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

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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, 211.765  
|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, 988.235
|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, 211.765  
|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, 289.655  
|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, 910.345  
|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, 988.235
|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, 503.23 (497.06)
|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 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

Revision as of 04:09, 2 June 2023

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 = 450¢). In the case of 8edo, L and s are the same size; in the case of 5ed8\7, s becomes so small it disappears (and all that remains are the five equal L's).

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

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 − s) by # and f "flat (F molle)".

The chain of perfect 4ths becomes: ... Ef Af Qf Ff Bf Df G C E A Q F B D ...

Thus the 13edIX gamut is as follows:

G/F# G#/Af A A#/Bf B/Cf C/B# C#/Qf Q Q#/Df D/Ef E/D# E#/Ff F/Gf G

The 18edIX gamut is notated as follows:

G F#/Af G# A Bf A#/Cf B C B#/Qf C# Q Df Q#/Ef D E D#/Ff E#/Gf F G

The 21edIX gamut:

G G# Af A A# Bf B B#/Cf C C# Qf Q Q# Df D D#/Ef E E# Ff F F#/Gf G

Names

The author suggests the name Neapolitan-oneirotonic.

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¢.

Notation (1/1 = G) name In L's and s's # generators up Notation of 2/1 inverse name In L's and s's
The 8-note MOS has the following intervals (from some root):
0 G perfect unison 0L + 0s 0 G “perfect” minor 9th 5L + 3s
1 C natural 4th 2L + 1s -1 Df minor 6th 3L + 2s
2 E major 7th 4L + 2s -2 Bf minor 3rd 1L + 1s
3 A major 2nd 1L + 0s -3 Ff diminished octave 4L + 3s
4 Q perfect 5th 3L + 1s -4 Qf diminished 5th 2L + 2s
5 F perfect octave 5L + 2s -5 Af minor 2nd 0L + 1s
6 B major 3rd 2L + 0s -6 Ef minor 7th 3L + 3s
7 D major 6th 4L + 1s -7 Cf diminished 4th 1L + 2s
The chromatic 13-note MOS (either 5L 8s, 8L 5s, or 13edIX) also has the following intervals (from some root):
8 G# augmented unison 1L - 1s -8 Gf diminished 9th 4L + 4s
9 C# augmented 4th 3L + 0s -9 Dff diminished 6th 2L + 3s
10 E# augmented 7th 5L + 1s -10 Bff diminished 3rd 0L + 2s
11 A# augmented 2nd 2L - 1s -11 Fff doubly diminished octave 3L + 4s
12 Q# augmented 5th 4L + 0s -12 Qff doubly diminished 5th 1L + 3s

Tuning ranges

Simple tunings

Table of intervals in the simplest Neapolitan-oneirotonic tunings:

Degree Size in 13edIX (basic) Size in 18edIX (hard) Size in 21edIX (soft) Note name on G #Gens up
unison 0\13, 0.00 0\18, 0.00 0\21, 0.00 G 0
minor 2nd 1\13, 100.00 1\18, 70.59 2\21, 126.32 Af -5
major 2nd 2\13, 200.00 3\18, 211.765 3\21, 189.47 A +3
minor 3rd 3\13, 300.00 4\18, 282.35 5\21, 315.79 Bf -2
major 3rd 4\13, 400.00 6\18, 423.53 6\21, 378.95 B +6
diminished 4th 5\18, 352.94 7\21, 442.105 Cf -7
natural 4th 5\13, 500.00 7\18, 494.12 8\21, 505.24 C +1
augmented 4th 6\13, 600.00 9\18, 635.29 9\21, 568.42 C# +9
diminished 5th 8\18, 564.71 10\21, 631.58 Qf -4
perfect 5th 7\13, 700.00 10\18, 705.88 11\31, 694.74 Q +4
minor 6th 8\13, 800.00 11\18, 776.47 13\21, 821.05 Df -1
major 6th 9\13, 900.00 13\18, 917.65 14\21, 884.21 D +7
minor 7th 12\18, 847.06 15\21, 947.37 Ef -6
major 7th 10\13, 1000.00 14\18, 988.235 16\21, 1017.53 E +2
diminished octave 11\13, 1100.00 15\18, 1052.82 18\21, 1136.84 Ff -3
perfect octave 12\13, 1200.00 17\18, 1200.00 19\21, 1200.00 F +5

Hypohard

Hypohard Neapolitan-oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.

Hypohard Neapolitan-oneirotonic can be considered " superpythagorean Neapolitan-oneirotonic". 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.

EDIXs that are in the hypohard range include 13edIX, 18edIX, and 31edIX.

The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various hypohard Neapolitan-oneiro tunings.

13edIX (basic) 18edIX (hard) 31edIX (semihard)
generator (g) 5\13, 500.00 7\18, 494.12 12\31, 496.55
L (3g - minor 9th) 2\13, 200.00 3\18, 211.765 5\31, 206.87
s (-5g + 2 minor 9ths) 1\13, 100.00 1\18, 70.59 2\31, 82.76

Intervals

Sortable table of major and minor intervals in hypohard Neapolitan-oneiro tunings:

Degree Size in 13edIX (basic) Size in 18edIX (hard) Size in 31edIX (semihard) Note name on G Approximate ratios[1] #Gens up
unison 0\13, 0.00 0\18, 0.00 0\31, 0.00 G 1/1 0
minor 2nd 1\13, 100.00 1\18, 70.59 2\31, 82.76 Af 21/20, 22/21 -5
major 2nd 2\13, 200.00 3\18, 211.765 5\31, 206.87 A 9/8 +3
minor 3rd 3\13, 300.00 4\18, 282.35 7\31, 289.655 Bf 13/11, 33/28 -2
major 3rd 4\13, 400.00 6\18, 423.53 10\31, 413.79 B 14/11, 33/26 +6
diminished 4th 5\18, 352.94 9\31, 372.41 Cf 5/4, 11/9 -7
natural 4th 5\13, 500.00 7\18, 494.12 12\31, 496.55 C 4/3 +1
augmented 4th 6\13, 600.00 9\18, 635.29 15\31, 620.69 C# 10/7, 18/13, 11/8 +9
diminished 5th 8\18, 564.71 14\31, 579.31 Qf 7/5, 13/9, 16/11 -4
perfect 5th 7\13, 700.00 10\18, 705.88 17\31, 703.45 Q 3/2 +4
minor 6th 8\13, 800.00 11\18, 776.47 19\31, 786.21 Df 52/33, 11/7 -1
major 6th 9\13, 900.00 13\18, 917.65 22\31, 910.345 D 56/33, 22/17 +7
minor 7th 12\18, 847.06 21\31, 868.97 Ef 5/3, 18/11 -6
major 7th 10\13, 1000.00 14\18, 988.235 24\31, 993.13 E 16/9 +2
diminished octave 11\13, 1100.00 15\18, 1052.82 26\31, 1075.86 Ff 11/6, 13/7, 15/8 -3
perfect octave 12\13, 1200.00 17\18, 1200.00 29\31, 1200.00 F 2/1 +5
  1. ↑ The ratio interpretations that are not valid for 18edo are italicized.

Hyposoft

Hyposoft Neapolitan-oneirotonic tunings (with generator between 8\21 and 5\13) have step ratios between 3/2 and 2/1. The 8\21-to-5\13 range of Neapolitan-oneirotonic tunings can be considered "meantone Neapolitan-oneirotonic”. 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 Neapolitan-oneirotonic are as follows in various hyposoft Neapolitan-oneiro tunings (13edIX not shown).

21edIX (soft) 34edIX (semisoft)
generator (g) 8\21, 505.26 13\34, 503.23
L (3g - minor 9th) 3\21, 189.47 5\34, 193.55
s (-5g + 2 minor 9ths) 2\21, 126.32 3\34, 116.19

Intervals

Sortable table of major and minor intervals in hyposoft tunings (13edIX not shown):

Degree Size in 21edIX (soft) Size in 34edIX (semisoft) Note name on G Approximate ratios #Gens up
unison 0\21, 0.00 0\34, 0.00 G 1/1 0
minor 2nd 2\21, 126.32 3\34, 116.19 Af 16/15 -5
major 2nd 3\21, 189.47 5\34, 193.55 A 10/9, 9/8 +3
minor 3rd 5\21, 315.79 8\34, 309.68 Bf 6/5 -2
major 3rd 6\21, 378.95 10\34, 387.10 B 5/4 +6
diminished 4th 7\21, 442.105 11\34, 425.81 Cf 9/7 -7
natural 4th 8\21, 505.24 13\34, 503.23 C 4/3 +1
augmented 4th 9\21, 568.42 15\34, 580.645 C# 7/5 +9
diminished 5th 10\21, 631.58 16\34, 619.355 Qf 10/6 -4
perfect 5th 11\31, 694.74 18\34, 696.77 Q 3/2 +4
minor 6th 13\21, 821.05 21\34, 812.90 Df 8/5 -1
major 6th 14\21, 884.21 23\34, 890.32 D 5/3 +7
minor 7th 15\21, 947.37 24\34, 929.03 Ef 12/7 -6
major 7th 16\21, 1017.53 26\34, 1006.45 E 9/5, 16/9 +2
diminished octave 18\21, 1136.84 29\34, 1122.58 Ff 27/14, 48/25 -3
perfect octave 19\21, 1200.00 31\34, 1200.00 F 2/1 +5

Parasoft to ultrasoft tunings

The range of Neapolitan-oneirotonic 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 Neapolitan-oneirotonic are as follows in various tunings in this range.

29edIX (supersoft) 37edIX
generator (g) 11\29, 507.69 14\37, 509.09
L (3g - minor 9th) 4\29, 184.615 5\37, 181.82
s (-5g + 2 minor 9ths) 3\29, 138.46 4\37, 145.455

Intervals

The intervals of the extended generator chain (-15 to +15 generators) are as follows in various softer-than-soft Neapolitan-oneirotonic tunings.

Degree Size in 29edIX (supersoft) Note name on G Approximate ratios #Gens up
unison 0\29, 0.00 G 1/1 0
chroma 1\29, 46.15 G# 33/32, 49/48, 36/35, 25/24 +8
diminished 2nd 2\29, 92.31 Aff 21/20, 22/21, 26/25 -13
minor 2nd 3\29, 138.46 Af 12/11, 13/12, 14/13, 16/15 -5
major 2nd 4\29, 184.615 A 9/8, 10/9, 11/10 +3
augmented 2nd 5\29, 230.77 A# 8/7, 15/13 +11
diminished 3rd 6\29, 276.92 Bff 7/6, 13/11, 33/28 -10
minor 3rd 7\29, 323.08 Bf 135/112, 6/5 -2
major 3rd 8\29, 369.23 B 5/4, 11/9, 16/13 +6
augmented 3rd 9\29, 415.385 B# 9/7, 14/11, 33/26 +14
diminished 4th 10\29, 461.54 Cf 21/16, 13/10 -7
natural 4th 11\29, 507.69 C 75/56, 4/3 +1
augmented 4th 12\29, 553.85 C# 11/8, 18/13 +9
doubly augmented 4th, doubly diminished 5th 13\29, 600.00 Cx, Qff 7/5, 10/7 -12
diminished 5th 14\29, 646.15 Qf 16/11, 13/9 -4
perfect 5th 15\29, 692.31 Q 112/75, 3/2 +4
augmented 5th 16\29, 738.46 Q# 32/21, 20/13 +12
diminished 6th 17\29, 784.615 Dff 11/7, 14/9 -9
minor 6th 18\29, 830.77 Df 13/8, 8/5 -1
major 6th 19\29, 876.92 D 5/3, 224/135 +7
augmented 6th 20\29, 923.08 D# 12/7, 22/13 -14
minor 7th 21\29, 969.23 Ef 7/4, 26/15 -6
major 7th 22\29, 1015.385 E 9/5, 16/9, 20/11 +2
augmented 7th 23\29, 1061.54 E# 11/6, 13/7, 15/8, 24/13 +10
doubly augmented 7th, doubly diminished octave 24\29, 1107.69 Ex, Fff 21/11, 25/13, 40/21 -11
diminished octave 25\29, 1153.85 Ff 64/33, 96/49, 35/18, 48/25 -3
perfect octave 26\29, 1200.00 F 2/1 +5
augmented octave 27\29, 1246.15 F# 33/16, 49/24, 72/35, 25/12 +13
doubly augmented octave, diminished 9th 28\29, 1292.31 Fx, Gf 21/10, 44/21, 52/25 -8

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.

Intervals

The intervals of the extended generator chain (-12 to +12 generators) are as follows in various Neapolitan-oneirotonic tunings close to 23edIX.

Degree Size in 23edIX

(superhard)

Note name on G Approximate ratios (23edIX) #Gens up
unison 0\23, 0.00 G 1/1 0
chroma 3\23, 163.63 G# 12/11, 11/10, 10/9 +8
minor 2nd 1\23, 54.545 Af 36/35, 34/33, 33/32, 32/31 -5
major 2nd 4\23, 218.18 A 9/8, 17/15, 8/7 +3
aug. 2nd 7\23, 381.82 A# 5/4 +11
dim. 3rd 2\23, 109.09 Bf 16/15 -10
minor 3rd 5\23, 272.73 B 7/6 -2
major 3rd 8\23, 436.36 B# 9/7, 14/11 +6
dim. 4th 6\23, 327.27 Cf 6/5 -7
nat. 4th 9\23, 490.91 C 4/3 +1
aug. 4th 12\23, 654.545 C# 16/11, 22/15 +9
double dim. 5th 7\23, 381.82 Qff 5/4 -12
dim. 5th 10\23, 545.455 Qf 15/11, 11/8 -4
perf. 5th 13\23, 709.09 Q 3/2 +4
aug. 5th 16\23, 872.73 Q# 5/3 +12
dim. 6th 11\23, 600.00 Dff 7/5, 24/17, 17/12, 10/7 -9
minor 6th 14\23, 763.64 Df 14/9, 11/7 -1
major 6th 17\23, 927.27 D 12/7 +7
minor 7th 15\23, 818.18 Ef 8/5 -6
major 7th 18\23, 981.82 E 7/4, 30/17, 16/9 +2
aug. 7th 21\23, 1145.455 E# 31/16, 64/33, 33/17, 35/18 +10
dim. octave 19\23, 1036.36 Ff 11/6, 20/11, 9/5 -11
perf. octave 22\23, 1200.00 F 2/1 -3
aug. octave 25\23, 1363.64 F# 24/11, 11/5, 20/9 +5
dim. ninth 20\23, 1090.91 J@ 15/8 -8

Ultrahard

Ultrapythagorean Neapolitan-oneirotonic is a rank-2 temperament in the pseudopaucitonic range. It represents the harmonic entropy minimum of the Neapolitan-oneirotonic spectrum where 7/4 is the major seventh.

In the broad sense, Ultrapyth can be viewed as any tuning that divides a 16/7 into 2 equal parts. 23edIX, 28edIX and 33edIX can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edIX and true Buzzard in terms of harmonies. 38edIX & 43edIX 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 48edIX is where it really comes into its own in terms of harmonies, providing not only excellent 7:8:9 melodies, but also 5/4 and archipelago harmonies, 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. 53edIX has an essentially perfect 7/4, 58edIX also gives three essentially perfect chains of third-comma meantone, while 63edIX has a double chain of essentially perfect quarter-comma meantone and gives about as low overall error as 83edIX does for the basic 4:6:7 triad. But beyond 83edIX, 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 Neapolitan-oneirotonic are as follows in various ultrapyth tunings.

38edIX 53edIX 63edIX Optimal (PNTE) Ultrapyth tuning JI intervals represented (2.3.5.7.13 subgroup)
generator (g) 15\38, 486.49 21\53, 484.615 25\63, 483.87 484.07 4/3
L (3g - minor 9th) 7/38, 227.03 10/53, 230.77 12/63, 232.26 231.51 8/7
s (-5g + 2 minor 9ths) 1/38, 32.43 1/53, 23.08 1/63, 19.355 21.05 50/49 81/80 91/90

Intervals

Sortable table of intervals in the Neapolitan-Dylathian mode and their Ultrapyth interpretations:

Degree Size in 38edIX Size in 53edIX Size in 63edIX Size in PNTE tuning Note name on G Approximate ratios #Gens up
1 0\38, 0.00 0\53, 0.00 0\63, 0.00 0.00 G 1/1 0
2 7/38, 227.03 10/53, 230.77 12/63, 232.26 231.51 A 8/7 +3
3 14\38, 454.05 20\53, 461.54 24\63, 464.52 463.03 B 13/10, 21/16 +6
4 15\38, 486.49 21\53, 484.615 25\63, 483.87 484.07 C 4/3 +1
5 22\38, 713.51 31\53, 715.385 37\63, 716.13 715.59 Q 3/2 +4
6 29\38, 940.54 41\53, 946.15 49\63, 948.39 947.10 D 26/15 +7
7 30\38, 972.97 42\53, 969.23 50\63, 967.74 968.15 E 7/4 +2
8 37\38, 1200.00 52\53, 1200.00 62\63, 1200.00 1199.66 F 2/1 +5

Modes

Neapolitan-Oneirotonic modes are named after cities in the Dreamlands.

Mode UDP Name
LLsLLsLs 7|0 Neapolitan-Dylathian (də-LA(H)TH-iən)
LLsLsLLs 6|1 Neapolitan-Illarnekian (ill-ar-NEK-iən)
LsLLsLLs 5|2 Neapolitan-Celephaïsian (kel-ə-FAY-zhən)
LsLLsLsL 4|3 Neapolitan-Ultharian (ul-THA(I)R-iən)
LsLsLLsL 3|4 Neapolitan-Mnarian (mə-NA(I)R-iən)
sLLsLLsL 2|5 Neapolitan-Kadathian (kə-DA(H)TH-iən)
sLLsLsLL 1|6 Neapolitan-Hlanithian (lə-NITH-iən)
sLsLLsLL 0|7 Neapolitan-Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"

Scale tree

Normalized Generator Cents L s L/s Comments
3\(8\7) 514.286 1 1 1.000
17\(45\40) 510 6 5 1.200
14\(37\33) 509.091 5 4 1.250
25\(66\59) 508.475 9 7 1.286
11\(29\26) 507.692 4 3 1.333
30\(79\71) 507.042 11 8 1.375
19\(50\45) 506.667 7 5 1.400
27\(71\64) 506.25 10 7 1.429
8\(21\19) 505.263 3 2 1.500 L/s = 3/2
29\(76\69) 504.348 11 7 1.571
21\(55\50) 504 8 5 1.600
34\(89\81) 503.704 13 8 1.625 Golden Neapolitan-oneirotonic
13\(34\31) 503.226 5 3 1.667
31\(81\74) 502.703 12 7 1.714
18\(47\43) 502.326 7 4 1.750
23\(60\55) 501.818 9 5 1.800
28\(73/67) 501.4925 11 6 1.833
33\(86\79) 501.265 13 7 1.857
38\(99\91) 501.099 15 8 1.875
43\(112\103) 500.971 17 9 1.889
5\(13\12) 500 2 1 2.000 Basic Neapolitan-oneirotonic

(generators smaller than this are proper)

42\(109\101) 499.01 17 8 2.125
37\(96\89) 498.876 15 7 2.143
32\(83\77) 498.701 13 6 2.167
27\(70\65) 498.4615 11 5 2.200
22\(57\53) 498.113 9 4 2.250
17\(44\41) 497.561 7 3 2.333
29\(75\70) 497.143 12 5 2.400
12\(31\29) 496.552 5 2 2.500
31\(80\75) 496 13 5 2.600
19\(49\46) 495.652 8 3 2.667
26\(67\63) 495.238 11 4 2.750
7\(18\17) 494.118 3 1 3.000 L/s = 3/1
30\(77\73) 493.151 13 4 3.250
23\(59\56) 492.857 10 3 3.333
16\(41\39) 492.308 7 2 3.500
25\(64\61) 491.803 11 3 3.667
9\(23\22) 490.909 4 1 4.000
20\(51\49) 489.796 9 2 4.500
11\(28\27) 488.889 5 1 5.000
24\(61\59) 488.136 11 2 5.500
13\(33\32) 487.5 6 1 6.000
2\5 480.000 1 0 → inf

See also

5L 3s (33/16-equivalent) - harmonic subminor ninth tuning

5L 3s (21/10-equivalent) - septimal chromatic minor ninth tuning

5L 3s (15/7-equivalent) - septimal diatonic minor ninth tuning

5L 3s (11/5-equivalent) - undecimal neutral ninth tuning

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