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User:Moremajorthanmajor/5L 3s (minor ninth-equivalent): Difference between revisions

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Created page with "The minor ninth of a diatonic scale has a '''5L 3s''' MOS structure with generators ranging from 2\5 (two degrees of 5ed8\7 = 548.6¢) to 3\8 (three degrees of 8edo =..."
 
 
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Line 3: Line 3:
Any edIX of an interval up to 8\7 with an interval between 450¢ and 548.6¢ has a 5L 3s scale. [[13edIX]] is the smallest edIX with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.
Any edIX of an interval up to 8\7 with an interval between 450¢ and 548.6¢ has a 5L 3s scale. [[13edIX]] is the smallest edIX with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.
==Standing assumptions==
==Standing assumptions==
The [[TAMNAMS]] system is used in this article to name 5L 3s<15/7> intervals and step size ratios and step ratio ranges.
The [[TAMNAMS]] system is used in this article to name 5L 3s<minor ninth> intervals and step size ratios and step ratio ranges.


The notation used in this article is G Mixolydian Mediant (LLsLLsLs) = GABCQDEFG, unless specified otherwise. We denote raising and lowering by a chroma (L &minus; s) by # and f "flat (F molle)".
The notation used in this article is G Mixolydian Mediant (LLsLLsLs) = GABCQDEFG, unless specified otherwise. We denote raising and lowering by a chroma (L &minus; s) by # and f "flat (F molle)".
Line 23: Line 23:
The author suggests the name '''Neapolitan'''-'''oneirotonic'''.
The author suggests the name '''Neapolitan'''-'''oneirotonic'''.
==Intervals==
==Intervals==
The table of Neapolitan-oneirotonic intervals below takes the fourth as the generator. Given the size of the fourth generator ''g'', any oneirotonic 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 1300 (for relative cents) if necessary (so you can use "''k''*''g'' % 1300" for search engines, for plugged-in values of ''k'' and ''g''). For example, since the major third is reached by six fourth generators, 18edIX's major third is 6*505.56 mod 1300 = 3033.33 mod 1300 = 433.33r¢.
The table of Neapolitan-oneirotonic intervals below takes the fourth as the generator. Given the size of the fourth generator ''g'', any oneirotonic 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 if necessary (so you can use "''k''*''g'' % ''x''" for search engines, for plugged-in values of ''k'' and ''g''). For example, since the major third is reached by six fourth generators, 18edIX's major third is 6*494.12 mod 1270.59 = 2964.71 mod 1270.59 = 423.53¢.
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
Line 177: Line 177:
|minor 2nd
|minor 2nd
|1\13, 100.00
|1\13, 100.00
|1\18, 70.59 (72.22)
|1\18, 70.59
|2\21, 126.32 (121.81)
|2\21, 126.32
|Af
|Af
| -5
| -5
Line 184: Line 184:
|major 2nd
|major 2nd
|2\13, 200.00
|2\13, 200.00
|3\18, 211.765 (216.67)
|3\18, 211.76
|3\21, 189.47 (185.71)
|3\21, 189.47
|A
|A
| +3
| +3
Line 191: Line 191:
|minor 3rd
|minor 3rd
|3\13, 300.00
|3\13, 300.00
|4\18, 282.35 (288.89)
|4\18, 282.35
|5\21, 315.79 (309.52)
|5\21, 315.79
|Bf
|Bf
| -2
| -2
Line 198: Line 198:
|major 3rd
|major 3rd
| rowspan="2" |4\13, 400.00
| rowspan="2" |4\13, 400.00
|6\18, 423.53 (433.33)
|6\18, 423.53
|6\21, 378.95 (371.43)
|6\21, 378.95
|B
|B
| +6
| +6
|-
|-
|diminished 4th
|diminished 4th
|5\18, 352.94 (361.11)
|5\18, 352.94
|7\21, 442.105 (433.33)
|7\21, 442.105
|Cf
|Cf
| -7
| -7
Line 211: Line 211:
|natural 4th
|natural 4th
|5\13, 500.00
|5\13, 500.00
|7\18, 494.12 (505.56)
|7\18, 494.12
|8\21, 505.26 (495.24)
|8\21, 505.24
|C
|C
| +1
| +1
Line 218: Line 218:
|augmented 4th
|augmented 4th
| rowspan="2" |6\13, 600.00
| rowspan="2" |6\13, 600.00
|9\18, 635.29 (650.00)
|9\18, 635.29
|9\21, 568.42 (557.14)
|9\21, 568.42
|C#
|C#
| +9
| +9
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|8\18, 564.71 (577.78)
|8\18, 564.71
|10\21, 631.58 (619.05)
|10\21, 631.58
|Qf
|Qf
| -4
| -4
Line 231: Line 231:
|perfect 5th
|perfect 5th
|7\13, 700.00
|7\13, 700.00
|10\18, 705.88 (722.22)
|10\18, 705.88
|11\31, 694.74 (680.95)
|11\31, 694.74
|Q
|Q
| +4
| +4
Line 238: Line 238:
|minor 6th
|minor 6th
|8\13, 800.00
|8\13, 800.00
|11\18, 776.47 (794.44)
|11\18, 776.47
|13\21, 821.05 (802.76)
|13\21, 821.05
|Df
|Df
| -1
| -1
Line 245: Line 245:
|major 6th
|major 6th
| rowspan="2" |9\13, 900.00
| rowspan="2" |9\13, 900.00
|13\18, 917.65 (938.89)
|13\18, 917.65
|14\21, 884.21 (866.67)
|14\21, 884.21
|D
|D
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|12\18, 847.06 (866.67)
|12\18, 847.06
|15\21, 947.37 (928.57)
|15\21, 947.37
|Ef
|Ef
| -6
| -6
Line 258: Line 258:
|major 7th
|major 7th
|10\13, 1000.00
|10\13, 1000.00
|14\18, 988.235 (1011.11)
|14\18, 988.24
|16\21, 1017.53 (990.48)
|16\21, 1017.53
|E
|E
| +2
| +2
Line 265: Line 265:
|diminished octave
|diminished octave
|11\13, 1100.00
|11\13, 1100.00
|15\18, 1052.82 (1083.33)
|15\18, 1052.82
|18\21, 1136.84 (1114.29)
|18\21, 1136.84
|Ff
|Ff
| -3
| -3
Line 272: Line 272:
|perfect octave
|perfect octave
|12\13, 1200.00
|12\13, 1200.00
|17\18, 1200.00 (1227.78)
|17\18, 1200.00
|19\21, 1200.00 (1178.19)
|19\21, 1200.00
|F
|F
| +5
| +5
Line 295: Line 295:
|generator (g)
|generator (g)
|5\13, 500.00
|5\13, 500.00
|7\18, 494.12 (505.56)
|7\18, 494.12
|12\31, 496.55 (503.23)
|12\31, 496.55
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|2\13, 200.00
|2\13, 200.00
|3\18, 211.765 (216.67)
|3\18, 211.765
|5\31, 206.87 (209.68)
|5\31, 206.87
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|1\13, 100.00
|1\13, 100.00
|1\18, 70.59 (72.22)
|1\18, 70.59
|2\31, 82.76 (83.87)
|2\31, 82.76
|}
|}
====Intervals====
====Intervals====
Line 330: Line 330:
|minor 2nd
|minor 2nd
|1\13, 100.00
|1\13, 100.00
|1\18, 70.59 (72.22)
|1\18, 70.59
|2\31, 82.76 (83.87)
|2\31, 82.76
|Af
|Af
|21/20, ''22/21''
|21/20, ''22/21''
Line 338: Line 338:
|major 2nd
|major 2nd
|2\13, 200.00
|2\13, 200.00
|3\18, 211.765 (216.67)
|3\18, 211.76
|5\31, 206.87 (209.68)
|5\31, 206.87
|A
|A
|9/8
|9/8
Line 346: Line 346:
|minor 3rd
|minor 3rd
|3\13, 300.00
|3\13, 300.00
|4\18, 282.35 (288.89)
|4\18, 282.35
|7\31, 289.655 (293.55)
|7\31, 289.66
|Bf
|Bf
|13/11, 33/28
|13/11, 33/28
Line 354: Line 354:
|major 3rd
|major 3rd
| rowspan="2" |4\13, 400.00
| rowspan="2" |4\13, 400.00
|6\18, 423.53 (433.33)
|6\18, 423.53
|10\31, 413.79 (419.355)
|10\31, 413.79  
|B
|B
|14/11, 33/26
|14/11, 33/26
Line 361: Line 361:
|-
|-
|diminished 4th
|diminished 4th
|5\18, 352.94 (361.11)
|5\18, 352.94
|9\31, 372.41 (377.42)
|9\31, 372.41  
|Cf
|Cf
|''5/4, 11/9''
|''5/4, 11/9''
Line 369: Line 369:
|natural 4th
|natural 4th
|5\13, 500.00
|5\13, 500.00
|7\18, 494.12 (505.56)
|7\18, 494.12
|12\31, 496.55 (503.23)
|12\31, 496.55
|C
|C
|4/3
|4/3
Line 377: Line 377:
|augmented 4th
|augmented 4th
| rowspan="2" |6\13, 600.00
| rowspan="2" |6\13, 600.00
|9\18, 635.29 (650.00)
|9\18, 635.29
|15\31, 620.69 (629.03)
|15\31, 620.69  
|C#
|C#
|''10/7, 18/13, 11/8''
|''10/7, 18/13, 11/8''
Line 384: Line 384:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|8\18, 564.71 (577.78)
|8\18, 564.71
|14\31, 579.31 (587.10)
|14\31, 579.31  
|Qf
|Qf
|''7/5, 13/9'', ''16/11''
|''7/5, 13/9'', ''16/11''
Line 392: Line 392:
|perfect 5th
|perfect 5th
|7\13, 700.00
|7\13, 700.00
|10\18, 705.88 (722.22)
|10\18, 705.88
|17\31, 703.45 (712.90)
|17\31, 703.45  
|Q
|Q
|3/2
|3/2
Line 400: Line 400:
|minor 6th
|minor 6th
|8\13, 800.00
|8\13, 800.00
|11\18, 776.47 (794.44)
|11\18, 776.47
|19\31, 786.21 (796.77)
|19\31, 786.21  
|Df
|Df
|52/33, 11/7
|52/33, 11/7
Line 408: Line 408:
|major 6th
|major 6th
| rowspan="2" |9\13, 900.00
| rowspan="2" |9\13, 900.00
|13\18, 917.65 (938.89)
|13\18, 917.65
|22\31, 910.345 (922.58)
|22\31, 910.34
|D
|D
|56/33, 22/17
|56/33, 22/17
Line 415: Line 415:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|12\18, 847.06 (866.67)
|12\18, 847.06
|21\31, 868.97 (880.645)
|21\31, 868.97  
|Ef
|Ef
|5/3, 18/11
|5/3, 18/11
Line 423: Line 423:
|major 7th
|major 7th
|10\13, 1000.00
|10\13, 1000.00
|14\18, 988.235 (1011.11)
|14\18, 988.24
|24\31, 993.13 (1007.45)
|24\31, 993.13  
|E
|E
|16/9
|16/9
Line 431: Line 431:
|diminished octave
|diminished octave
|11\13, 1100.00
|11\13, 1100.00
|15\18, 1052.82 (1083.33)
|15\18, 1052.82
|26\31, 1075.86 (1090.32)
|26\31, 1075.86  
|Ff
|Ff
|11/6, 13/7, 15/8
|11/6, 13/7, 15/8
Line 439: Line 439:
|perfect octave
|perfect octave
|12\13, 1200.00
|12\13, 1200.00
|17\18, 1200.00 (1227.78)
|17\18, 1200.00
|29\31, 1200.00 (1212.13)
|29\31, 1200.00  
|F
|F
|2/1
|2/1
Line 457: Line 457:
|-
|-
|generator (g)
|generator (g)
|8\21, 505.26 (495.24)
|8\21, 505.26
|13\34, 458.82
|13\34, 503.23
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|3\21, 189.47 (185.71)
|3\21, 189.47
|5\34, 193.55 (191.18)
|5\34, 193.55
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|2\31, 82.76 (83.87)
|2\21, 126.32
|3\34, 116.19 (114.71)
|3\34, 116.19
|}
|}
====Intervals====
====Intervals====
Line 474: Line 474:
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 21edIX (soft)
!Size in 21edIX (soft)
!Size in 34edo (semisoft)
!Size in 34edIX (semisoft)
! class="unsortable" |Note name on G
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios
! class="unsortable" |Approximate ratios
Line 487: Line 487:
|-
|-
|minor 2nd
|minor 2nd
|2\21, 126.32 (121.81)
|2\21, 126.32
|3\34, 116.19 (114.71)
|3\34, 116.19
|Af
|Af
|16/15
|16/15
Line 494: Line 494:
|-
|-
|major 2nd
|major 2nd
|3\21, 189.47 (185.71)
|3\21, 189.47
|5\34, 193.55 (191.18)
|5\34, 193.55  
|A
|A
|10/9, 9/8
|10/9, 9/8
Line 501: Line 501:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|5\21, 315.79 (309.52)
|5\21, 315.79
|8\34, 309.68 (305.88)
|8\34, 309.68  
|Bf
|Bf
|6/5
|6/5
Line 508: Line 508:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|6\21, 378.95 (371.43)
|6\21, 378.95
|10\34, 387.10 (382.35)
|10\34, 387.10  
|B
|B
|5/4
|5/4
Line 515: Line 515:
|-
|-
|diminished 4th
|diminished 4th
|7\21, 442.105 (433.33)
|7\21, 442.105
|11\34, 425.81 (420.59)
|11\34, 425.81  
|Cf
|Cf
|9/7
|9/7
Line 522: Line 522:
|-
|-
|natural 4th
|natural 4th
|8\21, 505.26 (495.24)
|8\21, 505.24
|13\34, 503.23 (497.06)
|13\34, 503.23
|C
|C
|4/3
|4/3
Line 529: Line 529:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 4th
|augmented 4th
|9\21, 568.42 (557.14)
|9\21, 568.42
|15\34, 580.645 (573.53)
|15\34, 580.645  
|C#
|C#
|7/5
|7/5
Line 536: Line 536:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|10\21, 631.58 (619.05)
|10\21, 631.58
|16\34, 619.355 (611.765)
|16\34, 619.355  
|Qf
|Qf
|10/6
|10/6
Line 543: Line 543:
|-
|-
|perfect 5th
|perfect 5th
|11\31, 694.74 (680.95)
|11\31, 694.74
|18\34, 696.77 (688.235)
|18\34, 696.77  
|Q
|Q
|3/2
|3/2
Line 550: Line 550:
|-
|-
|minor 6th
|minor 6th
|13\21, 821.05 (802.76)
|13\21, 821.05
|21\34, 812.90 (802.94)
|21\34, 812.90  
|Df
|Df
|8/5
|8/5
Line 557: Line 557:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6th
|major 6th
|14\21, 884.21 (866.67)
|14\21, 884.21
|23\34, 890.32 (879.41)
|23\34, 890.32  
|D
|D
|5/3
|5/3
Line 564: Line 564:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|15\21, 947.37 (928.57)
|15\21, 947.37
|24\34, 929.03 (917.65)
|24\34, 929.03  
|Ef
|Ef
|12/7
|12/7
Line 571: Line 571:
|-
|-
|major 7th
|major 7th
|16\21, 1017.53 (990.48)
|16\21, 1017.53
|26\34, 1006.45 (995.12)
|26\34, 1006.45  
|E
|E
|9/5, 16/9
|9/5, 16/9
Line 578: Line 578:
|-
|-
|diminished octave
|diminished octave
|18\21, 1136.84 (1114.29)
|18\21, 1136.84
|29\34, 1122.58 (1109.88)
|29\34, 1122.58  
|Ff
|Ff
|27/14, 48/25
|27/14, 48/25
Line 585: Line 585:
|-
|-
|perfect octave
|perfect octave
|19\21, 1200.00 (1178.19)
|19\21, 1200.00
|31\34, 1200.00 (1185.29)
|31\34, 1200.00
|F
|F
|2/1
|2/1
Line 602: Line 602:
|-
|-
|generator (g)
|generator (g)
|11\29, 507.69 (493.10)
|11\29, 507.69
|14\37, 509.09 (491.89)
|14\37, 509.09
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|4\29, 184.615 (179.31)
|4\29, 184.615
|5\37, 181.82 (175.68)
|5\37, 181.82
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|3\29, 138.46 (134.49)
|3\29, 138.46
|4\37, 145.455 (140.54)
|4\37, 145.455
|}
|}
====Intervals====
====Intervals====
Line 630: Line 630:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|chroma
|chroma
|1\29, 46.15 (44.83)
|1\29, 46.15
|G#
|G#
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
Line 636: Line 636:
|-
|-
|diminished 2nd
|diminished 2nd
|2\29, 92.31 (89.655)
|2\29, 92.31
|Aff
|Aff
|[[21/20]], [[22/21]], [[26/25]]
|[[21/20]], [[22/21]], [[26/25]]
Line 642: Line 642:
|-
|-
|minor 2nd
|minor 2nd
|3\29, 138.46 (134.49)
|3\29, 138.46
|Af
|Af
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
Line 648: Line 648:
|-
|-
|major 2nd
|major 2nd
|4\29, 184.615 (179.31)
|4\29, 184.615
|A
|A
|[[9/8]], [[10/9]], [[11/10]]
|[[9/8]], [[10/9]], [[11/10]]
Line 654: Line 654:
|-
|-
|augmented 2nd
|augmented 2nd
|5\29, 230.77 (224.14)
|5\29, 230.77
|A#
|A#
|[[8/7]], [[15/13]]
|[[8/7]], [[15/13]]
Line 660: Line 660:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 3rd
|diminished 3rd
|6\29, 276.92 (268.97)
|6\29, 276.92
|Bff
|Bff
|[[7/6]], [[13/11]], [[33/28]]
|[[7/6]], [[13/11]], [[33/28]]
Line 666: Line 666:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|7\29, 323.08 (313.79)
|7\29, 323.08
|Bf
|Bf
|[[135/112]], [[6/5]]
|[[135/112]], [[6/5]]
Line 672: Line 672:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|8\29, 369.23 (358.21)
|8\29, 369.23
|B
|B
|[[5/4]], [[11/9]], [[16/13]]
|[[5/4]], [[11/9]], [[16/13]]
Line 678: Line 678:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 3rd
|augmented 3rd
|9\29, 415.385 (403.45)
|9\29, 415.385
|B#
|B#
|[[9/7]], [[14/11]], [[33/26]]
|[[9/7]], [[14/11]], [[33/26]]
Line 684: Line 684:
|-
|-
|diminished 4th
|diminished 4th
|10\29, 461.54 (448.28)
|10\29, 461.54
|Cf
|Cf
|[[21/16]], [[13/10]]
|[[21/16]], [[13/10]]
Line 690: Line 690:
|-
|-
|natural 4th
|natural 4th
|11\29, 507.69 (493.10)
|11\29, 507.69
|C
|C
|[[75/56]], [[4/3]]
|[[75/56]], [[4/3]]
Line 696: Line 696:
|-
|-
|augmented 4th
|augmented 4th
|12\29, 553.85 (537.93)
|12\29, 553.85
|C#
|C#
|[[11/8]], [[18/13]]
|[[11/8]], [[18/13]]
Line 702: Line 702:
|-
|-
|doubly augmented 4th, doubly diminished 5th
|doubly augmented 4th, doubly diminished 5th
|13\29, 600.00 (582.76)
|13\29, 600.00
|Cx, Qff
|Cx, Qff
|[[7/5]], [[10/7]]
|[[7/5]], [[10/7]]
Line 708: Line 708:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|14\29, 646.15 (627.59)
|14\29, 646.15
|Qf
|Qf
|[[16/11]], [[13/9]]
|[[16/11]], [[13/9]]
Line 714: Line 714:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect 5th
|perfect 5th
|15\29, 692.31 (672.41)
|15\29, 692.31
|Q
|Q
|[[112/75]], [[3/2]]
|[[112/75]], [[3/2]]
Line 720: Line 720:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 5th
|augmented 5th
|16\29, 738.46 (717.24)
|16\29, 738.46
|Q#
|Q#
|[[32/21]], [[20/13]]
|[[32/21]], [[20/13]]
Line 726: Line 726:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 6th
|diminished 6th
|17\29, 784.615 (762.07)
|17\29, 784.615
|Dff
|Dff
|[[11/7]], [[14/9]]
|[[11/7]], [[14/9]]
Line 732: Line 732:
|-
|-
|minor 6th
|minor 6th
|18\29, 830.77 (806.90)
|18\29, 830.77
|Df
|Df
|[[13/8]], [[8/5]]
|[[13/8]], [[8/5]]
Line 738: Line 738:
|-
|-
|major 6th
|major 6th
|19\29, 876.92 (851.725)
|19\29, 876.92
|D
|D
|[[5/3]], [[224/135]]
|[[5/3]], [[224/135]]
Line 744: Line 744:
|-
|-
|augmented 6th
|augmented 6th
|20\29, 923.08 (896.55)
|20\29, 923.08
|D#
|D#
|[[12/7]], [[22/13]]
|[[12/7]], [[22/13]]
Line 750: Line 750:
|-
|-
|minor 7th
|minor 7th
|21\29, 969.23 (941.38)
|21\29, 969.23
|Ef
|Ef
|[[7/4]], [[26/15]]
|[[7/4]], [[26/15]]
Line 756: Line 756:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 7th
|major 7th
|22\29, 1015.385 (982.21)
|22\29, 1015.385
|E
|E
|[[9/5]], [[16/9]], [[20/11]]
|[[9/5]], [[16/9]], [[20/11]]
Line 762: Line 762:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 7th
|augmented 7th
|23\29, 1061.54 (1031.03)
|23\29, 1061.54
|E#
|E#
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
Line 768: Line 768:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|doubly augmented 7th, doubly diminished octave
|doubly augmented 7th, doubly diminished octave
|24\29, 1107.69 (1075.86)
|24\29, 1107.69
|Ex, Fff
|Ex, Fff
|[[21/11]], [[25/13]], [[40/21]]
|[[21/11]], [[25/13]], [[40/21]]
Line 774: Line 774:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished octave
|diminished octave
|25\29, 1153.85 (1120.69)
|25\29, 1153.85
|Ff
|Ff
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
Line 780: Line 780:
|-
|-
|perfect octave
|perfect octave
|26\29, 1200.00 (1165.52)
|26\29, 1200.00
|F
|F
|2/1
|2/1
Line 786: Line 786:
|-
|-
|augmented octave
|augmented octave
|27\29, 1246.15 (1210.345)
|27\29, 1246.15
|F#
|F#
|33/16, 49/24, 72/35, 25/12
|33/16, 49/24, 72/35, 25/12
Line 792: Line 792:
|-
|-
|doubly augmented octave, diminished 9th
|doubly augmented octave, diminished 9th
|28\29, 1292.31 (1255.13)
|28\29, 1292.31
|Fx, Gf
|Fx, Gf
|21/10, 44/21, 52/25
|21/10, 44/21, 52/25
Line 798: Line 798:
|}
|}
===Parahard===
===Parahard===
23edIX Neapolitan-oneiro combines the sound of the 32/15 minor ninth and the [[8/7]] whole tone. This is because 23edIX Neapolitan-oneirotonic a large step of 218.2¢, 22edIX's superpythagorean major second, which is both a warped Pythagorean [[9/8]] whole tone and a warped [[8/7]] septimal whole tone.
23edIX Neapolitan-oneiro combines the sound of the 32/15 minor ninth and the [[8/7]] whole tone. This is because 23edIX Neapolitan-oneirotonic has a large step of 218.2¢, 22edIX's superpythagorean major second, which is both a warped Pythagorean [[9/8]] whole tone and a warped [[8/7]] septimal whole tone.
====Intervals====
====Intervals====
The intervals of the extended generator chain (-12 to +12 generators) are as follows in various Neapolitan-oneirotonic tunings close to 23edIX.
The intervals of the extended generator chain (-12 to +12 generators) are as follows in various Neapolitan-oneirotonic tunings close to 23edIX.
Line 817: Line 817:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|chroma
|chroma
|3\23, 163.63 (169.565)
|3\23, 163.63
|G#
|G#
|12/11, 11/10, 10/9
|12/11, 11/10, 10/9
Line 823: Line 823:
|-
|-
|minor 2nd
|minor 2nd
|1\23, 54.545 (56.52)
|1\23, 54.545  
|Af
|Af
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
Line 829: Line 829:
|-
|-
|major 2nd
|major 2nd
|4\23, 218.18 (226.09)
|4\23, 218.18  
|A
|A
|[[9/8]], [[17/15]], [[8/7]]
|[[9/8]], [[17/15]], [[8/7]]
Line 835: Line 835:
|-
|-
|aug. 2nd
|aug. 2nd
|7\23, 381.82 (395.65)
|7\23, 381.82
|A#
|A#
|5/4
|5/4
Line 841: Line 841:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 3rd
|dim. 3rd
|2\23, 109.09 (113.04)
|2\23, 109.09
|Bf
|Bf
|16/15
|16/15
Line 847: Line 847:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|5\23, 272.73 (282.61)
|5\23, 272.73
|B
|B
|7/6
|7/6
Line 853: Line 853:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|8\23, 436.36 (452.17)
|8\23, 436.36
|B#
|B#
|9/7, 14/11
|9/7, 14/11
Line 859: Line 859:
|-
|-
|dim. 4th
|dim. 4th
|6\23, 327.27 (339.13)
|6\23, 327.27
|Cf
|Cf
|6/5
|6/5
Line 865: Line 865:
|-
|-
|nat. 4th
|nat. 4th
|9\23, 490.91 (508.70)
|9\23, 490.91
|C
|C
|4/3
|4/3
Line 871: Line 871:
|-
|-
|aug. 4th
|aug. 4th
|12\23, 654.545 (678.26)
|12\23, 654.545
|C#
|C#
|[[16/11]], [[22/15]]
|[[16/11]], [[22/15]]
Line 877: Line 877:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|double dim. 5th
|double dim. 5th
|7\23, 381.82 (395.65)
|7\23, 381.82
|Qff
|Qff
|5/4
|5/4
Line 883: Line 883:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 5th
|dim. 5th
|10\23, 545.455 (565.22)
|10\23, 545.455
|Qf
|Qf
|[[15/11]], [[11/8]]
|[[15/11]], [[11/8]]
Line 889: Line 889:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perf. 5th
|perf. 5th
|13\23, 709.09 (734.78)
|13\23, 709.09
|Q
|Q
|3/2
|3/2
Line 895: Line 895:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 5th
|aug. 5th
|16\23, 872.73 (904.35)
|16\23, 872.73
|Q#
|Q#
|5/3
|5/3
Line 901: Line 901:
|-
|-
|dim. 6th
|dim. 6th
|11\23, 600.00 (621.74)
|11\23, 600.00
|Dff
|Dff
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
Line 907: Line 907:
|-
|-
|minor 6th
|minor 6th
|14\23, 763.64 (791.30)
|14\23, 763.64
|Df
|Df
|14/9, 11/7
|14/9, 11/7
Line 913: Line 913:
|-
|-
|major 6th
|major 6th
|17\23, 927.27 (960.87)
|17\23, 927.27
|D
|D
|12/7
|12/7
Line 920: Line 920:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|15\23, 818.18 (847.83)
|15\23, 818.18
|Ef
|Ef
|8/5
|8/5
Line 926: Line 926:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 7th
|major 7th
|18\23, 981.82 (1017.39)
|18\23, 981.82  
|E
|E
|[[7/4]], [[30/17]], [[16/9]]
|[[7/4]], [[30/17]], [[16/9]]
Line 932: Line 932:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 7th
|aug. 7th
|21\23, 1145.455 (1186.96)
|21\23, 1145.455
|E#
|E#
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
Line 938: Line 938:
|-
|-
|dim. octave
|dim. octave
|19\23, 1036.36 (1073.91)
|19\23, 1036.36
|Ff
|Ff
|11/6, 20/11, 9/5
|11/6, 20/11, 9/5
Line 944: Line 944:
|-
|-
|perf. octave
|perf. octave
|22\23, 1200.00 (1243.48)
|22\23, 1200.00
|F
|F
|2/1
|2/1
Line 950: Line 950:
|-
|-
|aug. octave
|aug. octave
|25\23, 1363.64 (1413.04)
|25\23, 1363.64
|F#
|F#
|24/11, 11/5, 20/9
|24/11, 11/5, 20/9
Line 956: Line 956:
|-
|-
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 8-step
|dim. ninth
|20\23, 1090.91 (1130.435)
|20\23, 1090.91
|J@
|J@
|15/8
|15/8
Line 980: Line 980:
|-
|-
|generator (g)
|generator (g)
|15\38, 486.49  (513.16)
|15\38, 486.49   
|21\53, 484.615 (515.09)
|21\53, 484.615  
|25\63, 487.87 (515.87)
|25\63, 483.87  
|484.07 (515.52)
|484.07
|4/3
|4/3
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|7/38, 227.03 (239,47)
|7/38, 227.03  
|10/53, 230.77 (245.28)
|10/53, 230.77  
|12/63, 232.26 (247.62)
|12/63, 232.26  
|231.51 (246.55)
|231.51
|8/7
|8/7
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|1/38, 32.43 (34.21)
|1/38, 32.43  
|1/53, 23.08 (24.53)
|1/53, 23.08  
|1/63, 19.355 (20.635)
|1/63, 19.355  
|21.05 (22.41)
|21.05
|50/49 81/80 91/90
|50/49 81/80 91/90
|}
|}
Line 1,005: Line 1,005:
|-
|-
!Degree
!Degree
!Size in 38edo
!Size in 38edIX
!Size in 53edo
!Size in 53edIX
!Size in 63edo
!Size in 63edIX
!Size in PNTE tuning
!Size in PNTE tuning
!Note name on G
!Note name on G
Line 1,023: Line 1,023:
|-
|-
|2
|2
|7/38, 227.03 (239.47)
|7/38, 227.03  
|10/53, 230.77 (245.28)
|10/53, 230.77  
|12/63, 232.26 (247.62)
|12/63, 232.26  
|231.51 (246.55)
|231.51  
|A
|A
|8/7
|8/7
Line 1,032: Line 1,032:
|-
|-
|3
|3
|14\38, 454.05 (478.95)
|14\38, 454.05  
|20\53, 461.54 (490.57)
|20\53, 461.54  
|24\63, 464.52 (495.24)
|24\63, 464.52  
|463.03 (293.10)
|463.03
|B
|B
|13/10, 21/16
|13/10, 21/16
Line 1,041: Line 1,041:
|-
|-
|4
|4
|15\38, 486.49  (513.16)
|15\38, 486.49   
|21\53, 484.615 (515.09)
|21\53, 484.615  
|25\63, 483.87 (515.87)
|25\63, 483.87  
|484.07 (515.52)
|484.07
|C
|C
|4/3
|4/3
Line 1,050: Line 1,050:
|-
|-
|5
|5
|22\38, 713.51 (752.63)
|22\38, 713.51  
|31\53, 715.385 (760.38)
|31\53, 715.385  
|37\63, 716.13 (763.49)
|37\63, 716.13  
|715.59 (762.07)
|715.59
|Q
|Q
|3/2
|3/2
Line 1,059: Line 1,059:
|-
|-
|6
|6
|29\38, 940.54 (992.105)
|29\38, 940.54  
|41\53, 946.15 (1005.66)
|41\53, 946.15  
|49\63, 948.39 (1011.11)
|49\63, 948.39  
|947.10 (1008.63)
|947.10
|D
|D
|26/15
|26/15
Line 1,068: Line 1,068:
|-
|-
|7
|7
|30\38, 972.97 (1026.32)
|30\38, 972.97  
|42\53, 969.23 (1030.19)
|42\53, 969.23  
|50\63, 967.74 (1031.75)
|50\63, 967.74  
|968.15 (1031.03)
|968.15
|E
|E
|7/4
|7/4
Line 1,077: Line 1,077:
|-
|-
|8
|8
|37\38, 1200.00 (1265.57)
|37\38, 1200.00  
|52\53, 1200.00 (1275.47)
|52\53, 1200.00  
|62\63, 1200.00 (1280.645)
|62\63, 1200.00  
|1199.66 (1277.59)
|1199.66
|F
|F
|2/1
|2/1
Line 1,127: Line 1,127:
==Scale tree==
==Scale tree==
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="12" |Normalized Generator
! colspan="3" |Normalized Generator
!Cents
!Cents
!Relative Cents
!L
!L
!s
!s
Line 1,135: Line 1,134:
!Comments
!Comments
|-
|-
| colspan="3" |3\(8\7)|| || || || ||
| colspan="3" |3\8||514.286||1||1||1.000||
|
|
|
| ||514.286
|487.5||1||1||1.000||
|-
|-
|
| colspan="3" |17\45||510.000||6||5||1.200||
|
| || || || colspan="3" |17\(45\40)
|
|
|
| ||510
|491.111||6||5||1.200||
|-
|-
|
| colspan="3" |14\37||509.091||5||4||1.250||
|
| || || colspan="3" |14\(37\33)||
|
|
|
| ||509.091
|491.892||5||4||1.250||
|-
|-
|
| colspan="3" |25\66||508.475||9||7||1.286||
|
| || || || colspan="3" |25\(66\59)
|
|
|
| ||508.475
|492.424||9||7||1.286||
|-
|-
|
| colspan="3" |11\29||507.692||4||3||1.333||
|
| || colspan="3" |11\(29\26)|| ||
|
|
|
| ||507.692
|493.103||4||3||1.333||
|-
|-
|
| colspan="3" |19\50||506.667||7||5||1.400||
|
| || || || colspan="3" |30\(79\71)
|
|
|
| ||507.042
|493.671||11||8||1.375||
|-
|-
|
| colspan="3" |8\21||505.263||3||2||1.500||L/s = 3/2
|
| || || colspan="3" |19\(50\45)||
|
|
|
| ||506.667
|494||7||5||1.400||
|-
|-
|
| colspan="3" |29\76||504.348||11||7||1.571||
|
| || || || colspan="3" |27\(71\64)
|
|
|
| ||506.25
|494.366||10||7||1.429||
|-
|
|
| colspan="3" |8\(21\19)|| || ||
|
|
|
| ||505.263
|495.238||3||2||1.500||L/s = 3/2
|-
|
|
| || || || colspan="3" |29\(76\69)
|
|
|
| ||504.348
|496.053||11||7||1.571||
|-
|-
|
| colspan="3" |21\55||504.000||8||5||1.600||
|
| || || colspan="3" |21\(55\50)||
|
|
|
| ||504
|496.364||8||5||1.600||
|-
|-
|
| colspan="3" |34\89||503.704||13||8||1.625||Golden Neapolitan-oneirotonic
|
| || || || colspan="3" |34\(89\81)
|
|
|
| ||503.704
|496.629||13||8||1.625||Golden Neapolitan-oneirotonic
|-
|-
|
| colspan="3" |13\34||503.226||5||3||1.667||
|
| || colspan="3" |13\(34\31)|| ||
|
|
|
| ||503.226
|497.059||5||3||1.667||
|-
|-
|
| colspan="3" |31\81||502.703||12||7||1.714||
|
| || || || colspan="3" |31\(81\74)
|
|
|
| ||502.703
|497.531||12||7||1.714||
|-
|-
|
| colspan="3" |18\47||502.326||7||4||1.750||
|
| || || colspan="3" |18\(47\43)||
|
|
|
| ||502.326
|497.872||7||4||1.750||
|-
|-
|
| colspan="3" |23\60||501.818||9||5||1.800||
|
| || || || colspan="3" |23\(60\55)
|
|
|
| ||501.818
|498.333||9||5||1.800||
|-
|-
|
| colspan="3" |28\73
|
|501.493
|
|
|
|
| colspan="3" |28\(73/67)
|
|
|
|501.4925
|498.63
|11
|11
|6
|6
Line 1,294: Line 1,169:
|
|
|-
|-
|
| colspan="3" |33\86
|
|
|
|
|
|
| colspan="3" |33\(86\79)
|
|
|501.265
|501.265
|498.837
|13
|13
|7
|7
Line 1,311: Line 1,176:
|
|
|-
|-
|
| colspan="3" |38\99
|
|
|
|
|
|
|
| colspan="3" |38\(99\91)
|
|501.099
|501.099
|498.99
|15
|15
|8
|8
Line 1,328: Line 1,183:
|
|
|-
|-
|
| colspan="3" |43\112
|
|
|
|
|
|
|
|
| colspan="3" |43\(112\103)
|500.971
|500.971
|499.107
|17
|17
|9
|9
Line 1,345: Line 1,190:
|
|
|-
|-
|
| colspan="3" |5\13||500||2||1||2.000||Basic Neapolitan-oneirotonic
| colspan="3" |5\(13\12)|| || || ||
|
|
|
| ||500
|500||2||1||2.000||Basic Neapolitan-oneirotonic
(generators smaller than this are proper)
(generators smaller than this are proper)
|-
|-
|
| colspan="3" |42\109
|
|499.010
|
|
|
|
|
|
|
| colspan="3" |42\(109\101)
|499.01
|500.917
|17
|17
|8
|8
Line 1,371: Line 1,200:
|
|
|-
|-
|
| colspan="3" |37\96
|
|
|
|
|
|
|
| colspan="3" |37\(96\89)
|
|498.876
|498.876
|501.041
|15
|15
|7
|7
Line 1,388: Line 1,207:
|
|
|-
|-
|
| colspan="3" |32\83
|
|
|
|
|
|
| colspan="3" |32\(83\77)
|
|
|498.701
|498.701
|501.205
|13
|13
|6
|6
Line 1,405: Line 1,214:
|
|
|-
|-
|
| colspan="3" |27\70
|
|498.462
|
|
|
|
| colspan="3" |27\(70\65)
|
|
|
|498.4615
|501.429
|11
|11
|5
|5
Line 1,422: Line 1,221:
|
|
|-
|-
|
| colspan="3" |22\57||498.113||9||4||2.250||
|
| || || || colspan="3" |22\(57\53)
|
|
|
| ||498.113
|501.754||9||4||2.250||
|-
|-
|
| colspan="3" |17\44||497.561||7||3||2.333||
|
| || || colspan="3" |17\(44\41)||
|
|
|
| ||497.561
|502.273||7||3||2.333||
|-
|-
|
| colspan="3" |29\75||497.143||12||5||2.400||
|
| || || || colspan="3" |29\(75\70)
|
|
|
| ||497.143
|502.667||12||5||2.400||
|-
|-
|
| colspan="3" |12\31||496.552||5||2||2.500||
|
| || colspan="3" |12\(31\29)|| ||
|
|
|
| ||496.552
|503.226||5||2||2.500||
|-
|-
|
| colspan="3" |31\80||496.000||13||5||2.600||
|
| || || || colspan="3" |31\(80\75)
|
|
|
| ||496
|503.75||13||5||2.600||
|-
|-
|
| colspan="3" |19\49||495.652||8||3||2.667||
|
| || || colspan="3" |19\(49\46)||
|
|
|
| ||495.652
|504.082||8||3||2.667||
|-
|-
|
| colspan="3" |26\67||495.238||11||4||2.750||
|
| || || || colspan="3" |26\(67\63)
|
|
|
| ||495.238
|504.478||11||4||2.750||
|-
|-
|
| colspan="3" |7\18||494.118||3||1||3.000||L/s = 3/1
|
| colspan="3" |7\(18\17)|| || ||
|
|
|
| ||494.118
|505.556||3||1||3.000||L/s = 3/1
|-
|-
|
| colspan="3" |30\77
|
|
|
|
|
| colspan="3" |30\(77\73)
|
|
|
|493.151
|493.151
|506.4935
|13
|13
|4
|4
Line 1,511: Line 1,244:
|
|
|-
|-
|
| colspan="3" |23\59||492.857||10||3||3.333||
|
| || || || colspan="3" |23\(59\56)
|
|
|
| ||492.857
|506.78||10||3||3.333||
|-
|-
|
| colspan="3" |16\41||492.308||7||2||3.500||
|
| || || colspan="3" |16\(41\39)||
|
|
|
| ||492.308
|507.317||7||2||3.500||
|-
|-
|
| colspan="3" |25\64||491.803||11||3||3.667||
|
| || || || colspan="3" |25\(64\61)
|
|
|
| ||491.803
|507.8125||11||3||3.667||
|-
|-
|
| colspan="3" |9\23||490.909||4||1||4.000||
|
| || colspan="3" |9\(23\22)|| ||
|
|
|
| ||490.909
|508.696||4||1||4.000||
|-
|-
|
| colspan="3" |20\51||489.796||9||2||4.500||
|
| || || || colspan="3" |20\(51\49)
|
|
|
| ||489.796
|509.804||9||2||4.500||
|-
|-
|
| colspan="3" |11\28||488.889||5||1||5.000||
|
| || || colspan="3" |11\(28\27)||
|
|
|
| ||488.889
|510.714||5||1||5.000||
|-
|-
|
| colspan="3" |24\61
|
|
|
|
|
| colspan="3" |24\(61\59)
|
|
|
|488.136
|488.136
|511.475
|11
|11
|2
|2
Line 1,582: Line 1,263:
|
|
|-
|-
|
| colspan="3" |13\33||487.500||6||1||6.000||
|
| || || || colspan="3" |13\(33\32)
|
|
|
| ||487.5
|512.121||6||1||6.000||
|-
|-
| colspan="3" |2\5|| || || || ||
| colspan="3" |2\5||480.000||1||0||→ inf||
|
|
|
| ||480.000
|520||1||0||→ inf||
|}
|}


== See also ==
== See also ==
[[5L 3s (45/22-equivalent)]] - undecimal small diesis tuning
[[5L 3s (33/16-equivalent)]] - harmonic subminor ninth tuning  
[[5L 3s (33/16-equivalent)]] - harmonic subminor ninth tuning  
[[5L 3s (56/27-equivalent)]] - Archytas diatonic minor ninth tuning
[[5L 3s (25/12-equivalent)]] - classical chromatic minor ninth tuning
[[5L 3s (44/21-equivalent)]], [[5L 3s (208/99-equivalent)]] - Neogothic undecimal diatonic minor ninth tuning


[[5L 3s (21/10-equivalent)]] - septimal chromatic minor ninth tuning
[[5L 3s (21/10-equivalent)]] - septimal chromatic minor ninth tuning
[[5L 3s (32/15-equivalent)]] - classical diatonic minor ninth tuning
[[5L 3s (891/416-equivalent)]], [[5L 3s (189/88-equivalent)]] - Neogothic chromatic minor ninth tuning


[[5L 3s (15/7-equivalent)]] - septimal diatonic minor ninth tuning
[[5L 3s (15/7-equivalent)]] - septimal diatonic minor ninth tuning
[[5L 3s (243/112-equivalent)]] - Archytas chromatic minor ninth tuning


[[5L 3s (11/5-equivalent)]] - undecimal neutral ninth tuning
[[5L 3s (11/5-equivalent)]] - undecimal neutral ninth tuning
Retrieved from "https://en.xen.wiki/w/User:Moremajorthanmajor/5L_3s_(minor_ninth-equivalent)"