, Phrygian #9 (Theban)
|-
|-
| style="text-align:center;" |<nowiki>1|9</nowiki>
| |sLLLsLLsLLL
| |sLLLsLLsLLL
| style="text-align:center;" |<nowiki>1|9</nowiki>
| |(Great) Phrygian
| |(Great) Phrygian
|-
|-
| style="text-align:center;" |<nowiki>0|10</nowiki>
| |sLLsLLLsLLL
| |sLLsLLLsLLL
| style="text-align:center;" |<nowiki>0|10</nowiki>
| |(Great) Locrian
| |(Great) Locrian
|}
|}
===Cyclic Permutation order===
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.
{| class="wikitable"
{| class="wikitable"
|+
|+
!Spelling
|'''Mode'''
!'''Mode'''
|[[Modal UDP Notation|'''UDP''']] 1
![[Modal UDP Notation|'''UDP''']]
|[[Modal UDP Notation|'''UDP''']] 2
!'''Name'''
|'''Name 1'''
|'''Name 2'''
|-
|LLsLLsLLsLL
|<nowiki>6|4 b10</nowiki>
|<nowiki>4|6 #3</nowiki>
|(Great) Mixolydian f10
|(Great) Dorian #3
|-
|-
|GHJKLABCDEFG
|LLsLLsLLLLs
|<nowiki>8|2 b7</nowiki>
|<nowiki>6|4 #11</nowiki>
|(Great) Lydian f4 f7, Ionian f7 #11 (Distomian Dominant)
|(Great) Mixolydian #11
|-
|LLsLLLLsLLs
|<nowiki>10|0 b4</nowiki>
|<nowiki>8|2 #8</nowiki>
|(Great) Lydian f4 #8 (Tanagran f4)
|(Great) Lydian f4 #8, Ionian #8 #11 (Distomian #8)
|-
|LLLLsLLsLLs
|<nowiki>1|9 *b1</nowiki>
|<nowiki>10|0 #5</nowiki>
|(Great) Phrygian *f1
|(Great) Lydian #5 #8 (Tanagran #5)
|-
|LsLLsLLsLLL
|<nowiki>3|7 b9</nowiki>
|<nowiki>1|9 #2</nowiki>
|(Great) Aeolian f9
|(Great) Phrygian #2
|-
|LsLLsLLLLsL
|<nowiki>5|5 b6</nowiki>
|<nowiki>3|7 #10</nowiki>
|(Great) Mixolydian f3 f6, Dorian f6 #10 (Livadeian f6)
|(Great) Aeolian #10
|-
|sLLLLsLLsLL
|<nowiki>7|3 b3</nowiki>
|<nowiki>5|5 #7</nowiki>
|(Great) Ionian f3
|(Great) Mixolydian f3 #7, Dorian #7 #10 (Livadeian #7)
|-
|LLLsLLsLLsL
|<nowiki>9|1 b11</nowiki>
|<nowiki>7|3 #4</nowiki>
|(Great) Lydian f11
|(Great) Ionian #4
|-
|sLLsLLsLLLL
|<nowiki>0|10 b8</nowiki>
|<nowiki>9|1 *#1</nowiki>
|(Great) Locrian f8
|(Great) Lydian *#1
|-
|sLLsLLLLsLL
|<nowiki>2|8 b5</nowiki>
|<nowiki>0|10 #9</nowiki>
|(Great) Aeolian f2 f5, Phrygian f5 #9 (Theban f5)
|(Great) Locrian #9
|-
|LsLLLLsLLsL
|<nowiki>4|6 b2</nowiki>
|<nowiki>2|8 #6</nowiki>
|(Great) Dorian f2
|(Great) Aeolian f2 #6, Phrygian #6 #9 (Theban #6)
|}
 
===Cyclic Permutation order===
{| class="wikitable"
|+
!Spelling 1
!Spelling 2
!'''Mode'''
![[Modal UDP Notation|'''UDP''']]
!'''Name'''
|-
|GHJKLABCDEFG
|LABCDEFGHJKL
|LLsLLLsLLLs
|LLsLLLsLLLs
| style="text-align:center;" |<nowiki>8|2</nowiki>
|<nowiki>8|2</nowiki>
|(Great) Distomian
|(Great) Distomian
|-
|-
|HJKLABCDEFGH
|HJKLABCDEFGH
|ABCDEFGHJKLA
|LsLLLsLLLsL
|LsLLLsLLLsL
| style="text-align:center;" |<nowiki>5|5</nowiki>
|<nowiki>5|5</nowiki>
|(Great) Livadeian
|(Great) Livadeian
|-
|-
|JKLABCDEFGHJ
|JKLABCDEFGHJ
|BCDEFGHJKLAB
|sLLLsLLLsLL
|sLLLsLLLsLL
| style="text-align:center;" |<nowiki>2|8</nowiki>
|<nowiki>2|8</nowiki>
|(Great) Theban
|(Great) Theban
|-
|-
|KLABCDEFGHJK
|KLABCDEFGHJK
|CDEFGHJKLABC
|LLLsLLLsLLs
|LLLsLLLsLLs
| style="text-align:center;" |<nowiki>10|0</nowiki>
|<nowiki>10|0</nowiki>
|(Great) Tanagran
|(Great) Tanagran
|-
|-
|LABCDEFGHJKL
|LABCDEFGHJKL
|DEFGHJKLABCD
|LLsLLLsLLsL
|LLsLLLsLLsL
| style="text-align:center;" |<nowiki>7|3</nowiki>
|<nowiki>7|3</nowiki>
|(Great) Ionian
|(Great) Ionian
|-
|-
|ABCDEFGHJKLA
|ABCDEFGHJKLA
|EFGHJKLABCDE
|LsLLLsLLsLL
|LsLLLsLLsLL
| style="text-align:center;" |<nowiki>4|6</nowiki>
|<nowiki>4|6</nowiki>
|(Great) Dorian
|(Great) Dorian
|-
|-
|BCDEFGHJKLAB
|BCDEFGHJKLAB
|FGHJKLABCDEF
|sLLLsLLsLLL
|sLLLsLLsLLL
| style="text-align:center;" |<nowiki>1|9</nowiki>
|<nowiki>1|9</nowiki>
|(Great) Phrygian
|(Great) Phrygian
|-
|-
|CDEFGHJKLABC
|CDEFGHJKLABC
|GHJKLABCDEFG
|LLLsLLsLLLs
|LLLsLLsLLLs
| style="text-align:center;" |<nowiki>9|1</nowiki>
|<nowiki>9|1</nowiki>
|(Great) Lydian
|(Great) Lydian
|-
|-
|DEFGHJKLABCD
|DEFGHJKLABCD
|HJKLABCDEFGH
|LLsLLsLLLsL
|LLsLLsLLLsL
| style="text-align:center;" |<nowiki>6|4</nowiki>
|<nowiki>6|4</nowiki>
|(Great) Mixolydian
|(Great) Mixolydian
|-
|-
|EFGHJKLABCDE
|EFGHJKLABCDE
|JKLABCDEFGHJ
|LsLLsLLLsLL
|LsLLsLLLsLL
| style="text-align:center;" |<nowiki>3|7</nowiki>
|<nowiki>3|7</nowiki>
|(Great) Aeolian
|(Great) Aeolian
|-
|-
|FGHJKLABCDEF
|FGHJKLABCDEF
|KLABCDEFGHJK
|sLLsLLLsLLL
|sLLsLLLsLLL
| style="text-align:center;" |<nowiki>0|10</nowiki>
|<nowiki>0|10</nowiki>
|(Great) Locrian
|(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 -&gt; Theban -&gt; Tanagran -&gt; Ionian cyclic sequence will resemble the geography of ancient Greece.
==Scale tree==
{| class="wikitable"
{| class="wikitable"
|+
|+
! colspan="2" |Generator
!Spelling 1
!Normalized
!Spelling 2
!''ed19\12''
!'''Mode'''
![[Modal UDP Notation|'''UDP''']]
!'''Name'''
|-
|-
|3\8
|GHJKLABfCDEFG
|
|LABCDEFfGHJKL
|<u>720</u>
|LLsLLsLLLLs
|''712.5''
| style="text-align:center;" |<nowiki>8|2 b7</nowiki>
|(Great) Distomian Dominant
|-
|-
|19\51
|HJKLABfCDEFGH
|
|ABCDEFfGHJKLA
|<u>712.5</u>
|LsLLLLsLLsL
|''707.843''
| style="text-align:center;" |<nowiki>5|5 b6</nowiki>
|(Great) Livadeian f6
|-
|-
|
|JKLABfCDEFGHJ
|35\94
|BCDEFfGHJKLAB
|<u>711.864</u>
|sLLsLLLLsLL
|''707.447''
| style="text-align:center;" |<nowiki>2|8 b5</nowiki>
|(Great) Theban f5
|-
|-
|16\43
|KLABfCDEFGHJK
|
|CDEFfGHJKLABC
|<u>711.111</u>
|LLsLLLLsLLs
|''706.977''
| style="text-align:center;" |<nowiki>10|0 b4</nowiki>
|-
|(Great) Tanagran f4
|
|29\78
|<u>710.204</u>
|''706.41''
|-
|-
|13\35
|LABfCDEFGHJKL
|
|DEFfGHJKLABCD
|<u>709.091</u>
|LsLLLLsLLsL
|''705.714''
| style="text-align:center;" |<nowiki>7|3 b3</nowiki>
|(Great) Ionian f3
|-
|-
|
|ABfCDEFGHJKLA
|36\97
|EFfGHJKLABCDE
|<u>708.197</u>
|sLLLLsLLsLL
|705.155
| style="text-align:center;" |<nowiki>4|6 b2</nowiki>
|(Great) Dorian f2
|-
|-
|
|BfCDEFGHJKLABf
|23\62
|FfGHJKLABCDEFf
|<u>707.692</u>
|LsLLsLLsLLL
|''704.839''
| style="text-align:center;" |<nowiki>1|9 *b1</nowiki>
|(Great) Phrygian *f1
|-
|-
|
|CDEFGHJKLABfC
|33\89
|GHJKLABCDEFfG
|<u>707.143</u>
|LLLsLLsLLsL
|''704.494''
| style="text-align:center;" |<nowiki>9|1 b11</nowiki>
|(Great) Lydian f11
|-
|-
|
|DEFGHJKLABfCD
|43\116
|HJKLABCDEFfGH
|<u>706.849</u>
|LLsLLsLLsLL
|''704.31''
| style="text-align:center;" |<nowiki>6|4 b10</nowiki>
|(Great) Mixolydian f10
|-
|-
|10\27
|EFGHJKLABfCDE
|
|JKLABCDEFfGHJ
|<u>705.882</u>
|LsLLsLLsLLL
|''703.704''
| style="text-align:center;" |<nowiki>3|7 b9</nowiki>
|(Great) Aeolian f9
|-
|-
|
|FGHJKLABfCDEF
|27\73
|KLABCDEFfGHJK
|<u>704.348</u>
|sLLsLLsLLLL
|''702.738''
| style="text-align:center;" |<nowiki>0|10 b8</nowiki>
|(Great) Locrian f8
|}
{| class="wikitable"
!Spelling 1
!Spelling 2
!'''Mode'''
![[Modal UDP Notation|'''UDP''']]
!'''Name'''
|-
|-
|
|GHJKLABC#DEFG
|17\46
|LABCDEFG#HJKL
|<u>703.448</u>
|LLsLLsLLLLs
|''702.174''
|<nowiki>8|2 #8</nowiki>
|(Great) Distomian #8
|-
|-
|
|HJKLABC#DEFGH
|24\65
|ABCDEFG#HJKLA
|<u>702.439</u>
|LsLLLLsLLsL
|''701.5385''
|<nowiki>5|5 #7</nowiki>
|-
|(Great) Livadeian #7
|
|31\84
|<u>701.887</u>
|701.1905
|-
|7\19
|
|<u>700</u>
|''700''
|-
|-
|
|JKLABC#DEFGHJ
|60\163
|BCDEFG#HJKLAB
|<u>699.029</u>
|sLLLLsLLsLL
|''699.3865''
|<nowiki>2|8 #6</nowiki>
|(Great) Theban #6
|-
|-
|
|KLABC#DEFGHJK
|53\144
|CDEFG#HJKLABC
|<u>698.901</u>
|LLLLsLLsLLs
|''699.306''
|<nowiki>10|0 #5</nowiki>
|(Great) Tanagran #5
|-
|-
|
|LABC#DEFGHJKL
|46\125
|DEFG#HJKLABCD
|<u>698.734</u>
|LLLsLLsLLsL
|''699.2''
|<nowiki>7|3 #4</nowiki>
|(Great) Ionian #4
|-
|-
|
|ABC#DEFGHJKLA
|39\106
|EFG#HJKLABCDE
|<u>698.5075</u>
|LLsLLsLLsLL
|''699.057''
|<nowiki>4|6 #3</nowiki>
|(Great) Dorian #3
|-
|-
|
|BC#DEFGHJKLAB
|32\87
|FG#HJKLABCDEF
|<u>698.182</u>
|LsLLsLLsLLL
|698.852
|<nowiki>1|9 #2</nowiki>
|(Great) Phrygian #2
|-
|-
|
|C#DEFGHJKLABC#
|25\68
|G#HJKLABCDEFG#
|<u>697.674</u>
|sLLsLLsLLLL
|''698.529''
|<nowiki>9|1 *#1</nowiki>
|(Great) Lydian *#1
|-
|-
|
|DEFGHJKLABC#D
|18\49
|HJKLABCDEFG#H
|<u>696.77</u>
|LLsLLsLLLLs
|''697.959''
|<nowiki>6|4 #11</nowiki>
|(Great) Mixolydian #11
|-
|-
|
|EFGHJKLABC#DE
|29\79
|JKLABCDEFG#HJ
|<u>696</u>
|LsLLsLLLLsL
|''697.468''
|<nowiki>3|7 #10</nowiki>
|(Great) Aeolian #10
|-
|-
|
|FGHJKLABC#DEF
|40\109
|KLABCDEFG#HJK
|<u>695.652</u>
|sLLsLLLLsLL
|''697.248''
|<nowiki>0|10 #9</nowiki>
|(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 -&gt; Theban -&gt; Tanagran -&gt; Ionian cyclic sequence will resemble the geography of ancient Greece.
==Scale tree==
{| class="wikitable"
|+
!Generator
!Normalized
!Large step
!Small step
|-
|-
|11\30
|3\8
|
|<u>720.000</u>
|<u>694.737</u>
|1\5, <u>240.000</u>
|''696.667''
|0
|-
|-
|
|19\51
|26\71
|<u>712.500</u>
|<u>693.333</u>
|6\51, <u>225.000</u>
|''695.775''
|1\51, <u>37.500</u>
|-
|-
|15\41
|54\145
|
|<u>712.088</u>
|<u>692.308</u>
|17\145, <u>224.175</u>
|''695.122''
|3\145, <u>39.560</u>
|-
|-
|
|35\94
|34\93
|<u>711.864</u>
|<u>691.525</u>
|11\94, <u>223.729</u>
|''694.623''
|2\94, <u>40.678</u>
|-
|-
|19\52
|16\43
|
|<u>711.111</u>
|<u>690.909</u>
|5\43, <u>222.222</u>
|''694.231''
|1\43, <u>44.444</u>
|-
|-
|
|45\121
|42\115
|<u>710.526</u>
|<u>690.411</u>
|14\121, <u>221.053</u>
|''693.913''
|3\121, <u>47.368</u>
|-
|-
|23\63
|29\78
|
|<u>710.204</u>
|<u>690</u>
|9\78, <u>220.408</u>
|''693.651''
|2\78, <u>48.980</u>
|-
|42\113
|<u>709.859</u>
|13\113, <u>219.718</u>
|3\113, <u>50.704</u>
|-
|13\35
|<u>709.091</u>
|4\35, <u>218.182</u>
|1\35, <u>54.545</u>
|-
|49\132
|<u>708.434</u>
|15\132, <u>216.867</u>
|4\132, <u>57.831</u>
|-
|36\97
|<u>708.197</u>
|11\97, <u>216.393</u>
|3\97, <u>59.016</u>
|-
|23\62
|<u>707.692</u>
|7\62, <u>215.385</u>
|2\62, <u>61.538</u>
|-
|33\89
|<u>707.143</u>
|10\89, <u>214.286</u>
|3\89, <u>64.286</u>
|-
|43\116
|<u>706.849</u>
|13\116, <u>213.699</u>
|4\116, <u>65.753</u>
|-
|53\143
|<u>706.667</u>
|16\143, <u>213.333</u>
|5\143, <u>66.667</u>
|-
|63\170
|<u>706.542</u>
|19\170, <u>213.084</u>
|6\170, <u>67.290</u>
|-
|73\197
|<u>706.452</u>
|22\197, <u>212.903</u>
|7\197, <u>67.742</u>
|-
|-
|4\11
|10\27
|
|<u>705.882</u>
|<u>685.714</u>
|3\27, <u>211.765</u>
|''690.909''
|1\27, <u>70.588</u>
|}
|-
 
|47\127
|<u>705.000</u>
|14\127, <u>210.000</u>
|5\127, <u>75.000</u>
|-
|37\100
|<u>704.762</u>
|11\100, <u>209.524</u>
|4\100, <u>76.190</u>
|-
|27\73
|<u>704.348</u>
|8\73, <u>208.696</u>
|3\27, <u>78.261</u>
|-
|17\46
|<u>703.448</u>
|5\46, <u>206.897</u>
|2\46, <u>82.759</u>
|-
|41\111
|<u>702.857</u>
|12\111, <u>205.714</u>
|5\111, <u>85.714</u>
|-
|24\65
|<u>702.439</u>
|7\65, <u>204.878</u>
|3\65, <u>87.805</u>
|-
|31\84
|<u>701.887</u>
|9\84, <u>203.774</u>
|4\84, <u>90.566</u>
|-
|…
|…
|…
|…
|-
|59\160
|<u>700.990</u>
|17\160, <u>201.980</u>
|8\160, <u>95.050</u>
|-
|…
|…
|…
|…
|-
|122\331
|<u>700.478</u>
|35\331, <u>200.960</u>
|18\334, <u>97.608</u>
|-
|7\19
|<u>700.000</u>
|2\19, <u>200.000</u>
|1\19, <u>100.000</u>
|-
|123\334
|<u>699.526</u>
|35\334, <u>199.052</u>
|18\334, <u>102.370</u>
|-
|…
|…
|…
|…
|-
|63\163
|<u>699.029</u>
|17\163, <u>198.058</u>
|9\163, <u>104.854</u>
|-
|…
|…
|…
|…
|-
|32\87
|<u>698.182</u>
|9\87, <u>196.364</u>
|5\87, <u>109.091</u>
|-
|25\68
|<u>697.674</u>
|7\68, <u>195.349</u>
|4\68, <u>111.628</u>
|-
|43\117
|<u>697.297</u>
|12\117, <u>194.594</u>
|7\117, <u>113.514</u>
|-
|18\49
|<u>696.774</u>
|5\49, <u>193.548</u>
|3\49, <u>116.129</u>
|-
|29\79
|<u>696.000</u>
|8\79, <u>192.000</u>
|5\79, <u>120.000</u>
|-
|40\109
|<u>695.652</u>
|11\109, <u>191.304</u>
|7\109, <u>121.739</u>
|-
|51\139
|<u>695.455</u>
|14\139, <u>190.909</u>
|9\139, <u>122.727</u>
|-
|11\30
|<u>694.737</u>
|3\30, <u>189.474</u>
|2\30, <u>126.316</u>
|-
|59\161
|<u>694.118</u>
|16\161, <u>188.235</u>
|11\161, <u>129.412</u>
|-
|48\131
|<u>693.976</u>
|13\131, <u>187.952</u>
|9\131, <u>130.120</u>
|-
|37\101
|<u>693.750</u>
|10\101, <u>187.500</u>
|7\101, <u>131.250</u>
|-
|26\71
|<u>693.333</u>
|7\71, <u>186.667</u>
|5\71, <u>133.333</u>
|-
|41\112
|<u>692.958</u>
|11\112, <u>185.915</u>
|8\112, <u>135.211</u>
|-
|15\41
|<u>692.308</u>
|4\41, <u>184.615</u>
|3\41, <u>138.462</u>
|-
|34\93
|<u>691.525</u>
|9\93, <u>183.051</u>
|7\93, <u>142.373</u>
|-
|53\145
|<u>691.304</u>
|14\145, <u>182.609</u>
|11\145, <u>143.478</u>
|-
|19\52
|<u>690.909</u>
|5\52, <u>181.818</u>
|4\52, <u>145.455</u>
|-
|23\63
|<u>690.000</u>
|6\63, <u>180.000</u>
|5\63, <u>150.000</u>
|-
|4\11
|<u>685.714</u>
|1\11, <u>171.429</u>
|1\11, <u>171.429</u>
|}
 
== See also ==
== See also ==
[[8L 3s (3/1-equivalent)]]
[[8L 3s (3/1-equivalent)]]
[[16L 6s (80/9-equivalent)]]
[[16L 6s (352/39-equivalent)]]
[[16L 6s (64/7-equivalent)]]

Latest revision as of 19:28, 29 December 2024

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.[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.[2] 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
… … … …
122\331 700.478 35\331, 200.960 18\334, 97.608
7\19 700.000 2\19, 200.000 1\19, 100.000
123\334 699.526 35\334, 199.052 18\334, 102.370
… … … …
63\163 699.029 17\163, 198.058 9\163, 104.854
… … … …
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)

  1. ↑ 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 ..."
  2. ↑ Obikhod - Wikipedia. en.wikipedia.org. Retrieved July 28, 2021.
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