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{{Infobox MOS | {{Infobox MOS | ||
|Tuning=5L 3s<11/5>}} | |Tuning=5L 3s<11/5>}}{{MOS intro|Scale Signature=5L 3s<11/5>}} | ||
[[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. | |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 | |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 | |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 | |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. | |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. | |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. | |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. | |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. | |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. | |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. | |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 | !Size in 38ed11/5 | ||
!Size in | !Size in 53ed11/5 | ||
!Size in | !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. | |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. | | 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. | |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 | {{MOS tuning spectrum|Scale Signature=5L 3s<11/5>}} |