User:Moremajorthanmajor/5L 3s (11/5-equivalent): Difference between revisions

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{{Infobox MOS
{{Infobox MOS
|Tuning=5L 3s<11/5>}}The minor ninth of a diatonic scale has a '''5L 3s''' [[MOS]] structure with generators ranging from two degrees of [[5ed11/5]] = 511.9¢ to three degrees of [[8ed11/5]] = 546¢. In the case of 8ed11/5, L and s are the same size; in the case of 5ed11/5, s becomes so small it disappears (and all that remains are the five equal L's).
|Tuning=5L 3s<11/5>}}{{MOS intro|Scale Signature=5L 3s<11/5>}}


Any ed11/5 with an interval between 511.9¢ and 546¢ has a 5L 3s scale. [[13edIX]] is the smallest ed11/5 with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.
[[13ed11/5]] is the smallest ed11/5 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<11/5><ref>the rare simplest tuning for a diatonic minor ninth </ref> intervals and step size ratios and step ratio ranges.
The [[TAMNAMS]] system is used in this article to name 5L 3s<11/5><ref>the rare simplest tuning for a diatonic minor ninth </ref> intervals and step size ratios and step ratio ranges.


Line 34: Line 34:
!Notation of 2/1 inverse
!Notation of 2/1 inverse
!name
!name
!In L's and s's
! In L's and s's
|-
|-
| colspan="8" |The 8-note MOS has the following intervals (from some root):
| colspan="8" |The 8-note MOS has the following intervals (from some root):
Line 45: Line 45:
|G
|G
|“perfect” minor 9th
|“perfect” minor 9th
|5L + 3s
| 5L + 3s
|-
|-
|1
|1
Line 69: Line 69:
|major 2nd
|major 2nd
|1L + 0s
|1L + 0s
| -3
| -3
|Ff
|Ff
|diminished octave
|diminished octave
Line 93: Line 93:
|-
|-
|6
|6
|B
| B
|major 3rd
|major 3rd
|2L + 0s
|2L + 0s
Line 103: Line 103:
|7
|7
|D
|D
|major 6th
| major 6th
|4L + 1s
|4L + 1s
| -7
| -7
|Cf
|Cf
|diminished 4th
| diminished 4th
|1L + 2s
|1L + 2s
|-
|-
Line 116: Line 116:
|augmented unison
|augmented unison
|1L - 1s
|1L - 1s
| -8
| -8
|Gf
|Gf
|diminished 9th
|diminished 9th
|4L + 4s
|4L + 4s
|-
|-
|9
| 9
|C#
|C#
|augmented 4th
| augmented 4th
|3L + 0s
|3L + 0s
| -9
| -9
|Dff
|Dff
|diminished 6th
|diminished 6th
Line 133: Line 133:
|E#
|E#
|augmented 7th
|augmented 7th
|5L + 1s
| 5L + 1s
| -10
| -10
|Bff
|Bff
Line 144: Line 144:
|2L - 1s
|2L - 1s
| -11
| -11
|Fff
| Fff
|doubly diminished octave
|doubly diminished octave
|3L + 4s
|3L + 4s
Line 153: Line 153:
|4L + 0s
|4L + 0s
| -12
| -12
|Qff
| Qff
|doubly diminished 5th
|doubly diminished 5th
|1L + 3s
|1L + 3s
|}
|}
==Tuning ranges==
==Tuning ranges==
===Simple tunings===
=== Simple tunings===
Table of intervals in the simplest Neapolitan-oneirotonic tunings:
Table of intervals in the simplest Neapolitan-oneirotonic tunings:
{| class="wikitable right-2 right-3 right-4 sortable"
{| class="wikitable right-2 right-3 right-4 sortable"
Line 176: Line 176:
|0
|0
|-
|-
|minor 2nd
| minor 2nd
|1\13, 105.000
|1\13, 105.000
|1\18, 75.834
|1\18, 75.834
|2\21, 130.000  
|2\21, 130.000
|Af
|Af
| -5
| -5
Line 193: Line 193:
|3\13, 315.001
|3\13, 315.001
|4\18, 303.334
|4\18, 303.334
|5\21, 325.001  
|5\21, 325.001
|Bf
|Bf
| -2
| -2
Line 199: Line 199:
|major 3rd
|major 3rd
| rowspan="2" |4\13, 420.001
| rowspan="2" |4\13, 420.001
|6\18, 455.001  
|6\18, 455.001
|6\21, 390.001  
|6\21, 390.001
|B
|B
| +6
| +6
|-
|-
|diminished 4th
| diminished 4th
|5\18, 379.168
|5\18, 379.168
|7\21, 455.001
|7\21, 455.001
Line 211: Line 211:
|-
|-
|natural 4th
|natural 4th
|5\13, 525.002
| 5\13, 525.002
|7\18, 530.835
|7\18, 530.835
|8\21, 520.002  
|8\21, 520.002
|C
|C
| +1
| +1
Line 219: Line 219:
|augmented 4th
|augmented 4th
| rowspan="2" |6\13, 630.002
| rowspan="2" |6\13, 630.002
|9\18, 682.502  
|9\18, 682.502
|9\21, 585.002
|9\21, 585.002
|C#
|C#
Line 225: Line 225:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|8\18, 606.669  
|8\18, 606.669
|10\21, 650.002
|10\21, 650.002
|Qf
|Qf
| -4
| -4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect 5th
|perfect 5th
|7\13, 735.002
|7\13, 735.002
|10\18, 758.336
|10\18, 758.336
|11\21, 715.002
| 11\21, 715.002
|Q
|Q
| +4
| +4
Line 239: Line 239:
|minor 6th
|minor 6th
|8\13, 840.003
|8\13, 840.003
|11\18, 834.169  
|11\18, 834.169
|13\21, 780.002
|13\21, 780.002
|Df
|Df
Line 246: Line 246:
|major 6th
|major 6th
| rowspan="2" |9\13, 945.003
| rowspan="2" |9\13, 945.003
|13\18, 985.836  
|13\18, 985.836
|14\21, 845.003  
|14\21, 845.003
|D
|D
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
| minor 7th
|12\18, 910.003  
|12\18, 910.003
|15\21, 910.003  
|15\21, 910.003
|Ef
|Ef
| -6
| -6
Line 259: Line 259:
|major 7th
|major 7th
|10\13, 1050.003
|10\13, 1050.003
|14\18, 1061.670  
|14\18, 1061.670
|16\21, 975.003
|16\21, 975.003
|E
|E
Line 266: Line 266:
|diminished octave
|diminished octave
|11\13, 1155.004
|11\13, 1155.004
|15\18, 1137.5035
|15\18, 1137.504
|18\21, 1105.003  
|18\21, 1105.003
|Ff
| Ff
| -3
| -3
|-
|-
|perfect octave
|perfect octave
|12\13, 1260.004
|12\13, 1260.004
|17\18, 1289.171  
|17\18, 1289.171
|19\21, 1170.003  
|19\21, 1170.003
|F
|F
| +5
| +5
Line 297: Line 297:
|5\13, 525.002
|5\13, 525.002
|10\18, 758.336
|10\18, 758.336
|12\31, 496.55 (503.23)
|12\31, 496.55  
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|2\13, 210.001
|2\13, 210.001
|3\18, 227.501
|3\18, 227.501
|5\31, 206.87 (209.68)
|5\31, 206.87
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|1\13, 105.000
|1\13, 105.000
|1\18, 75.834
| 1\18, 75.834
|2\31, 82.76 (83.87)
|2\31, 82.76  
|}
|}
====Intervals====
====Intervals====
Line 326: Line 326:
|0\31, 0.00
|0\31, 0.00
|G
|G
|1/1
| 1/1
|0
|0
|-
|-
Line 332: Line 332:
|1\13, 105.000
|1\13, 105.000
|1\18, 75.834
|1\18, 75.834
|2\31, 88.065  
|2\31, 88.065
|Af
|Af
|21/20, ''22/21''
|21/20, ''22/21''
Line 343: Line 343:
|A
|A
|9/8
|9/8
| +3
| +3
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
Line 355: Line 355:
|major 3rd
|major 3rd
| rowspan="2" |4\13, 420.001
| rowspan="2" |4\13, 420.001
|6\18, 455.001  
|6\18, 455.001
|10\31, 440.324
|10\31, 440.324
|B
|B
Line 374: Line 374:
|C
|C
|4/3
|4/3
| +1
| +1
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 4th
|augmented 4th
| rowspan="2" |6\13, 630.002
| rowspan="2" |6\13, 630.002
|9\18, 682.502  
|9\18, 682.502
|15\31, 660.486
|15\31, 660.486
|C#
|C#
Line 385: Line 385:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|8\18, 606.669  
| 8\18, 606.669
|14\31, 616.4535
|14\31, 616.454
|Qf
|Qf
|''7/5, 13/9'', ''16/11''
|''7/5, 13/9'', ''16/11''
| -4
| -4
|-
|-
|perfect 5th
| perfect 5th
|7\13, 735.002
|7\13, 735.002
|10\18, 758.336
|10\18, 758.336
|17\31, 748.551
| 17\31, 748.551
|Q
|Q
|3/2
|3/2
Line 401: Line 401:
|minor 6th
|minor 6th
|8\13, 840.003
|8\13, 840.003
|11\18, 834.169  
|11\18, 834.169
|19\31, 836.6155
|19\31, 836.615
|Df
|Df
|52/33, 11/7
| 52/33, 11/7
| -1
| -1
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6th
| major 6th
| rowspan="2" |9\13, 945.003
| rowspan="2" |9\13, 945.003
|13\18, 985.836  
|13\18, 985.836
|22\31, 968.713
|22\31, 968.713
|D
| D
|56/33, 22/17
|56/33, 22/17
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|12\18, 910.003  
|12\18, 910.003
|21\31, 924.680
|21\31, 924.680
|Ef
|Ef
Line 422: Line 422:
| -6
| -6
|-
|-
|major 7th
| major 7th
|10\13, 1050.003
|10\13, 1050.003
|14\18, 1061.670  
|14\18, 1061.670
|24\31, 1056.778
|24\31, 1056.778
|E
|E
Line 432: Line 432:
|diminished octave
|diminished octave
|11\13, 1155.004
|11\13, 1155.004
|15\18, 1137.5035  
|15\18, 1137.5035
|26\31, 1144.84
|26\31, 1144.84
|Ff
|Ff
Line 440: Line 440:
|perfect octave
|perfect octave
|12\13, 1260.004
|12\13, 1260.004
|17\18, 1289.171  
|17\18, 1289.171
|29\31, 1276.939  
|29\31, 1276.939
|F
|F
|2/1
|2/1
Line 456: Line 456:
|-
|-
!
!
!21ed11/5 (soft)
! 21ed11/5 (soft)
!34ed11/5 (semisoft)
!34ed11/5 (semisoft)
|-
|-
|generator (g)
| generator (g)
|8\21, 520.002  
|8\21, 520.002
|13\34, 521.913  
|13\34, 521.913
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|3\21, 195.001
| 3\21, 195.001
|5\34, 200.736  
|5\34, 200.736
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|2\21, 130.000  
|2\21, 130.000
|3\34, 120.4415
|3\34, 120.442
|}
|}
====Intervals====
====Intervals====
Line 490: Line 490:
|-
|-
|minor 2nd
|minor 2nd
|2\21, 130.000  
| 2\21, 130.000
|3\34, 120.4415
|3\34, 120.442
|Af
|Af
|16/15
|16/15
Line 498: Line 498:
|major 2nd
|major 2nd
|3\21, 195.001
|3\21, 195.001
|5\34, 200.736  
|5\34, 200.736
|A
|A
|10/9, 9/8
|10/9, 9/8
| +3
| +3
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|5\21, 325.001  
|5\21, 325.001
|8\34, 321.1775
|8\34, 321.177
|Bf
|Bf
|6/5
|6/5
Line 511: Line 511:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|6\21, 390.001  
|6\21, 390.001
|10\34, 401.472  
|10\34, 401.472
|B
|B
|5/4
|5/4
Line 519: Line 519:
|diminished 4th
|diminished 4th
|7\21, 455.001
|7\21, 455.001
|11\34, 441.619  
|11\34, 441.619
|Cf
|Cf
|9/7
|9/7
Line 525: Line 525:
|-
|-
|natural 4th
|natural 4th
|8\21, 520.002  
|8\21, 520.002
|13\34, 521.913  
|13\34, 521.913
|C
|C
|4/3
| 4/3
| +1
| +1
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 4th
| augmented 4th
|9\21, 585.002
|9\21, 585.002
|15\34, 602.208  
|15\34, 602.208
|C#
| C#
|7/5
|7/5
| +9
| +9
Line 540: Line 540:
|diminished 5th
|diminished 5th
|10\21, 650.002
|10\21, 650.002
|16\34, 642.355  
| 16\34, 642.355
|Qf
|Qf
|10/6
|10/6
Line 547: Line 547:
|perfect 5th
|perfect 5th
|11\21, 715.002
|11\21, 715.002
|18\34, 722.649
| 18\34, 722.649
|Q
|Q
|3/2
|3/2
Line 560: Line 560:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6th
|major 6th
|14\21, 845.003  
|14\21, 845.003
|23\34, 923.385
|23\34, 923.385
|D
|D
|5/3
|5/3
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|15\21, 910.003  
|15\21, 910.003
|24\34, 963.532
|24\34, 963.532
|Ef
|Ef
Line 575: Line 575:
|major 7th
|major 7th
|16\21, 975.003
|16\21, 975.003
|26\34, 1043.826  
|26\34, 1043.826
|E
|E
|9/5, 16/9
|9/5, 16/9
Line 581: Line 581:
|-
|-
|diminished octave
|diminished octave
|18\21, 1105.003  
|18\21, 1105.003
|29\34, 1164.268  
|29\34, 1164.268
|Ff
|Ff
|27/14, 48/25
|27/14, 48/25
Line 588: Line 588:
|-
|-
|perfect octave
|perfect octave
|19\21, 1170.003  
|19\21, 1170.003
|31\34, 1244.563  
|31\34, 1244.563
|F
|F
|2/1
|2/1
Line 604: Line 604:
!37ed11/5
!37ed11/5
|-
|-
|generator (g)
| generator (g)
|11\29, 517.760  
|11\29, 517.760
|14\37, 516.488  
|14\37, 516.488
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|4\29, 188.276  
|4\29, 188.276
|5\37, 184.460  
| 5\37, 184.460
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
Line 633: Line 633:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|chroma
|chroma
|1\29, 47.069  
|1\29, 47.069
|G#
|G#
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
Line 639: Line 639:
|-
|-
|diminished 2nd
|diminished 2nd
|2\29, 94.138  
|2\29, 94.138
|Aff
|Aff
|[[21/20]], [[22/21]], [[26/25]]
|[[21/20]], [[22/21]], [[26/25]]
Line 650: Line 650:
| -5
| -5
|-
|-
|major 2nd
| major 2nd
|4\29, 188.276  
|4\29, 188.276
|A
|A
|[[9/8]], [[10/9]], [[11/10]]
|[[9/8]], [[10/9]], [[11/10]]
Line 658: Line 658:
|augmented 2nd
|augmented 2nd
|5\29, 235.346
|5\29, 235.346
|A#
| A#
|[[8/7]], [[15/13]]
|[[8/7]], [[15/13]]
| +11
| +11
Line 675: Line 675:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|8\29, 376.553  
|8\29, 376.553
|B
|B
|[[5/4]], [[11/9]], [[16/13]]
|[[5/4]], [[11/9]], [[16/13]]
| +6
| +6
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 3rd
| augmented 3rd
|9\29, 423.622
|9\29, 423.622
|B#
| B#
|[[9/7]], [[14/11]], [[33/26]]
|[[9/7]], [[14/11]], [[33/26]]
| +14
| +14
|-
|-
|diminished 4th
|diminished 4th
|10\29, 470.691  
|10\29, 470.691
|Cf
|Cf
|[[21/16]], [[13/10]]
|[[21/16]], [[13/10]]
Line 693: Line 693:
|-
|-
|natural 4th
|natural 4th
|11\29, 517.760  
|11\29, 517.760
|C
|C
|[[75/56]], [[4/3]]
|[[75/56]], [[4/3]]
Line 702: Line 702:
|C#
|C#
|[[11/8]], [[18/13]]
|[[11/8]], [[18/13]]
| +9
| +9
|-
|-
|doubly augmented 4th, doubly diminished 5th
|doubly augmented 4th, doubly diminished 5th
|13\29, 611.8985
|13\29, 611.898
|Cx, Qff
|Cx, Qff
|[[7/5]], [[10/7]]
|[[7/5]], [[10/7]]
Line 720: Line 720:
|Q
|Q
|[[112/75]], [[3/2]]
|[[112/75]], [[3/2]]
| +4
| +4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 5th
|augmented 5th
Line 735: Line 735:
|-
|-
|minor 6th
|minor 6th
|18\29, 847.244  
|18\29, 847.244
|Df
|Df
|[[13/8]], [[8/5]]
|[[13/8]], [[8/5]]
Line 747: Line 747:
|-
|-
|augmented 6th
|augmented 6th
|20\29, 941.382  
| 20\29, 941.382
|D#
|D#
|[[12/7]], [[22/13]]
|[[12/7]], [[22/13]]
Line 765: Line 765:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 7th
|augmented 7th
|23\29, 1082.590  
|23\29, 1082.590
|E#
|E#
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
Line 777: Line 777:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished octave
|diminished octave
|25\29, 1176.728  
|25\29, 1176.728
|Ff
| Ff
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
| -3
| -3
|-
|-
|perfect octave
|perfect octave
Line 802: Line 802:
===Parahard===
===Parahard===
23ed11/5 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.  
23ed11/5 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====
====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.
{| class="wikitable right-2 right-3 sortable"
{| class="wikitable right-2 right-3 sortable"
Line 816: Line 816:
|0\23, 0.00
|0\23, 0.00
|G
|G
|1/1
| 1/1
|0
|0
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
Line 826: Line 826:
|-
|-
|minor 2nd
|minor 2nd
|1\23, 59.348  
| 1\23, 59.348
|Af
|Af
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
| -5
| -5
|-
|-
|major 2nd
|major 2nd
|4\23, 237.392
| 4\23, 237.392
|A
|A
|[[9/8]], [[17/15]], [[8/7]]
|[[9/8]], [[17/15]], [[8/7]]
Line 850: Line 850:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|5\23, 296.740  
|5\23, 296.740
|B
|B
|7/6
| 7/6
| -2
| -2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
Line 867: Line 867:
| -7
| -7
|-
|-
|nat. 4th
| nat. 4th
|9\23, 534.132
|9\23, 534.132
|C
|C
Line 894: Line 894:
|13\23, 771.524
|13\23, 771.524
|Q
|Q
|3/2
| 3/2
| +4
| +4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
Line 901: Line 901:
|Q#
|Q#
|5/3
|5/3
| +12
| +12
|-
|-
|dim. 6th
|dim. 6th
Line 915: Line 915:
| -1
| -1
|-
|-
|major 6th
| major 6th
|17\23, 1008.916
|17\23, 1008.916
|D
|D
|12/7
| 12/7
| +7
| +7
|-
|-
Line 940: Line 940:
| +10
| +10
|-
|-
|dim. octave
| dim. octave
|19\23, 1127.612
|19\23, 1127.612
|Ff
|Ff
|11/6, 20/11, 9/5
| 11/6, 20/11, 9/5
| -11
| -11
|-
|-
Line 952: Line 952:
| -3
| -3
|-
|-
|aug. octave
| aug. octave
|25\23, 1483.700
|25\23, 1483.700
|F#
|F#
Line 963: Line 963:
|Gf
|Gf
|15/8
|15/8
| -8
| -8
|}
|}
===Ultrahard===
===Ultrahard===
Line 983: Line 983:
|-
|-
|generator (g)
|generator (g)
|15\38, 538.8175
|15\38, 538.817
|21\53, 540.850
| 21\53, 540.850
|25\63, 541.668
|25\63, 541.668
|484.07  
|484.07
|4/3
|4/3
|-
|-
Line 993: Line 993:
|10/53, 257.548
|10/53, 257.548
|12/63, 260.001
|12/63, 260.001
|231.51  
|231.51
|8/7
|8/7
|-
|-
Line 1,008: Line 1,008:
|-
|-
!Degree
!Degree
!Size in 38edIX
!Size in 38ed11/5
!Size in 53edIX
!Size in 53ed11/5
!Size in 63edIX
!Size in 63ed11/5
!Size in PNTE tuning
!Size in PNTE tuning
!Note name on G
!Note name on G
Line 1,025: Line 1,025:
|0
|0
|-
|-
|2
| 2
|7/38, 251.448
|7/38, 251.448
|10/53, 257.548
|10/53, 257.548
|12/63, 260.001
|12/63, 260.001
|231.51  
|231.51
|A
|A
|8/7
|8/7
| +3
| +3
|-
|-
|3
|3
Line 1,038: Line 1,038:
|20\53, 515.096
|20\53, 515.096
|24\63, 520.002
|24\63, 520.002
|463.03  
|463.03
|B
|B
|13/10, 21/16
|13/10, 21/16
Line 1,044: Line 1,044:
|-
|-
|4
|4
|15\38, 538.8175
|15\38, 538.817
|21\53, 540.850
|21\53, 540.850
|25\63, 541.668
|25\63, 541.668
|484.07  
|484.07
|C
|C
|4/3
|4/3
Line 1,056: Line 1,056:
|31\53, 798.399
|31\53, 798.399
|37\63, 801.669
|37\63, 801.669
|715.59  
|715.59
|Q
|Q
|3/2
|3/2
Line 1,065: Line 1,065:
|41\53, 1055.947
|41\53, 1055.947
|49\63, 1061.670
|49\63, 1061.670
|947.10  
|947.10
|D
|D
|26/15
|26/15
Line 1,071: Line 1,071:
|-
|-
|7
|7
|30\38, 1077.635
| 30\38, 1077.635
|42\53, 1081.7015
| 42\53, 1081.701
|50\63, 1083.337
|50\63, 1083.337
|968.15  
|968.15
|E
|E
|7/4
|7/4
| +2
| +2
|-
|-
|8
| 8
|37\38, 1329.083
|37\38, 1329.083
|52\53, 1339.249
|52\53, 1339.249
|62\63, 1343.3375
|62\63, 1343.337
|1199.66  
|1199.66
|F
|F
|2/1
|2/1
Line 1,129: Line 1,129:
|}
|}
==Scale tree==
==Scale tree==
{{Scale tree|5L 3s<11/5>}}
{{MOS tuning spectrum|Scale Signature=5L 3s<11/5>}}

Latest revision as of 17:59, 3 March 2025

↖ 4L 2s⟨11/5⟩ ↑ 5L 2s⟨11/5⟩ 6L 2s⟨11/5⟩ ↗
← 4L 3s⟨11/5⟩ 5L 3s (11/5-equivalent) 6L 3s⟨11/5⟩ →
↙ 4L 4s⟨11/5⟩ ↓ 5L 4s⟨11/5⟩ 6L 4s⟨11/5⟩ ↘ 
┌╥╥┬╥╥┬╥┬┐
│║║│║║│║││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLLsLs
sLsLLsLL
Equave 11/5 (1365.0 ¢)
Period 11/5 (1365.0 ¢)
Generator size(ed11/5)
Bright 3\8 to 2\5 (511.9 ¢ to 546.0 ¢)
Dark 3\5 to 5\8 (819.0 ¢ to 853.1 ¢)
Related MOS scales
Parent 3L 2s⟨11/5⟩
Sister 3L 5s⟨11/5⟩
Daughters 8L 5s⟨11/5⟩, 5L 8s⟨11/5⟩
Neutralized 2L 6s⟨11/5⟩
2-Flought 13L 3s⟨11/5⟩, 5L 11s⟨11/5⟩
Equal tunings(ed11/5)
Equalized (L:s = 1:1) 3\8 (511.9 ¢)
Supersoft (L:s = 4:3) 11\29 (517.8 ¢)
Soft (L:s = 3:2) 8\21 (520.0 ¢)
Semisoft (L:s = 5:3) 13\34 (521.9 ¢)
Basic (L:s = 2:1) 5\13 (525.0 ¢)
Semihard (L:s = 5:2) 12\31 (528.4 ¢)
Hard (L:s = 3:1) 7\18 (530.8 ¢)
Superhard (L:s = 4:1) 9\23 (534.1 ¢)
Collapsed (L:s = 1:0) 2\5 (546.0 ¢)

5L 3s⟨11/5⟩ is a 11/5-equivalent (non-octave) moment of symmetry scale containing 5 large steps and 3 small steps, repeating every interval of 11/5 (1365.0 ¢). Generators that produce this scale range from 511.9 ¢ to 546 ¢, or from 819 ¢ to 853.1 ¢.

13ed11/5 is the smallest ed11/5 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<11/5>[1] 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 13ed11/5 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 18ed11/5 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 21ed11/5 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 11/5.

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 13ed11/5) 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 13ed11/5 (basic) Size in 18ed11/5 (hard) Size in 21ed11/5 (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, 105.000 1\18, 75.834 2\21, 130.000 Af -5
major 2nd 2\13, 210.001 3\18, 227.501 3\21, 195.001 A +3
minor 3rd 3\13, 315.001 4\18, 303.334 5\21, 325.001 Bf -2
major 3rd 4\13, 420.001 6\18, 455.001 6\21, 390.001 B +6
diminished 4th 5\18, 379.168 7\21, 455.001 Cf -7
natural 4th 5\13, 525.002 7\18, 530.835 8\21, 520.002 C +1
augmented 4th 6\13, 630.002 9\18, 682.502 9\21, 585.002 C# +9
diminished 5th 8\18, 606.669 10\21, 650.002 Qf -4
perfect 5th 7\13, 735.002 10\18, 758.336 11\21, 715.002 Q +4
minor 6th 8\13, 840.003 11\18, 834.169 13\21, 780.002 Df -1
major 6th 9\13, 945.003 13\18, 985.836 14\21, 845.003 D +7
minor 7th 12\18, 910.003 15\21, 910.003 Ef -6
major 7th 10\13, 1050.003 14\18, 1061.670 16\21, 975.003 E +2
diminished octave 11\13, 1155.004 15\18, 1137.504 18\21, 1105.003 Ff -3
perfect octave 12\13, 1260.004 17\18, 1289.171 19\21, 1170.003 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 13ed11/5, 18ed11/5, and 31ed11/5.

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

13ed11/5 (basic) 18ed11/5 (hard) 31ed11/5 (semihard)
generator (g) 5\13, 525.002 10\18, 758.336 12\31, 496.55
L (3g - minor 9th) 2\13, 210.001 3\18, 227.501 5\31, 206.87
s (-5g + 2 minor 9ths) 1\13, 105.000 1\18, 75.834 2\31, 82.76

Intervals

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

Degree Size in 13ed11/5 (basic) Size in 18ed11/5 (hard) Size in 31ed11/5 (semihard) Note name on G Approximate ratios #Gens up
unison 0\13, 0.00 0\18, 0.00 0\31, 0.00 G 1/1 0
minor 2nd 1\13, 105.000 1\18, 75.834 2\31, 88.065 Af 21/20, 22/21 -5
major 2nd 2\13, 210.001 3\18, 227.501 5\31, 220.162 A 9/8 +3
minor 3rd 3\13, 315.001 4\18, 303.334 7\31, 308.227 Bf 13/11, 33/28 -2
major 3rd 4\13, 420.001 6\18, 455.001 10\31, 440.324 B 14/11, 33/26 +6
diminished 4th 5\18, 379.168 9\31, 396.292 Cf 5/4, 11/9 -7
natural 4th 5\13, 525.002 7\18, 530.835 12\31, 528.389 C 4/3 +1
augmented 4th 6\13, 630.002 9\18, 682.502 15\31, 660.486 C# 10/7, 18/13, 11/8 +9
diminished 5th 8\18, 606.669 14\31, 616.454 Qf 7/5, 13/9, 16/11 -4
perfect 5th 7\13, 735.002 10\18, 758.336 17\31, 748.551 Q 3/2 +4
minor 6th 8\13, 840.003 11\18, 834.169 19\31, 836.615 Df 52/33, 11/7 -1
major 6th 9\13, 945.003 13\18, 985.836 22\31, 968.713 D 56/33, 22/17 +7
minor 7th 12\18, 910.003 21\31, 924.680 Ef 5/3, 18/11 -6
major 7th 10\13, 1050.003 14\18, 1061.670 24\31, 1056.778 E 16/9 +2
diminished octave 11\13, 1155.004 15\18, 1137.5035 26\31, 1144.84 Ff 11/6, 13/7, 15/8 -3
perfect octave 12\13, 1260.004 17\18, 1289.171 29\31, 1276.939 F 2/1 +5
  1. ↑ the rare simplest tuning for a diatonic minor ninth

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).

21ed11/5 (soft) 34ed11/5 (semisoft)
generator (g) 8\21, 520.002 13\34, 521.913
L (3g - minor 9th) 3\21, 195.001 5\34, 200.736
s (-5g + 2 minor 9ths) 2\21, 130.000 3\34, 120.442

Intervals

Sortable table of major and minor intervals in hyposoft tunings (13ed11/5 not shown):

Degree Size in 21ed11/5 (soft) Size in 34ed11/5 (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, 130.000 3\34, 120.442 Af 16/15 -5
major 2nd 3\21, 195.001 5\34, 200.736 A 10/9, 9/8 +3
minor 3rd 5\21, 325.001 8\34, 321.177 Bf 6/5 -2
major 3rd 6\21, 390.001 10\34, 401.472 B 5/4 +6
diminished 4th 7\21, 455.001 11\34, 441.619 Cf 9/7 -7
natural 4th 8\21, 520.002 13\34, 521.913 C 4/3 +1
augmented 4th 9\21, 585.002 15\34, 602.208 C# 7/5 +9
diminished 5th 10\21, 650.002 16\34, 642.355 Qf 10/6 -4
perfect 5th 11\21, 715.002 18\34, 722.649 Q 3/2 +4
minor 6th 13\21, 780.002 21\34, 843.091 Df 8/5 -1
major 6th 14\21, 845.003 23\34, 923.385 D 5/3 +7
minor 7th 15\21, 910.003 24\34, 963.532 Ef 12/7 -6
major 7th 16\21, 975.003 26\34, 1043.826 E 9/5, 16/9 +2
diminished octave 18\21, 1105.003 29\34, 1164.268 Ff 27/14, 48/25 -3
perfect octave 19\21, 1170.003 31\34, 1244.563 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.

29ed11/5 (supersoft) 37ed11/5
generator (g) 11\29, 517.760 14\37, 516.488
L (3g - minor 9th) 4\29, 188.276 5\37, 184.460
s (-5g + 2 minor 9ths) 3\29, 141.207 4\37, 147.568

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 29ed11/5 (supersoft) Note name on G Approximate ratios #Gens up
unison 0\29, 0.00 G 1/1 0
chroma 1\29, 47.069 G# 33/32, 49/48, 36/35, 25/24 +8
diminished 2nd 2\29, 94.138 Aff 21/20, 22/21, 26/25 -13
minor 2nd 3\29, 141.207 Af 12/11, 13/12, 14/13, 16/15 -5
major 2nd 4\29, 188.276 A 9/8, 10/9, 11/10 +3
augmented 2nd 5\29, 235.346 A# 8/7, 15/13 +11
diminished 3rd 6\29, 282.415 Bff 7/6, 13/11, 33/28 -10
minor 3rd 7\29, 329.483 Bf 135/112, 6/5 -2
major 3rd 8\29, 376.553 B 5/4, 11/9, 16/13 +6
augmented 3rd 9\29, 423.622 B# 9/7, 14/11, 33/26 +14
diminished 4th 10\29, 470.691 Cf 21/16, 13/10 -7
natural 4th 11\29, 517.760 C 75/56, 4/3 +1
augmented 4th 12\29, 564.829 C# 11/8, 18/13 +9
doubly augmented 4th, doubly diminished 5th 13\29, 611.898 Cx, Qff 7/5, 10/7 -12
diminished 5th 14\29, 658.968 Qf 16/11, 13/9 -4
perfect 5th 15\29, 706.037 Q 112/75, 3/2 +4
augmented 5th 16\29, 753.106 Q# 32/21, 20/13 +12
diminished 6th 17\29, 800.175 Dff 11/7, 14/9 -9
minor 6th 18\29, 847.244 Df 13/8, 8/5 -1
major 6th 19\29, 894.313 D 5/3, 224/135 +7
augmented 6th 20\29, 941.382 D# 12/7, 22/13 -14
minor 7th 21\29, 988.451 Ef 7/4, 26/15 -6
major 7th 22\29, 1035.520 E 9/5, 16/9, 20/11 +2
augmented 7th 23\29, 1082.590 E# 11/6, 13/7, 15/8, 24/13 +10
doubly augmented 7th, doubly diminished octave 24\29, 1129.659 Ex, Fff 21/11, 25/13, 40/21 -11
diminished octave 25\29, 1176.728 Ff 64/33, 96/49, 35/18, 48/25 -3
perfect octave 26\29, 1223.797 F 2/1 +5
augmented octave 27\29, 1270.866 F# 33/16, 49/24, 72/35, 25/12 +13
doubly augmented octave, diminished 9th 28\29, 1317.935 Fx, Gf 21/10, 44/21, 52/25 -8

Parahard

23ed11/5 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 23ed11/5

(superhard)

Note name on G Approximate ratios (23edIX) #Gens up
unison 0\23, 0.00 G 1/1 0
chroma 3\23, 178.044 G# 12/11, 11/10, 10/9 +8
minor 2nd 1\23, 59.348 Af 36/35, 34/33, 33/32, 32/31 -5
major 2nd 4\23, 237.392 A 9/8, 17/15, 8/7 +3
aug. 2nd 7\23, 415.436 A# 5/4 +11
dim. 3rd 2\23, 118.696 Bf 16/15 -10
minor 3rd 5\23, 296.740 B 7/6 -2
major 3rd 8\23, 474.784 B# 9/7, 14/11 +6
dim. 4th 6\23, 356.088 Cf 6/5 -7
nat. 4th 9\23, 534.132 C 4/3 +1
aug. 4th 12\23, 712.176 C# 16/11, 22/15 +9
double dim. 5th 7\23, 415.436 Qff 5/4 -12
dim. 5th 10\23, 593.480 Qf 15/11, 11/8 -4
perf. 5th 13\23, 771.524 Q 3/2 +4
aug. 5th 16\23, 949.568 Q# 5/3 +12
dim. 6th 11\23, 652.828 Dff 7/5, 24/17, 17/12, 10/7 -9
minor 6th 14\23, 830.872 Df 14/9, 11/7 -1
major 6th 17\23, 1008.916 D 12/7 +7
minor 7th 15\23, 890.220 Ef 8/5 -6
major 7th 18\23, 1068.264 E 7/4, 30/17, 16/9 +2
aug. 7th 21\23, 1246.308 E# 31/16, 64/33, 33/17, 35/18 +10
dim. octave 19\23, 1127.612 Ff 11/6, 20/11, 9/5 -11
perf. octave 22\23, 1305.656 F 2/1 -3
aug. octave 25\23, 1483.700 F# 24/11, 11/5, 20/9 +5
dim. 8-step 20\23, 1186.960 Gf 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. 23ed11/5, 28ed11/5 and 33ed11/5 can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18ed11/5 and true Buzzard in terms of harmonies. 38ed11/5 & 43ed11/5 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 48ed11/5 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. 53ed11/5 has an essentially perfect 7/4, 58ed11/5 also gives three essentially perfect chains of third-comma meantone, while 63ed11/5 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 83ed11/5, 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.

38ed11/5 53ed11/5 63ed11/5 Optimal (PNTE) Ultrapyth tuning JI intervals represented (2.3.5.7.13 subgroup)
generator (g) 15\38, 538.817 21\53, 540.850 25\63, 541.668 484.07 4/3
L (3g - 11/5) 7/38, 251.448 10/53, 257.548 12/63, 260.001 231.51 8/7
s (-5g + 2 11/5s) 1/38, 35.921 1/53, 25.755 1/63, 21.667 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 38ed11/5 Size in 53ed11/5 Size in 63ed11/5 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, 251.448 10/53, 257.548 12/63, 260.001 231.51 A 8/7 +3
3 14\38, 502.896 20\53, 515.096 24\63, 520.002 463.03 B 13/10, 21/16 +6
4 15\38, 538.817 21\53, 540.850 25\63, 541.668 484.07 C 4/3 +1
5 22\38, 754.344 31\53, 798.399 37\63, 801.669 715.59 Q 3/2 +4
6 29\38, 1005.793 41\53, 1055.947 49\63, 1061.670 947.10 D 26/15 +7
7 30\38, 1077.635 42\53, 1081.701 50\63, 1083.337 968.15 E 7/4 +2
8 37\38, 1329.083 52\53, 1339.249 62\63, 1343.337 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

Scale tree and tuning spectrum of 5L 3s⟨11/5⟩
Generator(ed11/5) Cents Step ratio Comments
Bright Dark L:s Hardness
3\8 511.877 853.128 1:1 1.000 Equalized 5L 3s⟨11/5⟩
17\45 515.668 849.336 6:5 1.200
14\37 516.488 848.516 5:4 1.250
25\66 517.047 847.957 9:7 1.286
11\29 517.760 847.244 4:3 1.333 Supersoft 5L 3s⟨11/5⟩
30\79 518.356 846.648 11:8 1.375
19\50 518.702 846.303 7:5 1.400
27\71 519.086 845.918 10:7 1.429
8\21 520.002 845.003 3:2 1.500 Soft 5L 3s⟨11/5⟩
29\76 520.857 844.147 11:7 1.571
21\55 521.183 843.821 8:5 1.600
34\89 521.462 843.542 13:8 1.625
13\34 521.913 843.091 5:3 1.667 Semisoft 5L 3s⟨11/5⟩
31\81 522.409 842.595 12:7 1.714
18\47 522.768 842.237 7:4 1.750
23\60 523.252 841.753 9:5 1.800
5\13 525.002 840.003 2:1 2.000 Basic 5L 3s⟨11/5⟩
Scales with tunings softer than this are proper
22\57 526.844 838.160 9:4 2.250
17\44 527.388 837.616 7:3 2.333
29\75 527.802 837.203 12:5 2.400
12\31 528.389 836.615 5:2 2.500 Semihard 5L 3s⟨11/5⟩
31\80 528.939 836.065 13:5 2.600
19\49 529.287 835.717 8:3 2.667
26\67 529.703 835.301 11:4 2.750
7\18 530.835 834.169 3:1 3.000 Hard 5L 3s⟨11/5⟩
23\59 532.120 832.884 10:3 3.333
16\41 532.685 832.320 7:2 3.500
25\64 533.205 831.799 11:3 3.667
9\23 534.132 830.872 4:1 4.000 Superhard 5L 3s⟨11/5⟩
20\51 535.296 829.708 9:2 4.500
11\28 536.252 828.753 5:1 5.000
13\33 537.729 827.275 6:1 6.000
2\5 546.002 819.003 1:0 → ∞ Collapsed 5L 3s⟨11/5⟩
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